I also have the perspective of being the mother of a child who can't talk.

Andrew has to have a machine talk for him.

It works on exactly the same principal as a calulator: he searches for the words he wants to say, presses the button, and the machine speaks.

And it is a horrific handicap.

Far, far, far better he should simply open his mouth and produce words.

Spend five minutes at my house and you see, so very clearly, that the idea of turning simple arithmetic over to a machine is insane.

Constructivist math curricula that do not teach the algorithms to automaticity are giving children handicaps.

-- CatherineJohnson - 06 Jul 2005

No, we don't want kids to stop thinking, we just want to make automatic some processes so they can think at a higher level.

This is the essence of developing expertise, as I understand research in cognitive science and in neuroscience.

And these are fields I do understand reasonably well, from a sceince writer perspective.

-- CatherineJohnson - 06 Jul 2005

This is not onpoint, but relates to the use of symbolism as an aid to solving problems. alfred North Whitehead in his "Introuction to Logic" states: "By the aid of symbolism, we an make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain."

In a book on symbolic logic by Irving M. Copi, the author refers to the Whitehead quote and states: "From this point of view, paradoxially enough, logic is not concerned with developing our powers of thought, but with developing techniques that enable us to get along without thinking!"

One can substitute the word "math" for logic in the above sentence and it would still be true, I believe. In fact, this argument has been used in arguing for the use of algorithms. No, we don't want kids to stop thinking, we just want to make automatic some processes so they can think at a higher level. If you had to think about each step to take in order to walk, you literally could not think other thoughts and walk at the same time. Similarly, students subjected to the "reform math" follies of NCTM and others who are not properly taught algorithms for math operations are operating at a severe handicap. It always amazes me, however, to see many otherwise intelligent people arguing for NCTM's way of teaching math.

-- BarryGarelick - 06 Jul 2005

Did Mr. Bell suffer damage to the Broca's or Wernicke's area?

I'll check--

-- CatherineJohnson - 05 Jul 2005

From your evidence, it seems that brain researchers have reached a nearly unanimous conclusion that mathematics is not language. But I wonder if that conclusion is different from the conclusion that mathematics is not 'a' language.

My grasp of logic starts to fail me here (too many balls to keep in the air at once!) but I think, logically speaking, that the consensus of brain researchers does not (necessarily) tell us that math is not a language in the terms of other professionals who study language.

OTOH, I personally am happiest, and feel most confident about what I'm doing, when different fields start coming to the same conclusions.

I should say, too, that I've done some extremely quick skimming.

So I wouldn't take it as gospel that neuroscience & neuropsychology is in near-unanimous agreement that math isn't a language. (Though I'd be surprised to find that this is an area of debate.)

My questions now have to do with the expressive & communicative functions of language: obviously, people use language to describe & explain things to each other.

So is math a 'signal'?

A 'sign'?

In linguistic terms, what is it?

-- CatherineJohnson - 05 Jul 2005

Sounds like both, probably more to the Wernicke's area since he could come out with a little speech.

-- CarolynJohnston - 05 Jul 2005

Fascinating, Catherine. I've always been meaning to pick up a copy of Butterworth's book. Perhaps now I have a reason.

From your evidence, it seems that brain researchers have reached a nearly unanimous conclusion that mathematics is not language. But I wonder if that conclusion is different from the conclusion that mathematics is not 'a' language.

Anyway, Mr. Bell sounds like an interesting fellow. If mathematics can truly be seen as "a" language, then the only things that could explain Mr. Bell's peculiar abilities would be either (a) that they are not representative of the larger population or (b) he is not a female.

Female brains are often lateralized--linguistic ability (and perhaps other abilities) are stored on both sides of the brain. Males brains tend to be more specialized. This makes women often more capable of recovering from a stroke.

Did Mr. Bell suffer damage to the Broca's or Wernicke's area?

-- JdFisher - 05 Jul 2005

the language of numbers is not language

I skimmed my 3 books on the neuropsychology of mathematics this morning.

It seems there is strong agreement, amongst neuroscientists and cognitive psychologists, that math is not language and language is not math.

The title of this post, 'the language of numbers is not language,' comes from Brian Butterworth's book What Counts: How Every Brain is Hardwired for Math.

The idea that math is language comes from Jean Piaget. (Surprise!)

Noam Chomsky's in there, too.

Chomsky believes, or once believed, that 'number was just a special aspect of language.'

'Special aspect' is the critical modifier here.

The question of whether math is a language the way English is a language does not appear to have adherents outside ed schools.

Neither Piaget nor Chomsky appears to have argued that math is a language per se.

Their idea is that math depends on the same core logical-reasoning capacity language depends upon.

(As I say, I've been skimming.)

talking versus reading

• biologically primary cognitive skills (like talking)

• biologically secondary skills (like reading)

I've been needing this distinction for years.

Back when Christopher was in Kindergarten I spent a few months obsessively pouring over the research on how children learn to read.

All of the serious, peer-reviewed researchers, universally, would say things like, 'Learning to read is not natural.'

Learning to walk is natural; learning to talk is natural.

Learning to read is not natural.

I've spent several years being confused by this, because, obviously, you need a brain to read.

Plus from time to time I'd see brain-scan studies showing which parts of the brain are involved in dyslexia...which always made me think: Hey! Is that a left occipital temporal region Reading Module I see there?

The distinction I wasn't managing to intuit is the distinction between biologically primary skills, which children's brains are built to acquire and which children do acquire without being taught, and biologically secondary skills, which our brains are not specifically built to acquire, and which children do have to be taught.

Left to his own devices, your basic child is not going to learn to read.

However, when he does learn to read, he won't recruit just any part of his brain to handle this new skill. He will recruit the same part of the brain everyone else is recruiting. The fact that a certain part of the brain lights up when people read doesn't mean that that part of the brain was specifically built for reading. It means, as I understand it, that that part of the brain can be recruited to do the job.


off-topic: Of course, this leaves out hyperlexic kids and kids like me. I 'taught myself to read,' which is probably why I was always confused by the reading-isn't-natural idea.

Christopher 'taught himself to read,' too. Two weeks after his Kindergarten teacher told us he was at risk for dyslexia, because of his very poor handwriting, he burst into literacy. He just took off.

Then there's Andrew, who, back during the days of 9/11, was spelling out words like 'interpol warning' on the floor with his alphabet blocks. Once he spelled 'Osamy' and 'Somaly' on the refrigerator.

Somebody should probably come study my kids....


Back on topic, here is a nice summation of the distinction between primary & secondary skills:

Even though many of the neurobiological systems that support language also support reading (Luria, 1980), these systems have not evolved to automatically acquire reading skills.

estimation versus arithmetic

Math is not natural.

Children don't pick up mathematics the way they pick up walking and talking.

Estimating and approximating quantities, on the other hand, are natural. Animals do it, and all humans do it, too. Babies probably do it.

But people do not acquire knowledge of algebra in the same way they acquire knowledge of '2 cookies is better than 1 cookie.'

Here is Stanislas Dehaene, one of the major researchers in the field:

[The] human capacity for arithmetic finds its ultimate roots in a basic cerebral system for perception and mental manipulation of approximate numbers, very ancient in evolution. According to this theory, we share this system with many animal species, and it appears very early in human development, independently of language. Of course, it is a primitive system, capable only of basic computations such as estimation, comparison, addition and subtraction of approximate numbers. On this shared basis, various human cultures invent increasingly elaborate cultural tools such as Arabic symbols, counting routines, algorithms for exact addition, multiplication etc.

Thus, the origins of human arithmetic lie in both a universal core system of approximate quantity, and on various cultural tools for exact arithmetic.

does brain research tell us that math is a 'special branch' of language?

There is now reasonably extensive research on people who have suffered brain injuries that tells us math and language are separate and distinct.

We also seem to have a body of brain scan research showing the same thing.

This is what's known as converging lines of evidence, and it's important.

Researchers have studied people who, because of brain injury, have lost only their ability to do math.

Language is intact, memory is intact, logical reasoning is intact.

But math is gone.

There are also one or two cases of people who have lost everything but math.

Here's one:

Mr. Bell's language had almost completely disappeared. He was left being able to utter just a few stereotyped phrases, such as 'I don't know' and, curiously, 'Millionaire bub.' His understanding of speech or of written language was almost nonexistent. Nevertheless he was still pretty good at calculation, and could accurately add and subtract . . . He could also select the larger of two-and three-digit numbers, showing that he still understood about numbers as being ordered by size, and the way the Arabic numeral system worked.

I'm going to remember that.

update

I'm pretty sure that the (apparent) fact that math and language are two different things inside the brain does not mean they are necessarily two different things in philosophical or even linguistic terms.

But if you're coming at the question from a neuroscientific or cognitive science point of view, math is not a language.

(It's not dead, either!)


What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...)

StevenPinkerOnLearningMath
Dehaene on high quality neuro-gear

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
"the language of numbers is not language" 7-3-05

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• Posted by: CatherineJohnson
• LogDate: Jul 03, 2005 @ 09:27