kitchen table math, the sequel: number sense

Monday, March 28, 2011

number sense

from Number Sense with Whole Numbers
Some adults lack a sense of the "equal distance of 1" between the whole numbers (the numbers on a number line). Without this concept, addition makes no sense. All math including basic addition is learned by rote as a mechanical process.
I have a memory of Dehaene arguing that people have an "innate" number line inside their minds.



It was Dehaene:
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
Phone +33 1 69 86 78 73
Fax +33 1 69 86 78 16
Makes people who don't seem to possess an internal number line interesting.

I've been relying on number lines to explain things to myself and to C. ever since coming across Dehaene's work.

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


Lsquared said...

"Just as we cannot avoid seeing objects in color" ... "and at definite locations in space"

That's new to me--very nice. If you're color blind, of course, then you do avoid seeing objects in color. It makes it interesting when you meet someone without that number sense (number blind?). I don't know that I know any adults like that, but I've worked with at least one student like that. I'd really like to know what's going on inside their head when they think about numbers.

Catherine Johnson said...

What was your student like?

I have to make time to read Dehaene's revision ---- or maybe reading the precis will be enough.

Hainish said...

If you're color blind, of course, then you do avoid seeing objects in color.

Color blind people do see in color, they just don't have all the color categories normal-sighted people do. For example, to a person with red-green colorblindness, those color appear very similar. Ditto for a person with blue-yellow colorblindness.

And, if you're wondering why colorblindness occurs in color pairs, look up opponent process theory of color perception.

Anonymous said...

Aren't the various people in South America (the Piraha and some neo-sign-language speakers) who don't do numbers AT ALL?

Allison said...

Innate??? Innate number line?

I don't see my preschooler or his peers having any such thing. They have fingers. They count. Even mapping counting to an actual 1-to-1 correspondence to the number of things isn't innate. Watch toddlers.

Toddlers learn to "count" by reciting numbers. They know their numbers and can "count up" quite well without being able to keep track of how many objects they've got in front of them.

Watch them try to count 4 little ducks on the page, or 5 pigs. When their finger-index-motion speed isn't able to keep up with the cadence of their counting, they don't self correct. They may count 6 ducks on a page of 4 because they lost ability to match their motor (or cognitive) skills to the counting they do. Toddlers will both over and under count at this stage, because they don't yet have a mapping, spatial or otherwise, that fits their verbal stream of numbers.

Why would this happen if they've got an innate number line?

Even preschoolers have to count objects above 10 several times to be sure that they accounted for everything.

Further, what we know about most human physical processes is that we work on log scales. Our hearing, our sight/recognition of brightness, our perception of touch, our general notions of distance are all log scales. We simply do not process information linearly at all. We perceive distant objects closer than they are; we perceive louder noises as not much louder until they are double as loud as the prior one, etc. That's why it's so hard for us to comprehend really big numbers--because our neural processes take logs, and reduce 10^34 to something manageable.

So the closest thing we've got to an "innate" number line doesn't evenly space things at all, but compresses the distance between them the farther out we get.

Glen said...

I don't see any innate number line in the research. What I see is an innate ability to deal with spatial locations and a tendency people have to map abstract things they learn onto a spatial representation so other ideas can piggyback on our well-evolved spatial abilities. Our number lines, timelines, org charts, and other spatial metaphors differ from culture to culture (vertical, horizontal, left-to-right, right-to-left, back-to-forward), so I don't see number lines themselves as innate, only the ability to learn (or invent) and use such lines.

I'm not denying the innateness of subitization (directly perceiving small quantities such as 2 or 3 without counting, which even birds can do), but that's not a number line.

SteveH said...

"All math including basic addition is learned by rote as a mechanical process."

"All" because of a number line problem?

What I see is research that tries to make something much more than it is. It could be an effect rather than a cause, or it could be vapor. I've called it "brain research misdirection" in the past.

It couldn't possibly be a simple competence problem with the teaching and curricula. You won't get NSF money by studying math competence and low expectations in K-6.

hainish said...

Allison is right, there is no innate number line, with an equal distance between all integers, in people's minds.

There's a lot of research that shows that how far apart we innately perceive two integers depends on what those integers are. In other words, it's different for very small numbers than fr somewhat larger numbers.

What many people do have, though, is an intuitive number line. I know I do, and I know it's because my 1st grade teacher explicitly taught it in the first week of school.

Allison said...

Right, Hainish, that's the logarithmic scale humans use. Generally speaking, humans "feel" that the distance between 10^15 and 10^16 is about the distance between 100 and 1000, even though there are
9000000000000000 numbers between the first two and 900 between the latter two.

People have the same problem with percentiles. They think that the difference between 90% and 95% is about the same as the difference between 95% and 100%. But if someone is in the 90% range, then 10 in 100 people have surpassed them; at 95%, 5, or half that number has. Half that number again is 97.5%, where 2.5 people have surpassed it. The 99.99 percentile range means 1 in 10000, which is a whole lot different than 1 in 100. It means 99 100s in which there wasn't a 1.

Michael Weiss said...

People have the same problem with percentiles. They think that the difference between 90% and 95% is about the same as the difference between 95% and 100%.

Well, it depends on how you define "difference between". The number of people scoring between the 90th and 95th percentile is the same as the number of people scoring between the 95th and 100th percentile: It's exactly 5% if the population. (Incidentally, I was always taught not to use the % symbol for percentiles, in part because there is no 100th percentile, and in part to avoid the confusion between percent and percentile.)

Anonymous said...

allison's right of course:
there's no built-in number line...
and *counting* appears to be
more "natural" (or "intuitive"
or what have you) than measuring.

pre-algebra students actually
seem to *resist* number-line
explanations at least as often
as they embrace 'em...

anyhow, the phrase "number sense",
in most contexts i've encountered,
is edubabble: something somebody's
heard in a context where they knew
they were supposed to be impressed
and have therefore concluded
they should go around saying it
in hopes of impressing others
equally clueless. you might as
well go to church to learn math
as listen to this kind of drivel
and boy is it easy to find.

the connection between
is an enormous stumbling block
throughout the history of maths
(and again in many, i think most,
individual learners of the subject).

points are indivisible...
"one is one and all alone
and evermore shall be so"
as the verse has it;
"a point is that which has
no part" as euclid sez somewhere;
one could go on. there appears
*really* to be some intuitive notion
of "atomic" unbreakable things
closely related to the notion
of counting.

meanwhile, in the number *line*
model, it's necessary to define
a "unit distance". the unit
*interval* of course will *not*
be an indivisible monad but
will instead be subject to

and everybody *knows*
fractions are hard, right?

one need *different* notions
of "the number 1" for different
contexts. this confused the
ancients enough to slow down
the development of algebra
for a couple thousand years.

(greek "geometry" was of course
very highly developed centuries
before the christian era... including
a theory of proportion essentially
equivalent to the modern notions
of "the real number line"
(think "dedekind cuts").
but they failed to develop an
appropriate symbolism.
my [amateur; i was a pro
in math per se and math ed,
but never in history-and-
-philosophy of maths] investigations
lead me to believe that this
confusion is one important
reason [along with, e.g.,
resistance to the concept
of "negative numbers" and,
egad, "imaginary" numbers
{as if *all* numbers weren't
objects of our imaginations}].)

my main source is probably
jacob klein

the amazing underwood dudley
has some well-informed remarks
somewhere about the very
*different* mental representations
reported by individual users
of math's for the counting numbers:
"number forms".

i read 'em even longer ago than
klein, though, and am not inclined
to look 'em up for ya. just sayin'.

wait. i can't resist *this*:

"number forms are like sex".

vlorbik kibrolv
(his mark)

Kevin said...

It seems to me that while there is an innate understanding of small quantities as shown through experiment with infants showing that they can in fact recognize impossible situations that involve small quantities, it is governed by sets, not by the number line. The linear number line is an abstract concept that only gains a true mental representation around 6-8 years old although there does seem to be a natural logarithmic scale.