Innate Mathematical Ability

In my pre­sent se­quence of posts, I’m writ­ing about the na­ture of math­e­mat­i­cal abil­ity. My main rea­son for do­ing so is to provide in­for­ma­tion that can help im­prove math­e­mat­i­cal abil­ity.

Along the way, I’m go­ing to dis­cuss how peo­ple can’t im­prove their math­e­mat­i­cal abil­ity. This may seem an­ti­thet­i­cal to my goal. Fo­cus on in­nate abil­ity can lead to a sort of self-fulfilling proph­esy, where peo­ple think that their abil­ities are fixed and can’t be im­proved, which re­sults in them not im­prov­ing their abil­ities be­cause they think that do­ing so is pointless.

Carol Dweck has be­come well known for her growth mind­set /​ fixed mind­set frame­work. She writes:

“In a fixed mind­set stu­dents be­lieve their ba­sic abil­ities, their in­tel­li­gence, their tal­ents, are just fixed traits. They have a cer­tain amount and that’s that, and then their goal be­comes to look smart all the time and never look dumb. In a growth mind­set, stu­dents un­der­stand that their tal­ents and abil­ities can be de­vel­oped through effort, good teach­ing and per­sis­tence. They don’t nec­es­sar­ily think ev­ery­one’s the same or any­one can be Ein­stein, but they be­lieve ev­ery­one can get smarter if they work at it.” [...] This is im­por­tant be­cause in­di­vi­d­u­als with a “growth” the­ory are more likely to con­tinue work­ing hard de­spite set­backs...

As I’ll de­scribe in my next post, I’m broadly sym­pa­thetic with Dweck’s per­spec­tive. But it’s not an ei­ther-or situ­a­tion. Some abil­ities are in­nate and can’t be de­vel­oped, and other abil­ities can be.

One could ar­gue that this idea is too nu­anced for most peo­ple to ap­pre­ci­ate, so that it’s bet­ter to just not talk about in­nate abil­ity. This seems to me pa­ter­nal­is­tic and pa­tron­iz­ing. Peo­ple need to know which abil­ities are fixed and which can be de­vel­oped, so that they can fo­cus on de­vel­op­ing abil­ities that can in fact be de­vel­oped rather than wast­ing time and effort on de­vel­op­ing those that can’t be.

Work­ing to im­prove abil­ities that are fixed is unproductive

When I was in el­e­men­tary school, I would of­ten fall short of an­swer­ing all ques­tions cor­rectly on timed ar­ith­metic tests. Mul­ti­ple teach­ers told me that I needed to work on mak­ing fewer “care­less mis­takes.” I was puz­zled by the situ­a­tion – I cer­tainly didn’t feel as though I was be­ing care­less. In hind­sight, I see that my teach­ers were mostly mis­guided on this point. I imag­ine that their think­ing was:

“He knows how to do the prob­lems, but he still misses some. This is un­usual: stu­dents who know how to do the prob­lems usu­ally don’t miss any. When there’s a task that I know how to do and don’t do it cor­rectly, it’s usu­ally be­cause I’m be­ing care­less. So he’s prob­a­bly be­ing care­less.”

If so, their er­ror was in as­sum­ing that I was like them. I wasn’t miss­ing ques­tions that I knew how to do be­cause I was be­ing care­less. I was miss­ing the ques­tions be­cause my pro­cess­ing speed and short-term mem­ory are un­usu­ally low rel­a­tive to my other abil­ities. With twice as much time, I would have been able to get all of the prob­lems cor­rectly, but it wasn’t phys­i­cally pos­si­ble for me to do all of the prob­lems cor­rectly within the time limit based on what I knew at the time. (The situ­a­tion may have been differ­ent if I had had ex­po­sure to men­tal math tech­niques, which can sub­sti­tute for in­nate speed and ac­cu­racy.)

Even at that age, based on my in­tro­spec­tion, I sus­pected that my teach­ers were wrong in their as­sess­ment of the situ­a­tion, and so largely ig­nored their sug­ges­tion, while at the same time feel­ing faintly guilty, won­der­ing whether they were right and I was just ra­tio­nal­iz­ing. I made the right judg­ment call in that in­stance – mak­ing a sys­tem­atic effort to stop mak­ing “care­less er­rors” un­der time con­straints wouldn’t have been pro­duc­tive. To avoid such waste we need to delve into a dis­cus­sion of in­nate abil­ity.

In­tel­li­gence and in­nate math­e­mat­i­cal ability

I think that math­e­mat­i­cal abil­ity is best con­cep­tu­al­ized as the abil­ity to rec­og­nize and ex­ploit hid­den struc­ture in data. This defi­ni­tion is non­stan­dard, and it will take sev­eral posts to ex­plain my choice.

Ab­stract pat­tern recog­ni­tion ability

A large part of “in­nate math­e­mat­i­cal abil­ity” is “ab­stract pat­tern recog­ni­tion abil­ity,” which can be op­er­a­tional­ized as “the abil­ity to cor­rect an­swer Raven’s Ma­tri­ces type items.” Tests of Raven’s Ma­tri­ces type are per­haps the purest tests of IQ: the cor­re­la­tion be­tween perfor­mance on them and the g-fac­tor is ~0.8, as high as any IQ sub­test, and an­swer­ing the items doesn’t re­quire any sub­ject mat­ter knowl­edge. One ex­am­ple of an item is:

The test taker is asked to pick the choice that com­pletes the pat­tern. Peo­ple who are able to pick the cor­rect choice at all can usu­ally do so within 2 min­utes – the ques­tions have the char­ac­ter “ei­ther you see it or you don’t.” Most peo­ple can’t see the pat­tern in the above ma­trix. A small num­ber of peo­ple can see much more sub­tle pat­terns.

There’s fairly strong ev­i­dence that some­thing like 30% of what differ­en­ti­ates the best math­e­mat­i­ci­ans in the world from other math­e­mat­i­ci­ans is the in­nate abil­ity to see the sorts of pat­terns that are pre­sent in very difficult Raven’s ma­tri­ces type items. (I’ll make what I mean by “some­thing like 30%” more pre­cise in a fu­ture post.)

Fields Medal­ist Terry Tao was part of the Study of Math­e­mat­i­cally Pre­co­cious Youth (SMPY). Pro­fes­sor Ju­lian Stan­ley wrote:

On May 1985 I ad­ministered to [10 year old] Terry the Raven Pro­gres­sive Ma­tri­ces Ad­vanced, an un­timed test. He com­pleted its 36 8-op­tion items in about 45 min­utes. Whereas the av­er­age Bri­tish uni­ver­sity stu­dent scores 21, Terry scored 32. He did not miss any of the last, most difficult, 4 items. Also, when told which 4 items he had not an­swered cor­rectly, he was quickly able to find the cor­rect re­sponse to each. Few of SMPY’s ablest pro­tégés, mem­bers of its “700-800 on SAT-M Be­fore Age 13” group, could do as well.

Peo­ple like Terry are per­haps 1 in a mil­lion, but I’ve had the chance to tu­tor sev­eral chil­dren who are in his gen­eral di­rec­tion.

De­scrip­tions of mile­stones like “scored 760 on the math SAT at age 8” (as Terry did) usu­ally greatly un­der­state the abil­ity of these chil­dren when the mile­stone is in­ter­preted as “com­pa­rable to a high school stu­dent in the top 1%,” in that there’s a con­no­ta­tion that the child’s perfor­mance comes from the child hav­ing learned the usual things very quickly. The situ­a­tion is usu­ally closer to “the child hasn’t learned the usual things, but is able to get high scores by solv­ing ques­tions ththat high school stu­dents wouldn’t able to able to solve with­out hav­ing stud­ied alge­bra and ge­om­e­try.”

A im­pact of in­ter­act­ing with such a child can be over­whelming. I’ve re­peat­edly had the ex­pe­rience of teach­ing such a child a math­e­mat­i­cal topic typ­i­cally cov­ered only in grad­u­ate math courses, and one that I know well be­yond the level of text­book ex­po­si­tions, and the child re­spond­ing by mak­ing ob­ser­va­tions that I my­self had missed. The ex­pe­rience is sur­real, to the point that I wouldn’t have been sur­prised to learn that it had all been a dream 30 min­utes later.

I’ll give an ex­am­ple to give a taste of a visceral sense for it. In one of my high school classes, my teacher as­signed the prob­lem of eval­u­at­ing ‘x’ in the equa­tion be­low:

Tan­gen­tially, I don’t know why we were as­signed this prob­lem, which is of con­sid­er­able math­e­mat­i­cal in­ter­est, but also out­side of the usual high school cur­ricu­lum. In any case, I re­mem­ber puz­zling over it. Based on my ex­pe­riences with chil­dren similar to Terry, it seems likely that his 8-year old self would see how to an­swer it im­me­di­ately, with­out hav­ing ever seen any­thing like the prob­lem be­fore. Roughly speak­ing, an 8-year old child like Terry can rec­og­nize ab­stract pat­terns that very few (if any) of a group of 30 high school stu­dents with the math SAT score would be able to rec­og­nize.

In A Parable of Ta­lents, Scott Alexan­der wrote:

IQ is so im­por­tant for in­tel­lec­tual pur­suits that em­i­nent sci­en­tists in some fields have av­er­age IQs around 150 to 160. Since IQ this high only ap­pears in 110,000 peo­ple or so, it beg­gars co­in­ci­dence to be­lieve this rep­re­sents any­thing but a very strong filter for IQ (or some­thing cor­re­lated with it) in reach­ing that level. If you saw a group of dozens of peo­ple who were 7’0 tall on av­er­age, you’d as­sume it was a bas­ket­ball team or some other group se­lected for height, not a bunch of botanists who were all very tall by co­in­ci­dence.

Of the sci­ences, pure math is the one where in­nate ab­stract pat­tern abil­ity is most strongly cor­re­lated with suc­cess, and data sug­gest that many of the best math­e­mat­i­ci­ans in the world have in­nate ab­stract pat­tern recog­ni­tion pos­sessed by fewer than 1 in 10,000 peo­ple. Terry Tao’s in­nate ab­stract pat­tern recog­ni­tion abil­ity is much rarer than 1 in 10,000, per­haps 1 in 1 mil­lion: it’s ex­tremely im­prob­a­ble that some­one with such ex­cep­tional in­nate abil­ity would by chance also be some­one who would go on to do Fields Medal win­ning re­search.

In­ter­est­ingly, many math­e­mat­i­ci­ans are un­aware of this. Terry Tao him­self wrote:

A rea­son­able amount of in­tel­li­gence is cer­tainly a nec­es­sary (though not suffi­cient) con­di­tion to be a rea­son­able math­e­mat­i­cian. But an ex­cep­tional amount of in­tel­li­gence has al­most no bear­ing on whether one is an ex­cep­tional math­e­mat­i­cian.

It’s not en­tirely clear to me how some­body as math­e­mat­i­cally tal­ented as Tao could miss the ba­sic Bayesian prob­a­bil­is­tic ar­gu­ment that Scott Alexan­der gave, which shows that Tao’s own ex­is­tence is very strong ev­i­dence against his claim. But two hy­pothe­ses come to mind.

Ver­bal rea­son­ing ability

Like Grothendieck, like Scott Alexan­der, and like my­self, Tao has very un­even abil­ities, only in an en­tirely differ­ent di­rec­tion:

Yet at age 8 years 10 months, when he took both the SAT-M and the SAT-Ver­bal, Terry scored only 290 on the lat­ter. Just 9% of col­lege-bound male 12th-graders score 290 or less on SAT-V; a chance score is about 230. The dis­crep­ancy be­tween be­ing 10 points above the min­i­mum 99th per­centile on M and at the 9th per­centile on V rep­re­sents a gap of about 3.7 stan­dard de­vi­a­tions. Clearly, Terry did far bet­ter with the math­e­mat­i­cal rea­son­ing items (please see the Ap­pendix for ex­am­ples) than he did read­ing para­graphs and an­swer­ing com­pre­hen­sion ques­tions about them or figur­ing out antonyms, ver­bal analo­gies, or sen­tences with miss­ing words.

Was the “low­ness” of the ver­bal score (ex­cel­lent for one his age, of course) due to his lack of mo­ti­va­tion on that part of the test and/​or sur­prise at its con­tent? A year later, while this al­to­gether charm­ing boy was spend­ing four days at my home dur­ing early May of 1985, I ad­ministered an­other form of the SAT-V to him un­der the best pos­si­ble con­di­tions. His score rose to 380, which is the 31st per­centile. That’s a fine gain, but the M vs. V dis­crep­ancy was prob­a­bly as great as be­fore. Quite likely, on the SAT score scale his abil­ity had risen ap­pre­cia­bly above the 800 ceiling of SAT-M.

It’s likely that prin­ci­pal com­po­nent anal­y­sis would re­veal that Tao’s rel­a­tively low ver­bal scores re­flect still lower abil­ity on some as­pect of ver­bal abil­ity, which he was able to com­pen­sate for with his ab­stract pat­tern recog­ni­tion abil­ity, just as my rel­a­tively low math SAT score re­flected still lower short-term mem­ory and pro­cess­ing speed, which I was able to com­pen­sate for in other ways.

Aside from ab­stract pat­tern recog­ni­tion abil­ity, ver­bal rea­son­ing abil­ity is an­other ma­jor com­po­nent of in­nate math­e­mat­i­cal abil­ity. It’s re­flected in perfor­mance on the analo­gies sub­tests of IQ, which like Raven’s Ma­tri­ces, are among the IQ sub­tests that cor­re­late most strongly with the g-fac­tor.

Broadly, the more the­o­ret­i­cal an area of math is, the greater the role of ver­bal rea­son­ing is in un­der­stand­ing it and do­ing re­search in it. As one would pre­dict based on his math /​ ver­bal skew­ing, Tao’s math­e­mat­i­cal re­search is in ar­eas of math that are rel­a­tively con­crete, as op­posed to the­o­ret­i­cal. Ver­bal rea­son­ing abil­ity is also closely con­nected with metacog­ni­tion: aware­ness and un­der­stand­ing of one’s own thoughts. Tao’s ap­par­ent lack of aware­ness of the role of his ex­cep­tional ab­stract rea­son­ing abil­ity in his math­e­mat­i­cal suc­cess may be at­tributable to rel­a­tively low metacog­ni­tion.

[Edit: Some com­menters found the above para­graph con­fus­ing. I should clar­ify that the stan­dard that I have in mind here is ex­tremely high — I’m com­par­ing Tao with peo­ple such as Henri Poin­care, whose es­says are amongst the most pen­e­trat­ing analy­ses of math­e­mat­i­cal psy­chol­ogy.]

My own in­cli­na­tion is very much in the ver­bal di­rec­tion, as may be ev­i­dent from my posts. I used to think that it was a solely a mat­ter of prefer­ence, but af­ter read­ing the IQ liter­a­ture, I re­al­ized that prob­a­bly the rea­son that I have the prefer­ence is be­cause ver­bal rea­son­ing is what I’m best at, and we tend to en­joy what we’re best at the most.

Charles Spear­man, the re­searcher who dis­cov­ered the g-fac­tor found that the more in­tel­lec­tu­ally gifted some­body is, the less cor­re­lated his or her cog­ni­tive abil­ities, and that when one takes this van­tage point, Tao’s math /​ ver­bal abil­ity differ­en­tial is not so un­usual. For fur­ther de­tail, see Cog­ni­tive pro­files of ver­bally and math­e­mat­i­cally pre­co­cious stu­dents by Ben­bow and Minor.

I’ll have more to say about the role of ver­bal rea­son­ing abil­ity in math later on.

Is this all de­press­ing?

Another rea­son that Tao may have missed the ev­i­dence that his math­e­mat­i­cal suc­cess can be in large part at­tributed to his ex­cep­tional ab­stract rea­son­ing abil­ity is that he might have an ugh field around the sub­ject. Terry might find it dis­con­cert­ing that the main rea­son that many of his col­leagues at UCLA are un­able to pro­duce work that’s non­triv­ial rel­a­tive to his own is that he was born with a bet­ter brain (in some sense) than the brains of his col­leagues were. Such a per­spec­tive can feel de­hu­man­iz­ing.

An anal­ogy that may be offer fur­ther in­sight. Like Tao, Natalie Port­man is tal­ented on many differ­ent di­men­sions. But had she been less phys­i­cally at­trac­tive than the av­er­age woman (ac­cord­ing to the group con­sen­sus), she would not have been able to be­come Academy Award win­ning ac­tress. Women of similar tal­ent prob­a­bly failed where she suc­ceeded sim­ply be­cause they were less at­trac­tive than she is. If asked about the role of her phys­i­cal ap­pear­ance in her suc­cess, she would prob­a­bly feel un­com­fortable. One can imag­ine her giv­ing an ac­cu­rate an­swer, but one can also imag­ine her try­ing to min­i­mize the sig­nifi­cance of her ap­pear­ance as much as pos­si­ble. It might re­mind her of how painfully un­fair life can be.

But whether or not we be­lieve in the ex­is­tence and im­por­tance of in­di­vi­d­ual differ­ences in in­tel­li­gence, they’re there: we can’t make them go away by ig­nor­ing them. Fur­ther­more, if not for peo­ple with un­usu­ally high in­tel­li­gence, there would have been no Re­nais­sance and no in­dus­trial rev­olu­tion: Europe would still be in the dark ages, as would the rest of the world. We’re very lucky to have peo­ple with cog­ni­tive abil­ities like Tao’s, and he would have no rea­son to feel guilty about hav­ing be­ing priv­ileged. He’s given back to the com­mu­nity through efforts such as his blog. Even if one doubts the value of the­o­ret­i­cal re­search, one can still ap­pre­ci­ate the fact that his blog serves as a proof of con­cept show­ing how elite sci­en­tists in all fields could bet­ter com­mu­ni­cate their think­ing to their re­search com­mu­ni­ties.

To be continued

I’ll have more to say about in­nate later abil­ity, but I’ve said enough to move on to a dis­cus­sion of the con­nec­tion be­tween in­nate abil­ity and math­e­mat­i­cal abil­ity more gen­er­ally, with a view to­ward how it’s pos­si­ble to im­prove one’s math­e­mat­i­cal abil­ity.

Since peo­ple’s pri­mary ex­po­sure to math is gen­er­ally through school, in my next post I’ll dis­cuss math ed­u­ca­tion as it’s cur­rently prac­ticed.

My ba­sic premise is that math ed­u­ca­tion as it’s cur­rently prac­ticed is ex­tremely in­effi­cient for rea­sons that I touched on ear­lier on: what goes on in math classes in prac­tice is of­ten very similar to study­ing for in­tel­li­gence tests. Stu­dents and teach­ers are effec­tively try­ing to build abil­ities that are in fact fixed, rather than fo­cus­ing on de­vel­op­ing abil­ities that can be im­proved, just as I would have been if I were to have worked on mak­ing fewer “care­less mis­takes” in el­e­men­tary school. Things don’t have to be this way – math ed­u­ca­tion could in prin­ci­ple be much more en­rich­ing.

More soon.