Math Mania

Jim Holt has a nice article in The New Yorker: Numbers Guy: Are our brains wired for math?

It starts off, though, with the now-mandatory couple of pages of description, starting with a head injury victim and going on to MRI scans, of where exactly in the brain the number sense may or may not reside. I always skip over these sections of articles, perhaps because I lack the part of the brain that allows me to think three-dimensionally.

And, because I never seem to wind up missing anything important.

The NYT Magazine had a laugh last year at how credulous we are in the face of brain scan explanations:

"A paper published online in September by the journal Cognition shows that assertions about psychology — even implausible ones like “watching television improved math skills” — seem much more believable to laypeople when accompanied by images from brain scans."

This is not to say that this it won't eventually prove hugely useful for the layman to have a thorough understanding of brain anatomy, but I don't think that time has arrived yet.

But the second half of the article gets more interesting:

"Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first.

"Our number sense endows us with a crude feel for addition, so that, even before schooling, children can find simple recipes for adding numbers. If asked to compute 2 + 4, for example, a child might start with the first number and then count upward by the second number: “two, three is one, four is two, five is three, six is four, six.”

"But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.)

"The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 X 6, the reflex activation of 7 + 6 and 7 X 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 X 8, the error rate can exceed twenty-five per cent."

You just have to have the Times Table pounded into your head over and over as a kid, but our education system is against "rote learning," so school kids aren't usually forced to chant them like in the good old days. But kids actually kind of like rote learning. (It's adults who hate learning that way, and especially hate teaching that way.) Kids like singing the alphabet song, for instance. You'd think educators would find a times table rap that today's children would like.

"Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo."

I dunno. I spent the 1980s and 1990s in the business world, where arithmetic would be useful in practically every discussion, even if just for doing reality checks of ideas. Yet, I observed then that the only people who carried calculators around with them all the time were the computer geeks, who were way above average at doing arithmetic in their heads or on paper. Maybe it's now changed now that everybody has a cell phone, which can serve as a calculator (although, come to think of it, I never used my last cell phone as a calculator).

"This attitude might make Dehaene sound like a natural ally of educators who advocate reform math, and a natural foe of parents who want their children’s math teachers to go “back to basics.” But when I asked him about reform math he wasn’t especially sympathetic. “The idea that all children are different, and that they need to discover things their own way—I don’t buy it at all,” he said. “I believe there is one brain organization. We see it in babies, we see it in adults. Basically, with a few variations, we’re all travelling on the same road.”

Steven Pinker emphasizes that humans are awfully alike qualitatively, but not necessarily quantitatively. We all breathe oxygen, for example, but some people can function in the thin air above 20,000 feet and some people can't.

"He admires the mathematics curricula of Asian countries like China and Japan, which provide children with a highly structured experience, anticipating the kind of responses they make at each stage and presenting them with challenges designed to minimize the number of errors. “That’s what we’re trying to get back to in France,” he said. Working with his colleague Anna Wilson, Dehaene has developed a computer game called “The Number Race” to help dyscalculic children. The software is adaptive, detecting the number tasks where the child is shaky and adjusting the level of difficulty to maintain an encouraging success rate of seventy-five per cent."

When my kids were little, we bought a computer arithmetic drilling game in which you helped basketball star David Robinson (who scored 1300 on the SAT, old-style) beat the bad guys by getting the right answers. It had adaptive logic that gave you extra work on what you were having difficulty with. It was wiped out in the market place by games with more elaborate graphics that didn't adjust to errors.

In 1980, the military's AFQT entrance exam (which was used in The Bell Curve) was a discouraging 105 pages long. It was discovered years later that black males were particularly likely to give up early, which was one reason the white-black IQ gap in that test was a anomalously large 18.6 points. In 1997, a computerized version of the AFQT was introduced, which provides easier questions if you get a lot wrong. The white-black gap on that is only 14.7 points. So, this kind of software can be useful.

"Today, Arabic numerals are in use pretty much around the world, while the words with which we name numbers naturally differ from language to language. And, as Dehaene and others have noted, these differences are far from trivial. English is cumbersome. … Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)"

Interesting. Nobody is more number crazy than Hong Kongers -- just check out their gambling obsession.

But aren't the East Asian advantages in math ability rooted more on the visual side? Dan Seligman's intro to IQ, A Question of Intelligence has a fun chapter comparing the visual approach of the Japanese to the verbal approach of the Jews. A friend told me once that Leon Kamin, the left wing psychologist who wrote Not In Our Genes with Richard Lewontin and Steven Rose, refused to believe that some people used visual imagination to help them work with numbers. Kamin can do prodigious feats of mental arithmetic working wholly verbally in his head. Perhaps he's descended from a long line of kabbalists?

My published articles are archived at iSteve.com -- Steve Sailer