February 27, 2008

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

26 comments:

Varangy said...

(Bilinguals, it has been found, revert to the language they used in school when doing multiplication.)

So true. As a complete, natural bilingual (vocabulary, grammar, and accent) I have to revert to the language I learned mathematics in to do any sort of multiplication/division. My (non-bilingual) friends think I am retarded when they see me painfully using the other language to compute.

daveg said...

Interesting.

Spanish numbers are very long winded, many having three syllables.

The Spanish are not known for being math-centric.

Anonymous said...

Another interesting feature of Cantonese is that you can cover all of its tones by just reciting the numbers from 0-9. Each digit therefore has an additional distinguishing characteristic that takes no additional time to say. Whether this makes any difference or not, I have no idea, but as someone who's struggled with learning Cantonese for years (I live in Hong Kong) I can testify:

numbers=easy

all the other words=hard.

There's also a lot of numerological stuff deeply embedded in Chinese culture as well, so being quick with numbers therefore has high value.

braindead said...

As an engineer I work with math all day long, and study it to boot. Somehow I did not feel like reading the article, even though I consume most iSteve posts. I wonder why?

daveg said...

Could you test this by providing lists of numbers and measuring the rate of memorization with the verbal complexity of that list?

Might not be enough difference in the numbers (in English at least) to detect a difference.

There is only one single digit number with two syllables (seven) and the double digit number are pretty much the same (eleven and seventeen are longer tho).

Perhaps you could do the test in another language with more variety.

The first 20 French numbers are short, and they were great mathematicians.

Hibernia Girl said...

...multiplication facts must be stored in the brain verbally, as strings of words....

Not true for everybody -- at least it's not true for me.

The multiplication tables -- which I learned by rote memorization -- are burned into my brain visually. At school we had these maths workbooks that included little pictorial representations of the numbers -- dots like on a die. That's how I recall my times tables -- e.g. 8 (dots) x 8 (dots) = 64 (dots).

I haven't stored the multiplication facts verbally -- I've stored them visually -- the same way that my thinking process works -- pictures first, words afterwards.

And I'm not East Asian. ;-)

Not surprisingly, I didn't do very well in higher maths courses! Basic algebra ok -- geometry & trig, terrific -- beyond that, not so good.

Kai Carver said...

French digits are also mono-syllabic: un, deux, trois, quatre, cinq, six, sept, huit, neuf, dix, better than English (seven) and German (sieben). And France has a great mathematical tradition! QED.

Of course, Russian has almost as many polysyllabic digits (adin, tchetirye, vosiem, dieviat, diesiat) as Spanish (uno, cuatro, cinco, siete, ocho, nueve), yet they also have a great mathematical tradition. And anyway, mathematicians don't deal with digits much. Plus, if you take zéro into account, French is tied by German, which has the lovely monosyllabic null.

Clearly, further study is needed :-)

IMR said...

I'm pretty sure it was the British psychologist Alan Baddeley who did experiments comparing the short-term-memory digit spans of Cantonese and English-speaking Hong Kong schoolkids. The Cantonese speakers could recall an average of 9 or 10 digits, against an average of 7 for the English-speakers. The Cantonese-speakers also had better school grades in maths and showed better scores on simple tests of mental arithmetic.

Topiary Utopia said...

The stuff about number names sounds like a limited version of the Sapir-Whorf hypothesis.

Jeremiah said...

I would imagine another factor that creeps in as the math becomes more complex is working memory - the ability to mentally track a series of items and manipulate them without losing the data due to delay. This is one of the four smaller factors assessed by the WAIS-III (the most common IQ test administered by psychologists in the clinical setting). The other three are verbal comprehension, perceptual organization, and processing speed. The two subtests that load most heavily on working memory are "Digit Span" (being given a series of numbers, which you have to repeat - first forwards, then backwards), and "Matrix Reasoning." This is a relatively new subtest, in which a scrambled series of numbers and letters are provided, and the examinee has to mentally rearrange the items and give them back, with numbers first (in sequential order), then letters (in alphabetical order). As you can imagine, the most important ability is to be able to "hold" the data, and be able to manipulate it, without losing any due to memory erosion. This taps memory more than mathematics, but I'd imagine its quite important when discussing more complicated math problems (though the use of paper, chalkboard, etc. would reduce its importance).

What's fascinating to watch when administering these tests are the strategies people will employee in order to attempt to provide answers (harder to assess for the tests administered only verbally, but still occasionally observable). The more intelligent people tend to recognize they need to memorize the data first, before they can manipulate it. You can see them mouthing the original data to themselves over and over until they've "got it," then they go about moving the data into the proper order.

Eric said...

I doubt gambling is a big help to mathmatics. I used to play a lot of poker, and the number of Asian "scientific" players is very small. Far outnumbered by the ones who think bringing a little golden frog figurine will guarentee a winning session.

Eastern Europeans and Russians, though... almost all of them are mathmatical players. Eventually I realized what happended to all those unemployed nuclear engineers in the '90s. Who knew a PhD in nuclear physics would be such good training for calculating pot odds?

Maybe I should get back into the game now that the Russians are gearing up for Cold War II and those guys are safely ensconced back in Siberian weapons factories.

Truth said...

Steve:

You just insinuated that David Robinson, with his 1300 SAT and Naval Academy degree was smart. You know what that means.

Bill said...

If I learned a phone number in Chinese, I had to think in Chinese to remember it.

As for the visual/verbal aspects of math, I think there must be a lot of idiosyncracies. I scored higher on the verbal than math SAT, but I was clearly stronger in visual math like geometry than in algebra, which I detested. Actually, I had no problem rotating graphs or shapes in my head, but writing out the proofs was not easy. But that might be related to a mild disability I have (neurological in origin -- go figure) that makes it painful to write with pencil and paper.

Anyway, I believe I read something recently that suggested that apes' mathematical skills are on par with humans'.

Ah, here it is:

"It shows when you take language away from a human, they end up looking just like monkeys in terms of their performance,"

Anonymous said...


Another interesting feature of Cantonese is that you can cover all of its tones by just reciting the numbers from 0-9.


Ummm, the same is true in Mandarin.

I daresay the same is true in Thai and Laotian. (Although I cannot be sure.)

Your point?

I know a Cantonese speaker who is also a math teacher in the US and I am pretty sure that she does multiplication in English.

However, when she and I count Mahjong scores, we do so in Cantonese (and I am a native English speaker).

Anonymous said...


If I learned a phone number in Chinese, I had to think in Chinese to remember it.


The same here.

However, if I concentrate, I can translate them from English to Cantonese and vice versa.

Sideways said...

Interesting, my wife is quadingual, with Chinese being her first language and English being her fourth. She learned math in Indonesian. She still does math entirely in Chinese, which would seem to make her an exception.


You just insinuated that David Robinson, with his 1300 SAT and Naval Academy degree was smart. You know what that means.


I suspect it means your understanding of Steve's motivations are deeply flawed.

Anonymous said...

If you are strongly analytical, there is no way you are satisfied with rote learning. You want to know why, what are the patterns and principles. Rote I suspect, produces people who do not connect the dots very well, but are competent at mid brow tasks and professions.

David Davenport said...

Steve,

There's also a divergence between arithmetic skill and quickness versus adeptness in geometry or map reading, and between grasp of more abstract mathematics and deftness in arithmetic -- i.e., adding, dividing, multiplying, and dividing.

This divergence tends to split along male/female lines.

Some math professors are vulnerable to arithmetic errors, and I once took a course taught by a Mechanical Engineering prof. who was great at multiplying or dividing large numbers in his head, but who didn't fully grasp the tensor algebra he was trying to teach. I suppose the gender stereotype doesn't always hold.

Reg Cæsar said...

This number-name business would suggest the Danes are the worst mathematicians in Europe, an idea which has no other evidence. The sons of Gorm manage to combine the place-swapping "four-and-twenty" quirk of the Germans, Dutch and old English, with the "score" system of French and Italian (quatre-vingt for 80.) So 55 comes out as "five-and-half-the-third-score". Swedes and Norwegians have told me they can understand Danish quite easily but can't count at all in it.

Don't forget the Japanese, with their three-syllable native numbers-- hitotsu, futotsu, etc. Those are the mathephobes who miniaturized the world! Overcompensation, perhaps?

canson said...

"If you are strongly analytical, there is no way you are satisfied with rote learning. You want to know why, what are the patterns and principles. Rote I suspect, produces people who do not connect the dots very well, but are competent at mid brow tasks and professions."

I disagree. Rote learning if correctly applied will benefit everyone. Those that aren't so bright get some useful knowledge and disciplne. The brighter sort are able to increase speed and accuracy for routine tasks allowing time for more elevated thinking. Memorization saves time on the basics.

KlaosOldanburg said...

"If you are strongly analytical, there is no way you are satisfied with rote learning. You want to know why, what are the patterns and principles. Rote I suspect, produces people who do not connect the dots very well, but are competent at mid brow tasks and professions."

i disagree. go to a higher math class (set theory, linear algebra, differential equations etc.). it's all purely theoretical, abstract material (just like sociology, har dee har har). you will find students who know their multiplication tables (know them because they've been burned into memory) better than anyone else on campus.

linux nerds, who memorize enormous amounts of obscure technical details about the kernel, and how to use a command line interface to manipulate it, also excel at computer programming, which is extremely abstract as well.

if your thesis were correct, the janitor would be better at multiplication than the calculus teacher.

it's sad how afraid we've become of hard work.

dearieme said...

I was tested for number-memory as a freshman. Mine was lousy and I was told I'd never get a degree. But I, forgive me, proved to be a star undergraduate. Perhaps my result related to the fact that I came from a tiny town where the phone numbers had just three digits? I had never in my life had to remember a number of any length.

Anonymous said...

Don't forget the Japanese, with their three-syllable native numbers-- hitotsu, futotsu, etc.

hitotsu, futotsu,... mean first, second,... not generally used in calculations

i*chi, ni, san, shi, go, ro*ku, shi*chi, ha*chi, kyu, ju are the Japanese words for 1-10 used in calculations

Justin Halter said...

The guy was totally wrong about kids using calculators.

Don't let your children use calculators in elementary school, and don't let your kid's teacher allow them in the classroom either. Learning algebra is about a zillion times easier if you know the basic algorithms for numbers, especially fraction operations and even long division.

Using calculators raises mathematical cripples who can't even estimate or check for reasonableness. I taught junior high math, I saw first hand the damage that calculators can do to children's math abilities.

Chris said...

>"But aren't the East Asian advantages in math ability rooted more on the visual side?"

>"i*chi, ni, san, shi, go, ro*ku, shi*chi, ha*chi, kyu, ju are the Japanese words for 1-10 used in calculations"

So, the Japanese have the same superior math ability as the Chinese while using a number system even more cumbersome than English. That would kind of cast doubt on the article's implication that the math facility disparity is linguistic in origin, and suggest that it might be innate.

Markku said...

a*b = min(a,b)*10 - min(a,b)*(10-max(a,b))

That's my algoritm.

For example,

7*8 = 7*10 - 7*(10-8) = 70-14=56.

Or,

14*12 = 12*10 - (10-14)*12 = 12*10 - (-4)*12 = 120 + 48 = 168.

Or,

765*34 = (700+60+5)*(30+4)=
= 21000 + 1400 + 1800 + 20
= 21000 + 3200 + 20
= 24220

Now it is obvious why having the multiplication table burned in your memory is extremely useful. Otherwise you run out of short-term memory slots very fast when multiplying even two- or three digit numbers.