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View Full Version : Is any part of mathematics innate? Even counting and simple operations?


Spectre of Pithecanthropus
11-15-2007, 05:19 PM
I was thinking recently about a couple of docudramas I saw years ago about the life of Bill Sackter, starring Mickey Rooney. At the time the story opens, Sackter is in his fifties, and has lived most of his life in institutions. In his youth, he'd been diagnosed as what was then termed imbecilism, which means a rather significant degree of mental handicap. Rooney played the character about as you would expect--a good human being who was, however, very dim.

There was one scene where he's being tested in the simplest arithmetic, and he's unable to do it. I'm talking about questions like, "If you have two apples and give one to me, how many do you have left?" and so on.

Granted it's a docudrama, not fact. But on looking up the real Bill Sackter, I learned that he was eventually tested out at near normal intelligence, and I'm sure the same sort of thing happened to many real people back then. Which brings me to my question. I can understand how somebody who lived his life might never have learned to read. But mathematics at the level of 1+1 or 2-1 seems so obvious that it's hard to comprehend anyone of near normal or greater intelligence not to grasp it immediately. I was already reading a little bit when I started school, but even so when I got to the first grade and had my first lessons in reading, it seemed a bit daunting. The teacher pointed out that "airplane" looks a little bit like an airplane--and I wondered if we were going to have to learn words by their shapes. That seemed daunting much as I love to read now.

But I had no trouble with that level of math. It was like it was innate in me, and what makes it all the more strange is that in later grades I showed no special ability in math--rather the reverse!

Frylock
11-15-2007, 05:31 PM
I was thinking recently about a couple of docudramas I saw years ago about the life of Bill Sackter, starring Mickey Rooney. At the time the story opens, Sackter is in his fifties, and has lived most of his life in institutions. In his youth, he'd been diagnosed as what was then termed imbecilism, which means a rather significant degree of mental handicap. Rooney played the character about as you would expect--a good human being who was, however, very dim.

There was one scene where he's being tested in the simplest arithmetic, and he's unable to do it. I'm talking about questions like, "If you have two apples and give one to me, how many do you have left?" and so on.

Granted it's a docudrama, not fact. But on looking up the real Bill Sackter, I learned that he was eventually tested out at near normal intelligence, and I'm sure the same sort of thing happened to many real people back then. Which brings me to my question. I can understand how somebody who lived his life might never have learned to read. But mathematics at the level of 1+1 or 2-1 seems so obvious that it's hard to comprehend anyone of near normal or greater intelligence not to grasp it immediately. I was already reading a little bit when I started school, but even so when I got to the first grade and had my first lessons in reading, it seemed a bit daunting. The teacher pointed out that "airplane" looks a little bit like an airplane--and I wondered if we were going to have to learn words by their shapes. That seemed daunting much as I love to read now.

But I had no trouble with that level of math. It was like it was innate in me, and what makes it all the more strange is that in later grades I showed no special ability in math--rather the reverse!

Answering the question posed to the character in the film takes more than math skills. You need, for example, familiarity with the conversational pragmatics of test taking. Related to this, you also need to have been habituated into the practice of considerting hypotheticals in a particular way. ("How many would you have left? I don't know. Did you eat any? Did anyone else take any? How should I know?" These are questions you and I would hardly even think to ask, but you can imagine someone who had not been answering these kinds of questions since they were a young child might be at a loss as to which of these possibilities his questioner really meant to point out. It takes training to know both that a hypothetical means to specify nothing over and above what it actually says, and also to know what exactly constitutes "nothing over and above what it actually says.") More basically, you need to know how to imagine situations in the first place. Someone who is very handicapped, or has not had enough experience doing this in the context of a conversational interaction with someone else, may not be able, from one moment to the next, to remember (or bring forth to salience) how many apples the hypothetical person was supposed to have had in the first place.

-FrL-

gazpacho
11-15-2007, 05:31 PM
It is totally learned. Once you learn counting you basically have learned addition and subtraction especially adding and subtracting one.

yelimS
11-15-2007, 05:47 PM
I think there's at least one tribe somewhere that doesn't use numbers. They had a quite strange way of getting about concepts such as a boat being loaded to capacity, I seem to recall. Cite, somebody, pretty please.

WhyNot
11-15-2007, 05:57 PM
No, it's not innate even for those of a normal intelligence. As gazpacho said, you learn it while learning counting. My daughter, at 2.5, is starting to grok very very simple addition: "Mommy has one Cheerio and Caileigh has two Cheerios. How many Cheerios all together?" She just started to realize that she can count them literally all together, that one and two aren't two separate realities. But she still has to go, "One, two...three!", pointing to each one in turn; she can't look at them in two separate piles and make them into one group in her head.

Subtraction is the same: "Caileigh, here are three Cheerios. If I take one, how many are left?" Well, she just has to count them all over again.

There's no way she could conceptualize a general or representative one or two yet. It has to be an actual thing she can count - Cheerio, cat, mark on a paper.

ftg
11-15-2007, 09:55 PM
Yes, basic counting is innate in humans (and up to small numbers in many other animals). The person in the OP is "special".

There's several books on this topic. An easy read is:

"The Number Sense: How the Mind Creates Mathematics" by Stanislas Dehaene

and if you really want to get deep try:

"Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being" by George Lakoff and Rafael Nuńez.

(There's another one that I can't recall right now.)

Oh, and the old Piaget sense of number at a certain age stuff is bull. The kids were thrown off by being asked (to them) stupid questions.

dtilque
11-15-2007, 10:06 PM
I think there's at least one tribe somewhere that doesn't use numbers. They had a quite strange way of getting about concepts such as a boat being loaded to capacity, I seem to recall. Cite, somebody, pretty please.
You're probably thinking of the Pirahă (http://en.wikipedia.org/wiki/Pirah%C3%A3_people)

CutterJohn
11-16-2007, 12:38 AM
My niece, being two, can barely talk, and certainly can't do even elementary 1+1=2 arithmetic. However, she is perfectly capable of realizing when one of her two boo's is not there, or when told to get her father and I a beer(i love kids!) will get two beers from the fridge and bring them to us, but will only get one when one of us asks.


People have some simple innate counting skills, as do many animals. More abstract concepts, like multiplication and division, are learned.

Saint Cad
11-16-2007, 01:35 AM
From the research I'm doing for my EdD in math education, the idea of a 1-1 correspondence seems to be inate and an important precursor to counting.

WhyNot
11-16-2007, 09:28 AM
I guess, based on other responses, that I need some clarification of what "any part of mathematics" and "innate" mean.

Does "any part of mathematics" include the notion that "One" = that thing there and also that thing over there, and also the concept that "one" and "two" refer to different things, must go in that order to arrive at meaning, and that "two" refers to both the second item and the total number of things? Is there a more fundamental "part of mathematics" before that I'm skipping?

Does "innate" mean "without any teaching whatsoever, we'd expect to see the concepts displayed"?

BlinkingDuck
11-16-2007, 09:40 AM
I was taught back in school that most primitive cultures have 1,2,3,many. Since there is many different types of many they needed to denote how many many.

Like there are leaves on the tree, that is how many....

Like there are fingers on the hand, that is how many...

Koxinga
11-16-2007, 10:17 AM
I was taught back in school that most primitive cultures have 1,2,3,many. Since there is many different types of many they needed to denote how many many.

IIRC, it goes something like: one, two, three, many, many-one, many-two, many-three, many-many, many-many-one, many-many-two, many-many-three, many-many-many, many-many-many-one, many-many-many-two, many-many-many-three, LOTS.

Or maybe I'm thinking of trolls.

Trunk
11-16-2007, 10:29 AM
IIRC, it goes something like: one, two, three, many, many-one, many-two, many-three, many-many, many-many-one, many-many-two, many-many-three, many-many-many, many-many-many-one, many-many-many-two, many-many-many-three, LOTS.

Or maybe I'm thinking of trolls.
So, even though you have a cite many posts above yours, you're going to call a guy a troll in a GQ thread for repeating hearsay.

Nice.

Koxinga
11-16-2007, 10:37 AM
So, even though you have a cite many posts above yours, you're going to call a guy a troll in a GQ thread for repeating hearsay.

Nice.

Ya ever get that feeling that *somebody's* being whooshed, but you're not sure if it's you or the other guy?

Flander
11-16-2007, 10:39 AM
I remember hearing studies that children up to a certain age can't tell the quantity of something unless they count every single item. I'm trying to find an eloquent way to explain this. Basically, if you lined up a string of cheerios like this:

O O O O O

And then one like this:

O O O O O

They will say the second example has more. I did this experiment with a friend's nephew. He said the second line had more. Then I asked him to count both (which both had 5 cheerios). He was baffled.

Frylock
11-16-2007, 11:02 AM
IIRC, it goes something like: one, two, three, many, many-one, many-two, many-three, many-many, many-many-one, many-many-two, many-many-three, many-many-many, many-many-many-one, many-many-many-two, many-many-many-three, LOTS.

Or maybe I'm thinking of trolls.

I'm guessing that's a Prachett reference.

Am I right?

-FrL-

Frylock
11-16-2007, 11:03 AM
So, even though you have a cite many posts above yours, you're going to call a guy a troll in a GQ thread for repeating hearsay.

Nice.

Nobody ain't called nobody no troll no how.

-FrL-

Frylock
11-16-2007, 11:05 AM
I remember hearing studies that children up to a certain age can't tell the quantity of something unless they count every single item. I'm trying to find an eloquent way to explain this. Basically, if you lined up a string of cheerios like this:

O O O O O

And then one like this:

O O O O O

They will say the second example has more. I did this experiment with a friend's nephew. He said the second line had more. Then I asked him to count both (which both had 5 cheerios). He was baffled.

I remember when I was a little kid seeing a Sesame Street segment where Bert proved to Ernie that no matter how he arranged the cookies on the plate, there would always be the same number. He couldn't eat a cookie, then rearrange them, and end up with the same amount he started with.

I must have already passed the developmental stage you mention because I remember thinking the toddler equivalent of "What kind of an idiot do they think I am?"

-FrL-

don't ask
11-16-2007, 11:08 AM
If my memory still works I am pretty sure that the "one, two, three, many.." and it didn't become any more complicated, was attributed to an Australian Aborigine tribe in one of the first few volumes of The Guinness Book of World Records.

Omi no Kami
11-16-2007, 11:14 AM
You're probably thinking of the Pirahă (http://en.wikipedia.org/wiki/Pirah%C3%A3_people)

The Pirahă language is extremely cool, but it's worth noting that the guy who did a lot of the original fieldwork (which the wikipedia article is based one) made some pretty unusual claims, many of which haven't been supported by other research.

BlinkingDuck
11-16-2007, 11:22 AM
IIRC, it goes something like: one, two, three, many, many-one, many-two, many-three, many-many, many-many-one, many-many-two, many-many-three, many-many-many, many-many-many-one, many-many-many-two, many-many-many-three, LOTS.

Or maybe I'm thinking of trolls.

LOL

Only if you're counting goats.

Trunk
11-16-2007, 11:40 AM
Ya ever get that feeling that *somebody's* being whooshed, but you're not sure if it's you or the other guy?It appeared that BlinkingDuck made a comment about "primitive people" that has been labeled as racist by some, but is considered controversial in some circles.

It further appeared that you were calling him out on it, and begging the question.

I'll try to keep straight the reading habits and shared experiences of your and other posters. Perhaps you could draw me a diagram.

Frylock
11-16-2007, 12:00 PM
It appeared that BlinkingDuck made a comment about "primitive people" that has been labeled as racist by some, but is considered controversial in some circles.

It further appeared that you were calling him out on it, and begging the question.

I'll try to keep straight the reading habits and shared experiences of your and other posters. Perhaps you could draw me a diagram.

You thought Koxinga was calling BlinkingDuck a troll. But even on the account you've given of your reading of the situation, it wouldn't have made any sense for Koxinga to call BlinkingDuck a troll. You don't call someone a troll for using terminology believed by some to be racist. You call someone a troll when you mean to accuse them of posting specifically for the sole purpose of sabotaging the intent of the thread in some way. Even on your account, Koxinga didn't think BlinkingDuck was doing that.

So I don't think you can chalk this up to your not knowing some obscure reference in the OP or something. I think you just don't know what "Troll" means. Or else, you inexplicably were convinced that Koxinga didn't know what "Troll" means. (And still yet inexplicably failed to mention this in your criticism of him.)

-FrL-

Malachi Throne
11-16-2007, 12:07 PM
As famously recorded by Plato, Socrates was able to teach an uneducated slave the value of the square root of 2 though his Socratic Method of asking leading questions.
So, both of those thought that math was inherent knowledge.

Trunk
11-16-2007, 12:11 PM
You thought Koxinga was calling BlinkingDuck a troll. But even on the account you've given of your reading of the situation, it wouldn't have made any sense for Koxinga to call BlinkingDuck a troll. You don't call someone a troll for using terminology believed by some to be racist. You call someone a troll when you mean to accuse them of posting specifically for the sole purpose of sabotaging the intent of the thread in some way. Even on your account, Koxinga didn't think BlinkingDuck was doing that.

So I don't think you can chalk this up to your not knowing some obscure reference in the OP or something. I think you just don't know what "Troll" means. Or else, you inexplicably were convinced that Koxinga didn't know what "Troll" means. (And still yet inexplicably failed to mention this in your criticism of him.)

-FrL-
I thought that Koxinga was calling BD a troll for raising the issue of "primitive people can only count to three".

Frylock
11-16-2007, 12:29 PM
I thought that Koxinga was calling BD a troll for raising the issue of "primitive people can only count to three".

I know.

How could someone think that constitutes being a troll?

-FrL-

Frylock
11-16-2007, 12:31 PM
As famously recorded by Plato, Socrates was able to teach an uneducated slave the value of the square root of 2 though his Socratic Method of asking leading questions.
So, both of those thought that math was inherent knowledge.

"Thought" being the operative word here. Socrates asked some awfully leading questions in that dialogue. The extent to which Plato realized this is debated.

-FrL-

Trunk
11-16-2007, 01:21 PM
I know.

How could someone think that constitutes being a troll?

-FrL-
Because it's been the subject of heated back-and-forths and name calling and the argument devolves into personal, not scientific, interpretations of the same data.

This has gone on a ridiculous amount of words.

Jamaika a jamaikaiaké
11-16-2007, 04:27 PM
This is a pretty good article about the Piraha (http://www.newyorker.com/reporting/2007/04/16/070416fa_fact_colapinto)

psiekier
11-16-2007, 04:47 PM
It is totally learned. Once you learn counting you basically have learned addition and subtraction especially adding and subtracting one.I will second this statement.

My three-year-old son has learned to count to 20, but to him numbers are little more than words that come in a pre-arranged sequence. When he was a little younger, he was completely unable to grasp the concept of enumeration – namely, that the words corresponded to numbers which stood for quantities – but as time has progressed, he has learned to apply larger and larger numbers to quantities he sees.

He’s up to about four items now. Any more than that and he just starts reciting the remaining numbers through ten regardless of how many there actually are.

ftg
11-17-2007, 01:14 PM
I remember hearing studies that children up to a certain age can't tell the quantity of something unless they count every single item. ...

These tests have been considered debunked by some. What appears to happen is that the kids of a certain age really can tell they are the same. But the kids are puzzled by the question and in order to please the adult answer the way they do. See the books I listed above.

------------

If you want primitive, look at English speaking people: Once, twice, thrice. What word comes next?

aldiboronti
11-17-2007, 01:23 PM
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If you want primitive, look at English speaking people: Once, twice, thrice. What word comes next?

Why, frice, of course. Everybody knows that. :)