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.
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 1⁄10,000 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.
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:
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:
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:
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:
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:
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.