"It has often been said (and I think Keith Devlin reiterates the point in his book "The Math Gene") that mathematicians do not THINK linguistically."

I don't know if this is true, but I do think it's true that mathematicians don't think linguistically when they are doing math.

One thing I've noticed when programming computers is that if I try to insert comments into the program as I'm going along writing the program, then when I read them afterward, they look like gibberish. I can't even reconstruct what I was thinking, usually. My brain was really in some non-linguistic place. Math is like that, only more so.

But mathematicians do learn to convey ideas to each other, but it's a very high level function that they generally pick up in grad school, and one I don't think kids should have to bother with as they are trying to learn the fundamentals. Simply showing their work will reveal whether they know what's going on or not.

-- CarolynJohnston - 14 Nov 2005

Before I knew anything about constructivism per se, I was aware that our schools was asking the kids to put everything in writing.

From the get-go, that struck me as wrong.

I kept thinking: is a kid who's good at math always good at English, too?

You should take a look at Arthur Reber's book one of these days. It was life-altering for me. Incredible. (I think the title is ... IMPLICIT KNOWLEDGE. Oxford.)

I'll look it up later--

-- CatherineJohnson - 16 Aug 2005

I would guess that the real problem with "breaking into" mathematics lies in the difficulties we have trying to teach and learn an 'alinguistic' thought process linguistically

Well, let me tell you.....as a writer, I have had this thought many, many times.

-- CatherineJohnson - 16 Aug 2005

It has often been said (and I think Keith Devlin reiterates the point in his book "The Math Gene") that mathematicians do not THINK linguistically.

Thus, although a brilliant mathematician can be, well, brilliant, he has a difficult time capturing his thought processes into language.

If I were allowed to submit a radical hypothesis here, I would guess that the real problem with "breaking into" mathematics lies in the difficulties we have trying to teach and learn an 'alinguistic' thought process linguistically.

There has even been some research into the English language itself as a possible barrier to the effective learning of mathematics.

-- JdFisher - 16 Aug 2005

best performing students, part 3

The question of how our top students compare to everyone else's top students has made me realize I need to be paying attention to this. My goal as a homeschooler-on-the-side is for Christopher to be able to major in a math-related subject in college if he chooses, which apparently means he should be able to score a 625 or higher on TIMSS.

So I'm going to start scouting information on all ranges of student achievement, and posting it here.

Here's my first:

Researchers determined which items students who achieved at the various levels on the total test were likely to get right. Then they placed the items on a scale from 200 to 750. So we have a pretty good idea of what the best students know that others have difficulty with.

Only the top 10 percent of 9-year-olds were likely to get this math item right. Students had to explain their answers verbally, symbolically or pictorially.

In the first part they had to indicate that 20 is twice as large as 10 or that 10 is half of 20. 10 percent of third graders and 21percent of fourth graders did this. A small number of students (less than 1 percent in any country ) received credit for satisfactory explanations even though they did not give a yes or no response to whether Julia was right.

U.S. percentages were 13 percent at third grade and 25 percent at fourth grade.

For the second part, only 6 percent of third graders and 15 percent of fourth graders responded correctly. 6 percent of U.S. third graders and 17 percent of U.S. 4th graders got credit. However, 30 percent or more got credit in Japan, Korea and Singapore.

But I tend to think he would have been thrown by the sight of the numbers '10' and '20.'

As well, I'd say this problem imposes a high cognitive load. You have to keep Juanita and Amanda straight in your mind, unless you've developed seriously good informal chart-making skills, which Christopher has not done now and certainly had not done in 4th grade.

update: Christopher's answer

His first impulse, as I feared, was to say 'yes,' Amanda is right.

He obviously had the 'environmental dependency' effect of seeing the numbers '10' and '20' and thinking: 1/2.

But then he corrected himself, and said, confidently, that Juanita is right and Amanda is wrong. (Nice to see that the Designated Stupid Person concept has spread to TIMSS, too.)

His explanation was a bit strangled, but it was right. He said, 'Well, if there's 1 girl for every 2 boys, then there's 1 girl and 2 boys, then 2 girls and 4 boys, then 3 girls and 6 boys...'

This is pretty interesting, because I think he had a 'number sense' or 'pattern' way of getting this answer. In other words, I think he got the answer without really knowing why or how he got it. He just knew it. Juanita's correct statement of the problem instantly became his statement of the problem; he didn't have to do any adding or subtracting or logical reasoning to test Juanita's statement.

Then, when I asked him to explain why Juanita was right, he explained how her answer would work as a kind of Fancy Skip Counting Mechanism. If you kept counting up by 2-to-1 ratios, eventually you'd hit 30 kids, and your ratio would be 10 girls, 20 boys.

After he gave this illustration I asked him, 'how many girls and how many boys would there be in the class' (forgetting that in fact THE PROBLEM TELLS YOU THIS UP FRONT) and Christopher said, instantly, '10 girls and 20 boys.'

When I asked him how he knew (TIMSS should just have 'Catherine' be the Designated Stupid Person) he said, 'I just knew it.'

Apparently he had forgotten the fact that we'd been given this information, too. Like mother like son.

In any case.....this is something I was talking to Carolyn about the other night: what is the relationship of implicit knowledge to expertise when you're talking about math?

Certainly in every other field (I think) implicit knowledge is a sign that you're getting good at what you do, because you don't have to think about it. You 'just know it.'

But math has been confusing for me in this realm.....our friend Fred was here a few weekends ago, and I asked him to take a look at a RUSSIAN MATH problem that was stumping me. Fred is a Big Brain; he went to Yale undergrad, then got a Ph.D. in experimental psychology at Stanford, I think it was; then got a law degree at Yale; then clerked for the Supreme Court.

So I hope you're impressed.

Anyway, Fred was keenly interested in math when he went to college, but pretty quickly found out that pure mathematics wasn't going to be for him.

anti-constructivist digression