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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
INSERM U.334
Service Hospitalier Frédéric Joliot
CEA/DSV
4 Place du Général Leclerc
91401 cedex Orsay
France
The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition
13 comments:
I could hypothesize many things, but is this some grand genetic key to math or the simple realization that this concept is important and schools must ensure that it is taught and understood? If schools do teach the concept, is this a greater hurdle to overcome than every other fundamental concept in math? I'm not big on these things because the assumption is that the problems in math are much more complicated than simple competence.
What happens in schools that directly teach the concept of a number line? Do these problems disappear? One could claim a genetic cause for almost everything, but the implication is that the issue still exists after a proper introduction to the material. Perhaps the hypothesis is simply that some kids "see" the number line and that some actually have to be directly taught.
I simply don't buy the argument. And I wonder why it's useful?
Does it mean that people who don't have a numerical sense are genetically inferior?
Also, how does this impact what you should be doing in the classroom? Wouldn't it make more sense to take a Kumon-like approach and just teach math as we know it in small steps and not worry about some grand genetic scheme? Or take a DI approach and try different ways of teaching and only use what works, and capture that method in a script. Either approach is more effective than starting with a grand genetic theory.
We've discussed this before and I'm not anymore persuaded now. (I should find the post.) I think it's unlikely to be true, since we have overwhelming evidence that our brain structures work with *log* scales, not linear scales. Our ears and eyes feed us data in log scales. Probably our pain scales are log. Our mental impression of how big a number is fits a log scale (which is why we have such a terrible time understanding just how much bigger a trillion is than a billion) too. And in the other direction, for small numbers, it's why we have such trouble understanding the difference between, say, 99% and 99.5%.
But so what--what if we do organize numbers physically in a cartesian coordinate system? so what? Whether or not it's innate, our Math System organizes numbers in a cartesian coordinate system, so YES, we should teach kids the number line because that's the way we express math.
--http://kitchentablemath.blogspot.com/2011/03/number-sense.html
we didn't believe it here, either.
our prior conversation about this:
http://kitchentablemath.blogspot.com/2011/03/number-sense.html
the philosophy-of-mind stuff...
including but not at all limited to
the pop-neurology that inexplicably
grips the minds of so many of
our contemporaries...
is mostly irrelevant in considering
how best to create workable useful
elementary math curricula.
we are not mad scientists and
those children are not skinner's pigeons.
so mostly i pay it no mind.
the previous thread
(teach the # line in 1st grade)
pointed at wu pointing at the # line
specifically as a site for refuting the
there-is-no-alternative-to-the-spiral
approach to (failing to) teach fractions.
"you have to be willing to accept
multiple meanings for it at the outset"
is... talking for me now not wu...
a profoundly *anti-mathematical*
viewpoint. the importance of *definitions*
in mathematics... *particularly* at the
elementary level... is almost impossible
to overstate. you can hide it but you
can't run from it: it's everywhere and
it's not going away.
somebody said something about "rulers"...
the kind you measure with... upthread
or in the next one over. looks like maybe
a good tool for introducing number lines.
but "somewhere along the line" (as we
unselfconsciously say having once
internalized the time-as-moving-point
model)... and the sooner the better...
one should forthrightly admit that
over the generations it's been noticed
(time and again)
that by fixing *one* carefully-constructed
formalization as the *definition*
of a certain symbol (or phrase or
what have you) one has *simplified*
the situation to the point where
it becomes much *easier* to say
useful or interesting things about
that hitherto-only-vaguely-understood
thing.
mathphobes are always claiming to
shy away because something one
is trying to clarify for them is
"too complicated" when in my view
the opposite is actually the case:
what it actually *is* is "unfamiliar"
(and that's the way they want it).
the correct answer to "how do you
get to carnegie hall" isn't *complicated*
(or unfamiliar: "practice, practice,
practice", of course)... but most
of take it for granted we'll be unable
to use it despite it's simplicity.
so they'll drag out some canard
about perfect-practice-makes-perfect
and pretend there's something about
a great performer's *brain* or something.
hohum.
owen thomas said...
"but most of [us]
take it for granted we'll be unable
to use it despite [its] simplicity."
yeesh.
i hope i usually edit my posts
better than this: two mistakes
in less than half a sentence.
if i'm in such a big hurry to
get somewhere, why am i
typing out rambles on the
doggone internet in the first place?
yay number lines!
Wouldn't it make more sense to take a Kumon-like approach and just teach math as we know it in small steps and not worry about some grand genetic scheme?
I think Kumon teaches the number line, but I'll have to check.
Looks like Kumon teaches the number line (judging by the Kumon books for sale in bookstores)
autistic child & Kumon number lines
I think it's unlikely to be true, since we have overwhelming evidence that our brain structures work with *log* scales, not linear scales
That's not true for counting.
Wish to heck I could remember the term for knowing immediately that 3 is different from 2....darn.
Will try to look it up.
Where number is concerned, the brain is built to see small, linear numbers correctly and without training. True with animals, too.
Oh!
Here it is: subitize.
<< Evidence has been given that, up to number 3, human beings link almost immediately objects and their corresponding numbers, without any appearance of counting: we do not need to count, we subitize. >>
BUT -- after the number 3, we may have a logarithmic number line:
<< Dehaene says the research suggests that a logarithmic number line might be an intuitive mathematical concept, whereas the idea of a linear number line might have to be learned. >>
http://www.scientificamerican.com/article.cfm?id=a-natural-log
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