kitchen table math, the sequel: reading and math, math and reading

Monday, June 27, 2011

reading and math, math and reading

via Marginal Revolution:
To an economist, the most important fact to know about mental abilities is that across large populations, different mental abilities are positively correlated.1 In other words, people who are above average in arithmetic tend to perform above average in vocabulary tests, block puzzles, as well as in memorizing lists of numbers and then repeating them in reverse order. This positive correlation conflicts with the commonsense view that abilities are negatively correlated, that for instance while some people are good in math, others are good in verbal tasks.

This positive correlation is at the heart of the psychological concept of intelligence. Quantitatively, it is at the heart of psychometric methods that extract a principal component from a broad variety of mental ability subtests. This principal component is formally called g, or the g factor. Lay persons, and most routine psychological research, refer to IQ instead of g, but it is worth keeping the concept of g in mind. Across thousands of studies on the correlation across mental abilities across populations, no one has yet found a reliable negative correlation.

This fact should strengthen our priors when we come across new mental tasks and we ask ourselves, “Will high IQ groups be better than average at this new mental task?” For every mental task so far that involves any level of sophistication, the answer has been yes.
National IQ and National Productivity: The Hive Mind Across Asia
Garett Jones
January 2011
[Published in Asian Development Review, June 2011]

Serendipity!

Since last Thursday, I've been trying to explain to C. that a 190-point gap between your reading score & your math score is ridiculous. Ridiculous because a 190-point gap between a reading score & a math score is not consistent with the way a person's brain is built. On IQ tests, everything goes with everything; if you're good on one task, you're good on another.

C's math scores, I keep telling him, like my math scores when I was his age, are a measure of the math teaching and math curriculum he's had, not of his innate ability to do math. His math tutor told him the same thing this morning, but C. claimed later not to have heard. So I was in need of fresh ammunition, and now here it is.

I love the internet.

37 comments:

Catherine Johnson said...

C. got an 800 on reading.

800!

I've never even met anyone who had an 800 on reading.

What a blast ---- it's like winning the lottery.

SIX-TEN ON MATH.

CRAZY!

gasstationwithoutpumps said...

Correlations in the population still allow for outliers. There are dyslexics who are excellent at math and mathophobes who excel at reading tests—they are just much less common than the ones who do similarly well at both.

It is common for educational psychologists to use large discrepancies in scores as diagnostic information for learning disorders. (Though further testing is needed to distinguish learning disorders from bad teaching.)

Anonymous said...

Agree with gasstationwithoutpumps. Averages are instructive, but outliers are common. Plus, some people are fine with cultivating all of their talents, but some get to the point where they invest much more heavily on one side (math or reading/writing) than the other.

SATVerbalTutor. said...

You can tell C. that I beat him: 220 point gap between CR and math (800/580), and I actually knew a couple of people with even larger gaps (800/560, 800/490!). I had horrific math teachers all through high school; the worst teacher I had in my life was for geometry, and I don't think I ever recovered math-wise from the experience of being in his class. I know that I could have been very good at math with some halfway decent instruction, but alas, I never even got that.

Linda Seebach said...

Catherine said, "I've never even met anyone who had an 800 on reading."
How would anyone know that? People who get 800's on SAT tests are usually smart enough to know that mentioning that in a social context will be resented by anyone present whose score was lower, which is usually everybody.

Anonymous said...

LOL at Linda's remark. I'd agree -- though word got out about my son's 800 CR (it was January's test) pretty well amongst the kids, but his other two were over 700 too.

It helped that his older brother who had to take it twice to get all of them over 700 (it was again the math that was just below and he is a math-y kid) told him to just do what I said in terms of practice so he didn't have to take it twice!

One of our local universities has that (plus decent grades, of course) as their criteria for the honors college.

Of course, I'll sign this anonymous...

kcab said...

Good for C!

Independent George said...

I scored a 420 on my Writing achievement test (waaaay back in 1994). I had a 720 on the SAT-Verbal, and a 5 on the AP English test, but I scored 420 on the writing test.

Apparently, my writing style is not conducive to their scoring method.

Catherine Johnson said...

It is common for educational psychologists to use large discrepancies in scores as diagnostic information for learning disorders.

Right - and autism is an interesting disorder in that you get 'scatter' (is that the term?? I'm forgetting now.)

For years I've been trying to tell people that large discrepancies in scores like SAT aren't 'natural,' citing IQ exams.

But I can't get anyone to listen because there is a universal belief in math brains versus verbal brains.

People who believe in 'g' (or in IQ) still believe it's possible to be extremely bright in one realm and extremely not-bright in another.

Autistic people can show that pattern -- and that's one of the signs that they are autistic!

I remember Donna Williams writing that her scores on IQ subscales ranged all the way from 'retarded' to 'genius.'

Catherine Johnson said...

some people are fine with cultivating all of their talents, but some get to the point where they invest much more heavily on one side (math or reading/writing) than the other.

right!

That is the issue here, not biology.

My desire, as a parent (and as a person who wasn't taught much math in high school herself), is that K-12 should see its job as 'being my child's frontal lobes': the school should do everything in its power to make sure students reach their full potential in all subjects being taught.

I don't want it to be left to 11 year olds to 'decide' they'd like to 'focus' their efforts on their verbal talents at the expense of their math talents.

"g" is general intelligence; it applies generally.

If a student is high in 'g,' then he should be achieving well in all K-12 subjects -- and if he or she is **not** achieving well across the board the school should figure out how to improve his or her achievement.

Crimson Wife said...

One of the reasons why 800's on the verbal section were very rare prior to re-centering is that getting a single question wrong used to knock someone all the way down to a 750. I'm *STILL* annoyed about that 15+ years later!

Catherine Johnson said...

C's score validates my perception when we were dealing with our middle school: we needed to get him tested for the **opposite** reason our autistic children needed testing.

We needed to bring the school objective evidence that C's IQ was more than adequate to the challenge of learning algebra & Earth science in 8th grade.

Catherine Johnson said...

getting a single question wrong used to knock someone all the way down to a 750

Right - I had a 720 in high school, which I think meant 2 questions wrong. Didn't learn that 'til a few years ago, thought.

Catherine Johnson said...

fyi--

from C's score report for May test, where he scored 730 on reading:

Section 2 (23 questions):
missed 1, left 1 blank

Section 5 (25 questions):
missed 2

Section 9 (19 questions):
missed 2

So: on the May 7 test, you could miss 6 out of 67 and get a 730.

Catherine Johnson said...

Catherine said, "I've never even met anyone who had an 800 on reading."
How would anyone know that?


I have ways of making you talk.

Catherine Johnson said...

(That was a joke.)

Catherine Johnson said...

I just looked up the Raw Scores for the May 2011 test.

You could have a raw score of 64 and get an 800 on reading.

For people who haven't made a study of the SAT, you get 1 point for each correct answer & lose 1/4 points for each wrong answer, so that means:

* you could miss 2 and get an 800
* you could omit 3 and get an 800
* you could miss 1 and omit 2 and get an 800

Catherine Johnson said...

On math, the only raw score that gave you an 800 was a 54 out of 54.

You could get a 790 if you got 53 right.

A raw score of 52 puts you down to a 760; a raw score of 51 puts you at 740; a raw score of 49 is 710; a raw score of 48 is 700.

Glen said...

There's a mathematical issue here that is worth considering. The quote is apparently referring to the strong correlation among group averages, not among individual scores. They are different.

Assume the SAT measures achievement, where achievement is IQ times attention. Take a group of people with identical IQs but who allocate their attention differently. Those who allocate equal attention to math and reading should get the group average math and reading scores. Those who allocate extra attention to math have less to allocate to other things, so they should score higher than the group average in math and lower than the group average in, for example, reading.

Over the group, individual differences in attention allocation should cancel out, leaving the group average scores in both math and reading primarily dependent on group average IQ. High-IQ groups should get high average scores in both math and reading, while low-IQ groups get low scores in both. The correlation of GROUP AVERAGE math scores to group average reading scores should be (and is) high. A group's average math score is a good predictor of its average reading score.

But correlation of individual math-to-reading scores should be somewhat less, because individual scores also depend on attention allocation. Picture your math score on one side of a seesaw and your reading score on the other. You IQ raises or lowers the midpoint, raising or lowering both math and reading together, creating some correlation. But your attention allocation determines the slant, and it is the slant that determines the size of the gap.

Even perfect correlation of group average math to group average reading scores would not imply that individual seesaws should be level or that a strong slant was any kind of outlier, just that individuals are as likely to slant one way as the other.

Groups with higher average IQs have advantages in all cognitive skills that depend on IQ. But within a circle of peers of roughly equal IQ, an individual who was significantly better than his peers in one area, having no IQ advantage, would have to achieve it by focusing on that area at the expense of others. To rise above equally smart peers, you have to specialize.

I've left out complicating factors such as the tendency of entire groups to specialize, individual differences in work ethic, the effects of test prep, and so on, but result of this simple seesaw model would be that groups that are better than the average group at one cognitive skill tend to be better at others, while individuals who are better than their average peer at one skill tend to be worse at others. The idea that people who are better at this tend to be worse at that may be both false and true, depending on whether "people" means groups or individuals within a peer group.

gasstationwithoutpumps said...

Glen's analysis here is a good one for group correlation versus indiviusal correlation, but I believe that the original result was about individual correlations, which makes his analysis irrelevant.

Glen said...

The original article cited is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

The Marginal Revolution blog, citing the above, suggests that we have an answer to the question, "Will high IQ groups be better than average at this new mental task?" It does not ask about high IQ individuals, but other comments he makes suggest he isn't making any distinction.

Then our KTM comments discuss correlation between an individual's math and reading scores in light of the above study.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a reminder relevant. I also think that a look at the differences might help clarify some issues and resolve some apparent contradictions.

Glen said...

The original article cited is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

The Marginal Revolution blog, citing the above, suggests that we have an answer to the question, "Will high IQ groups be better than average at this new mental task?" It does not ask about high IQ individuals, but other comments he makes suggest he isn't making any distinction.

Then our KTM comments discuss correlation between an individual's math and reading scores in light of the above study.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a reminder relevant. I also think that a look at the differences might help clarify some issues and resolve some apparent contradictions.

Glen said...

The original article cited is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

The Marginal Revolution blog, citing the above, suggests that we have an answer to the question, "Will high IQ groups be better than average at this new mental task?" It does not ask about high IQ individuals, but other comments he makes suggest he isn't making any distinction.

Then our KTM comments discuss correlation between an individual's math and reading scores in light of the above study.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a reminder relevant. I also think that a look at the differences might help clarify some issues and resolve some apparent contradictions.

Glen said...

The original article cited is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

The Marginal Revolution blog, citing the above, suggests that we have an answer to the question, "Will high IQ groups be better than average at this new mental task?" It does not ask about high IQ individuals, but other comments he makes suggest he isn't making any distinction.

Then our KTM comments discuss correlation between an individual's math and reading scores in light of the above study.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a reminder relevant. I also think that a look at the differences might help clarify some issues and resolve some apparent contradictions.

Glen said...

Test post. Sorry, something strange going on.

Glen said...

The cited paper is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

The Marginal Revolution blog, citing the above, claims to answer the question, "Will high IQ groups be better than average at this new mental task?" It does not ask about high IQ individuals, but other comments he makes suggest he isn't making any distinction.

Then our KTM comments discuss correlation between an individual's math and reading scores in light of the above study.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a reminder relevant. I also think that a look at the differences might help clarify some issues and resolve some apparent contradictions.

Glen said...

The original article cited is called "National IQ and National Productivity". National IQ is a group average. National Productivity is a group average. The name suggests that the article is about the correlation between the two.

Subsequent discussion has dealt with both group and individual correlation, seeming to conflate the two.

Under the circumstances, I think there is enough possibility of confusion between different types of correlation to make a note about the differences relevant. I also think that a consideration of the features of the different statistical views might help clarify some issues and resolve some apparent contradictions.

Catherine Johnson said...

The quote is apparently referring to the strong correlation among group averages, not among individual scores.

I'll have to go back and look, but for the record I've always read that the correlation is for individuals.

The reason I pulled this quote out & posted is that I can never get anyone to believe me when I tell them this!

People believe deeply that English and math are two completely different animals: if you're good in one, you're not good in the other, and vice versa.

It's impossible to get teachers (or parents or students!) to see a gap like C's as evidence of a serious flaw in his math education rather than as a natural and unalterable state of affairs.

Catherine Johnson said...

I wonder if there's any particular talent when it comes to English versus math-----?

I read a fascinating study today about artists & "stereo blindness" that I'll try to get posted; I wondered immediately whether stereo blindness would help with math (or at least with geometry).

I guess at the moment I can imagine 'deficits' -- not 'talents' -- being a defining difference or 'edge.'

If two people have the same 'g,' then the person with the useful deficit/difference has the advantage.

Spoken like a true mother of autistic kids!

Glen said...

I'll have to go back and look, but for the record I've always read that the correlation is for individuals.

There is no doubt in my mind that g exists, and g is, by definition, the factor(s) underlying the high correlation in INDIVIDUAL performance of various novel (untrained) tasks. Countless studies have confirmed its existence. I'll just call it IQ. It's solidly supported by evidence.

I was pointing out some distinctions that I think were getting lost. IQ, which is domain general, moves (almost) all cognitive abilities in the same direction. Attention, which is domain specific, moves one at the expense of others. Achievement is the product of both.

The SAT purports to measure achievement. At the group level, individual differences in interest tend to cancel out, leaving IQ to drive group achievement, raising or lowering math and reading together. In state-by-state comparisons, math and reading scores should match each other quite closely. (If they don't, something may be wrong.)

But that doesn't say anything about what math-reading gap would be normal for an individual, where attention (how the IQ is allocated) plays a major role in achievement. That the Marginal Revolution blogger was using a study of national IQ vs national productivity to inform his conclusions about individual specialization seemed evidence of, or at least liable to cause, some confusion.

People believe deeply that English and math are two completely different animals: if you're good in one, you're not good in the other, and vice versa.

And I wanted to point out that this is not right or wrong but dependent on the base of comparison. Again, it is about achievement. IQ raises or lowers math and reading together, so if you compare an individual to a group with a wide range of IQs, his IQ should dominate in determining how both his math and reading compare to the group. In such a group, he'll tend to be good, bad, or average in both.

But among a group of peers, say his high school class, the IQ variance is probably much lower. In an environment where everyone has roughly the same IQ, it is how you allocate that IQ (attention) that tends to dominate your achievement relative to those peers. To be significantly better than they are at one thing, you may have to be worse at something else.

Since most of us pay more attention to our peers than to distant, diverse strangers, we'll tend to see the "better at one means worse at the other" view. It's not wrong; it's just not the whole story.

Glen said...

I'll have to go back and look, but for the record I've always read that the correlation is for individuals.

There is no doubt in my mind that g exists, and g is, by definition, the factor(s) underlying the high correlation in INDIVIDUAL performance of various novel (untrained) tasks. Countless studies have confirmed its existence. I'll just call it IQ. It's solidly supported by evidence.

I was pointing out some distinctions that I think were getting lost. IQ, which is domain general, moves (almost) all cognitive abilities in the same direction. Attention, which is domain specific, moves one at the expense of others. Achievement is the product of both.

The SAT purports to measure achievement. At the group level, individual differences in interest tend to cancel out, leaving IQ to drive group achievement, raising or lowering math and reading together. In state-by-state comparisons, math and reading scores should match each other quite closely. (If they don't, something may be wrong.)

But that doesn't say anything about what math-reading gap would be normal for an individual, where attention (how the IQ is allocated) plays a major role in achievement. That the Marginal Revolution blogger was using a study of national IQ vs national productivity to inform his conclusions about individual specialization seemed evidence of, or at least liable to cause, some confusion.

(cont'd)

Glen said...

People believe deeply that English and math are two completely different animals: if you're good in one, you're not good in the other, and vice versa.

And I wanted to point out that this is not right or wrong but dependent on the base of comparison. Again, it is about achievement. IQ raises or lowers math and reading together, so if you compare an individual to a group with a wide range of IQs, his IQ should dominate in determining how both his math and reading compare to the group. In such a group, he'll tend to be good, bad, or average in both.

But among a group of peers, say his high school class, the IQ variance is probably much lower. In an environment where everyone has roughly the same IQ, it is how you allocate that IQ (attention) that tends to dominate your achievement relative to those peers. To be significantly better than they are at one thing, you may have to be worse at something else.

Since most of us pay more attention to our peers than to distant, diverse strangers, we'll tend to see the "better at one means worse at the other" view. It's not wrong; it's just not the whole story.

kcab said...

But to me, this looks like you are not talking about "above average" vs "below average", but about "way above average" vs "above average". At least, I think that's where a 610 lands. I don't think it's uncommon at all for someone who is outstanding in one area to be very good, but not at the top, in another. And then there's chance, as well as personal interest.

Though, my (ancient) SAT scores were almost exactly the same in the two areas.

Catherine Johnson said...

But among a group of peers, say his high school class, the IQ variance is probably much lower. In an environment where everyone has roughly the same IQ, it is how you allocate that IQ (attention) that tends to dominate your achievement relative to those peers.

right

my issue with the K-12 education C. has received (in an expensive public school system and in a Jesuit high school) is that the schools naturalized the discrepancy in his achievement by attributing it to ability.

SAT math is not high level math; it's algebra 1, a bit of algebra 2, and some geometry. Yes, the problems are tricky, but they get less tricky with practice.

At some point all of us specialize, which means we become good at what we focus on and we don't become good at what we don't focus on.

For my purposes, K-12 should have given my son a very solid education in the 'basics,' which -- again for my purposes -- includes high school math.

The schools our son has attended have been happy to have the kids 'specialize' at age 11 or even earlier.

The moment a child shows a lack of interest in math, it's OK for him to 'focus' on English.

That was terrible for C, who was going to do well in English and history with or without a school -- and who in middle school tested far beyond the reading curriculum he'd had at school.

Just the other day, when I told a teacher C's scores, she said, "C. always struggled in math."

That was the line: C. struggles in math.

I don't think I've ever used the word "IQ" in talking about C, but I think I'll go ahead and do so now: when a kid has an IQ as high as C's, he should not be "struggling" in ARITHMETIC, and he should certainly not be **defined** as struggling in arithmetic.

We've had the same mishegoss at his parochial high school.

At the end of this school year, the teacher actually told him, in front of the entire class, "You don't belong in this class."

"This class" being the class the school placed him in, for pete's sake.

Catherine Johnson said...

I see the mirror-image problem with kids who aren't academically talented.

With C, teachers and administrators simply assumed he was 'bad at math' because he was so much better at English/history.

Another boy I know, who has a natural interest in and liking for math that C. does not, is assumed to be 'bad at math' because he's bad at other subjects, too.

The school has no category for high achievement (or strong achievement or pretty good achievement or anything else that's positive) in a kid whom they see as globally not academically talented.

It's crazy!

Glen's point that achievement involves ability and attention is exactly right: this boy may have below-average ability, but he started out with above-average attention in the subject of math.

In the school of my dreams, kids' interests would be exploited and supported; you'd do everything in your power to make sure slower students really do reach their full potential in the subjects they have an affinity for.

Catherine Johnson said...

What makes me crazy - SORRY! I'M RANTING! - is the apparently universal perception that high school math is the pinnacle of math achievement!

I'm sorry: it is simply NOT THE CASE that only a MATH GENIUS can handle high school trigonometry.

Catherine Johnson said...

I think that's where a 610 lands. I don't think it's uncommon at all for someone who is outstanding in one area to be very good, but not at the top, in another

Absolutely - but that should come later!

The time to 'specialize' is not age 7 or 8!