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.
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.
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.
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.
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”?
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…
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?”
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.
The Pirahã 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.