To get a sense of what I mean by “stereotypically good at math”, think about the abilities involved in solving tricky puzzles or competition-style problems. Or, consider the comments section of Eliezer’s Drawing Two Aces post, full of people who got the right answer (I didn’t, which resulted in this post). The idea isn’t exactly well-defined, but seems to involve some combination of powerful short-term memory and an ability to quickly identify the particular abstraction that the poser of a concrete problem is attempting to refer to.
The implication of the contrast between Ahab and Billy, on my account, isn’t what you perhaps think. I don’t necessarily deny that some kind of “native” difference could be responsible for Billy’s greater difficulties relative to Ahab. The fact that Billy manages to get an “A”, however, means that anyone with Billy’s level of “native math ability” can’t invoke that to explain why they didn’t get an “A”. Billy may have other native abilities that such an individual may lack, but they won’t be specifically math-related, and instead will be general things like “the ability to overcome akrasia”, etc.
Notwithstanding the above, “lack of native math ability” is still a fake explanation. Whatever “math ability” is, it is reducible. I want to know in detail what goes through Billy’s mind as he attempts to solve an algebra problem, and how it differs from what goes on in Ahab’s mind. Once we know this, we can try to determine what causes this difference: is Billy’s IQ just lower than Ahab’s (which would be a general problem, not a math-specific one), does he lack certain pieces of information that Ahab has (easily fixable), or is he executing particular cognitive habits that prevent him from processing the same information as efficiently as Ahab (fixable via training)?
Teasing this out a little more, bullet points of my own:
If B learns math at a slower pace than A, then it can literally be the case that B will never understand math as well as A. At suitable slow (but common) learning paces, it can be impractical and unrewarding for B to study math. And I think there might even be large numbers of mentally normal human beings walking around for whom this pace is so slow that it’s misleading to call it a “learning pace” at all, e.g. too slow for progress they make one day to stick the next.
I’m sure that “math ability,” like anything else, is reducible, but in these kinds of brain-and-behavior cases it might “reduce” to thousands of different factors that don’t have much to do with each other. In that case it wouldn’t be very easy to give advice on how to be better at math, beyond “arrange each of those thousands of factors in a way favorable to math ability.”
Even if “math ability” has a more satisfying explanation than the kind in 2., so that it’s possible to give good advice on how to improve it, I think that this is not a solved problem. Specifically I still think that your proposed advice (“you simply haven’t bothered to update some aspect of your identity since childhood”) is no good.
In the meantime “lack of math ability” seems to me to be a perfectly good label for a real phenomenon, though I guess I agree with you that it is not an explanation for that phenomenon.
1. I disagree that a slower learning pace is less rewarding. On the contrary, learning is most rewarding when there is time to do it properly, and the frustration many people experience in school settings results from the pressure to (appear-to-) learn things more quickly than their natural pace.
(I owe to Michael Vassar the observation that there is something inherently contradictory and unrealistic about expecting people to learn calculus in a semester when they required five years to learn arithmetic.)
2. It might seem like it could be that complicated, but it turns out not to be. In practice (as revealed by teaching experience), “lacking math ability” usually reduces to something like “I flinch and run away when I realize that I will have to carry out more than two or three steps (especially if there is recursion involved), instead of just gritting my teeth and carrying them out.”
3. Most people don’t even try updating their identities; while on the other hand, I myself have updated my skill-related identities on a number of occasions, and it worked. (Math happens to be an example.)
4. It would be best to have a label that conveys more information about the cause(s) of the phenomenon.
Several points to make in reply:
To get a sense of what I mean by “stereotypically good at math”, think about the abilities involved in solving tricky puzzles or competition-style problems. Or, consider the comments section of Eliezer’s Drawing Two Aces post, full of people who got the right answer (I didn’t, which resulted in this post). The idea isn’t exactly well-defined, but seems to involve some combination of powerful short-term memory and an ability to quickly identify the particular abstraction that the poser of a concrete problem is attempting to refer to.
The implication of the contrast between Ahab and Billy, on my account, isn’t what you perhaps think. I don’t necessarily deny that some kind of “native” difference could be responsible for Billy’s greater difficulties relative to Ahab. The fact that Billy manages to get an “A”, however, means that anyone with Billy’s level of “native math ability” can’t invoke that to explain why they didn’t get an “A”. Billy may have other native abilities that such an individual may lack, but they won’t be specifically math-related, and instead will be general things like “the ability to overcome akrasia”, etc.
Notwithstanding the above, “lack of native math ability” is still a fake explanation. Whatever “math ability” is, it is reducible. I want to know in detail what goes through Billy’s mind as he attempts to solve an algebra problem, and how it differs from what goes on in Ahab’s mind. Once we know this, we can try to determine what causes this difference: is Billy’s IQ just lower than Ahab’s (which would be a general problem, not a math-specific one), does he lack certain pieces of information that Ahab has (easily fixable), or is he executing particular cognitive habits that prevent him from processing the same information as efficiently as Ahab (fixable via training)?
Teasing this out a little more, bullet points of my own:
If B learns math at a slower pace than A, then it can literally be the case that B will never understand math as well as A. At suitable slow (but common) learning paces, it can be impractical and unrewarding for B to study math. And I think there might even be large numbers of mentally normal human beings walking around for whom this pace is so slow that it’s misleading to call it a “learning pace” at all, e.g. too slow for progress they make one day to stick the next.
I’m sure that “math ability,” like anything else, is reducible, but in these kinds of brain-and-behavior cases it might “reduce” to thousands of different factors that don’t have much to do with each other. In that case it wouldn’t be very easy to give advice on how to be better at math, beyond “arrange each of those thousands of factors in a way favorable to math ability.”
Even if “math ability” has a more satisfying explanation than the kind in 2., so that it’s possible to give good advice on how to improve it, I think that this is not a solved problem. Specifically I still think that your proposed advice (“you simply haven’t bothered to update some aspect of your identity since childhood”) is no good.
In the meantime “lack of math ability” seems to me to be a perfectly good label for a real phenomenon, though I guess I agree with you that it is not an explanation for that phenomenon.
1. I disagree that a slower learning pace is less rewarding. On the contrary, learning is most rewarding when there is time to do it properly, and the frustration many people experience in school settings results from the pressure to (appear-to-) learn things more quickly than their natural pace.
(I owe to Michael Vassar the observation that there is something inherently contradictory and unrealistic about expecting people to learn calculus in a semester when they required five years to learn arithmetic.)
2. It might seem like it could be that complicated, but it turns out not to be. In practice (as revealed by teaching experience), “lacking math ability” usually reduces to something like “I flinch and run away when I realize that I will have to carry out more than two or three steps (especially if there is recursion involved), instead of just gritting my teeth and carrying them out.”
3. Most people don’t even try updating their identities; while on the other hand, I myself have updated my skill-related identities on a number of occasions, and it worked. (Math happens to be an example.)
4. It would be best to have a label that conveys more information about the cause(s) of the phenomenon.
.