kitchen table math, the sequel: 8/7/11 - 8/14/11

Saturday, August 13, 2011

Stanislas Dehaene on the mental number line

My hypothesis is that number sense qualifies as a biologically determined category of knowledge. I propose that the foundations of arithmetic lie in our ability to mentally represent and manipulate numerosities on a mental “ number line ”, an analogical representation of number ; and that this representation has a long evolutionary history and a specific cerebral substrate. “ Number appears as one of the fundamental dimensions according to which our nervous system parses the external world. Just as we cannot avoid seeing objects in color (an attribute entirely made up by circuits in our occipital cortex, including area V4) and at definite locations in space (a representation reconstructed by occipito-parietal neuronal projection pathways), in the same way numerical quantities are imposed on us effortlessly through the specialized circuits of our inferior parietal lobe. The structure of our brain defines the categories according to which we apprehend the world through mathematics. ” (TNS, p. 245).
Précis of “ The number sense ”
Stanislas Dehaene
Service Hospitalier Frédéric Joliot
4 Place du Général Leclerc
91401 cedex Orsay
The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition

The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition

teach the number line in 1st grade

Several times over the past years I've come across the idea that humans possess an innate number line inside our minds. At least, that's how I interpret of the snippets of research I've read.

Not long after encountering the possibility that number lines have a privileged place in math learning, I read H. Wu's revelatory definition of a fraction as a point on the number line:
The following is a new approach to the teaching of fractions. It is not new in the sense of introducing new concepts; the subject is too old for that. Rather, it is new in the way the various skills and concepts are introduced and woven together. Whereas it is traditional to ask you to believe that the concept of a fraction is so profound that you have to be willing to accept multiple meanings for it at the outset, we merely ask you to accept one clear-cut definition of a fraction (as a point on the number line), and use reasoning to deduce as logical consequences all other meanings of this concept.
On the Teaching of Fractions (pdf file)
H. Wu
I've been relying heavily on number lines for self teaching and reteaching for several years now.

David Geary's new longitudinal study seems to add further evidence that number lines are important:
The researchers also found that first-graders who understood the number line and how to place numbers on the line and who knew some basic facts showed faster growth in math skills than their counterparts during the next five years.
MU Psychology Study Finds Key Early Skills for Later Math Learning
Long-term study shows students must know about numbers at beginning of first grade
July 11, 2011

The News in Connecticut

Connecticut has been plagued by nothing but bad news these days on the education front. In case you missed it, here's a quick re-cap on Throwing Curves.

Wednesday, August 10, 2011

yet another solution to the problem we don't have

Next up: Angry Birds in the classroom. Rovio plans to publish a line of educational books for children ages four to their teens. Embracing the Finnish education system—which emphasizes play in learning—they're also designed as a counterpoint to the strict "Tiger Mom" approach to child rearing, says Wibe Wagemans, a Rovio branding executive.

He says one of the books will illustrate geometry lessons by launching a bird character through the air...

'Angry Birds' Spreads Wings
WSJ | August 10, 2010
E.D. Hirsch needs to get with a branding executive.

it's come to this, part 2

Sad Guys on the Trading Floor

(probably not appropriate for children)

All Children Are Capable of Greatness

From the Kumon website:

At the heart of the Kumon Method is the belief that all children are capable of greatness. With the help of their parents, family and friends, children can develop in ways that will humble and amaze you.

Kumon’s founder, Toru Kumon, believed every child has the potential to learn far beyond his or her parents’ expectation. “It’s our job as educators,” Kumon said, “Not to stuff knowledge into children as if they were merely empty boxes, but to encourage each child to want to learn, to enjoy learning and be capable of studying whatever he or she may need to or wish to in the future.” Children who learn through the Kumon Method not only acquire more knowledge, but also the ability to learn on their own.

Last week Catherine and I visited the Kumon headquarters.

I bring back some Kumon lore:

  • Kumon started in 1954, when 2nd grader Takeshi Kumon came home from school with a crumpled up math test stuffed in his backpack. I find it hilarious, by the way, that the "crumpled math test" is this universal experience that transcends continents and generations.
  • Today, there are 4.2 million children studying Kumon in 46 countries.

Turns out, there's also an adult Kumon workbook, Train Your Brain: 60 Days to a Better Brain, and it has sold millions of copies. From the introduction:

Through my research, I found that simple calculations could activate the brain more effectively than any other activity. I also discovered that the best way to activate the largest regions of the brain was to solve these calculations quickly.

Cross-posted on Perfect Score Project

A Questionable Question

I haven't been doing much in the way of SAT Critical Reading lately. I've got bigger fish to fry.

That said, I don't want to lose momentum.....and, if the truth be told, I've come to love the Critical Reading sections (and yes, I am telling the truth).

Thanks in large part to my marathon lunch dates with Erica Meltzer, I rarely get a reading question wrong these days. You can click on this page to see myrenditions of her Critical Reading "recipes" (i.e. don't blame Erica if you don't understand. I take full responsibility for the translation.)

But every once in a while, I come across a question that stumps me.

Take, for example, the following:

Flummoxed, I answered incorrectly. I knew my answer was wrong, but I couldn't see a right answer.

Ok, STOP reading before you see the explanation below, and tell me:

  • A) Which one would you choose?
  • B) Which one do you think I picked?

I'm obsessed, determined, and like a dog with a bone: I asked nearly everyone I know, "is this question legit?"

PWNtheSAT 's response made the most sense (to me):

Tough question, but it's legit. You can't infer A through D, because they're all too specific. You can't really ever infer a phrase was "first used" unless the author comes right out and says it directly. There's no mention of "college educated" women, and WWII is really only mentioned to establish a setting. So you COULD get it by elimination if you're careful.

The real reason the answer is legit, though, can best be illustrated with analogy.

What would you think if you read that "some people ALREADY had internet access in 1985," or "Springsteen was ALREADY a local hero in New Jersey before he broke nationally"?

The implication, when you use "already" in this sense, is that something is ahead of the curve. "Male chauvinist" is a common phrase today, but it clearly wasn't then or the author wouldn't have felt the need to say "already." So the implication is that in 1945, use of the phrase was rare, but it's commonplace today.

Illustrations by Jennifer Orkin Lewis

Cross posted on the Perfect Score Project

Tuesday, August 9, 2011

in which I stumble upon a learning style that may actually exist

Barry pointed me to a terrific post by Haim, one of the regulars on Math Teach, who writes:
As it happens, a good friend of mine did his dissertation on learning styles some fifteen years ago. He did his study in the context of math education at the college level. I still clearly remember the frisson when he told me that 30 yrs of scientific literature on the subject was thoroughly consistent: not a shred of (real) evidence to support the hypothesis of learning styles.

His study did not perturb the consistency. At the time, I remember thinking "Gosh, after 30 yrs of nothing maybe it's time to look for another fishing hole."

Since then, of course, I have been deeply impressed by the evident fact that 30 yrs of nothing makes not the slightest impression on the Education Mafia. They carry on, exuberantly, as if learning styles are real.
Just a few days after reading Haim, and quite by accident, I came across the possibility that learning styles may actually exist (!)

Just not the learning styles we've heard so much about for lo these 30 years.

Here is Michael J. Frank on  “Go” and “NoGo” Learning and the Basal Ganglia:
We tested healthy college students who were given low doses of three different drugs: a drug that enhances the release of dopamine, a drug that reduces the release of dopamine, and a placebo; each student was tested in each of the three conditions.


We hypothesized that all participants would learn to choose A over B, but that they would do so on different bases, depending on which drug they had been given. When dopamine levels are elevated, we hypothesized, participants should learn to choose symbol A, which had received the most positive feedback (that is, they should learn “Go” to A). But they should be relatively impaired in learning to avoid (NoGo) symbol B....


We found a striking effect of the different dopamine medications on this positive versus negative learning bias....While on placebo, participants performed equally well at choose-A and avoid-B test choices. But when their dopamine levels were increased, they were more successful at choosing the most positive symbol A and less successful at avoiding B. Conversely, lowered dopamine levels were associated with the opposite pattern: worse choose-A performance but more-reliable avoid-B choices. Thus the dopamine medications caused participants to learn more or less from positive versus negative outcomes of their decisions.

These research discoveries raise the intriguing question of whether individual differences in learning from positive versus negative outcomes of decisions can be found even in nonmedicated healthy people. Indeed, although on average our study participants taking the placebo showed roughly equal choose-A and avoid-B performance, individual participants still performed better at one or the other; we refer to these subgroups as positive or negative learners....[A] mutation in another gene previously shown to control the density of (NoGo) D2 receptors in the basal ganglia predicted the extent to which participants learned from negative decision outcomes. Together, these results provide more-specific confirmation of our model’s suggestion that Go and NoGo learning depends on the D1 and D2 receptors.
I am gobsmacked.

Until the moment I read Michael Frank's research, I believed that positive reinforcement was better than negative. Always. Turns out the truth of the matter may be it depends.

Choose A or Avoid B.

The lady or the tiger.

it's come to this

I now live in a state of suspense over what the Fed will say at its next meeting.

In the department of silver linings, I am finally reading Reinhart and Rogoff's This Time Is Different

This Time Is Different: Eight Centuries of Financial Folly

Codswallop, part 2

Education Needs a Digital Up-Grade

by Virginia Heffernan
If you have a child entering grade school this fall, file away just one number with all those back-to-school forms: 65 percent.

Chances are just that good that, in spite of anything you do, little Oliver or Abigail won’t end up a doctor or lawyer — or, indeed, anything else you’ve ever heard of. According to Cathy N. Davidson, co-director of the annual MacArthur Foundation Digital Media and Learning Competitions, fully 65 percent of today’s grade-school kids may end up doing work that hasn’t been invented yet.


Ms. Davidson herself was appalled not long ago when her students at Duke, who produced witty and incisive blogs for their peers, turned in disgraceful, unpublishable term papers. But instead of simply carping about students with colleagues in the great faculty-lounge tradition, Ms. Davidson questioned the whole form of the research paper.“What if bad writing is a product of the form of writing required in school — the term paper — and not necessarily intrinsic to a student’s natural writing style or thought process?” She adds: “What if ‘research paper’ is a category that invites, even requires, linguistic and syntactic gobbledygook?”

What if, indeed. After studying the matter, Ms. Davidson concluded, “Online blogs directed at peers exhibit fewer typographical and factual errors, less plagiarism, and generally better, more elegant and persuasive prose than classroom assignments by the same writers.”


A classroom suited to today’s students should deemphasize solitary piecework.


The new classroom should teach the huge array of complex skills that come under the heading of digital literacy. And it should make students accountable on the Web, where they should regularly be aiming, from grade-school on, to contribute to a wide range of wiki projects.
Number one: I write books, and I write a blog. Books are harder.

Number two: Kathleen Porter-Magee deals with the forget-the-past-teach-the-future folderol.

Number three: As always, I object to other people telling me what my kids must spend their childhood doing -- and, more importantly, not doing -- at school. Especially seeing as how other people's folderol means I have to pay for a Jesuit high school because the public schools I am also paying for are assigning posters in Honors English. If Cathy N. Davidson and Virginia Heffernan want their kids contributing to a wide range of wiki projects starting at the age of 5, fine. Leave my kids out of it.

extra credit: Does that 65% figure apply to Smart Boards?

the founder, chairman, and CEO of Netflix has a really bad idea
speaking of technology and stagnant scores
oh brave new world!
codswallop, part 2