Thanks for the thanks, but I’ll also comment on iarwain1′s discipline getting this done. This journey began about 7 months ago, and through all the difficulties life has thrown, the progress has been consistent.
I’ll also comment that the grading system above is very difficult, and were I running this particular course, it would’ve been upwards 100%.
In any reasonable class, the score should have exceeded 100%.
Perhaps in any ordinary class at a North American institution. But I don’t think such grading schemes are reasonable—there’s more to reach for (and more humility, and much finer discrimination) when 80% is difficult to achieve.
Yeah, bearing in mind that the mapping from percentage scores to letter grades or the equivalent is pretty much arbitrary, I much prefer systems where some of the problems presented are really hard and the grade boundaries placed correspondingly lower. It allows for more ambition and more flexibility, and perhaps more importantly it’s just more interesting than a system where you get a perfect score if you don’t screw up each of twenty virtually identical basic exercises. I still have fond memories of a high-school physics class where I once earned an A on a test with a score of 57%. (The median was somewhere in the 30s.)
That presumes it’s real difficulty rather than busywork or pointless procedural stuff, though, which is harder to design and in some fields harder to grade: in mathematics you can grade only on the final answer (with partial credit if you e.g. obviously lost a sign somewhere), but that’s not true for something like e.g. physics lab notebooks.
One problem with this is that the amount of effort you can spend on a set of problems of this sort is nearly unbounded. If the problems are simple enough that a decent understanding of the subject leads you to get them all correct, you will only have to spend as much time as it takes to finish the assignment, and then you’re done. If a decent understanding of the subject only leads you to get 50% correct, then you’ll probably be in a position where you can spend another hour and raise that to 55%, and another hour for 57%, etc. You don’t know (until after the fact) how much you need to get correct to actually get a good grade, so you’re stuck not knowing how much effort is reasonable.
Furthermore, if it’s graded on a curve, this will result in a race to the bottom where everyone spends an extra two hours for that 7% advantage over everyone else and since everyone’s spent it, the overall effect is just that everyone spent an extra two hours for little benefit.
And woe be it if you have two such assignments at the same time. Not only do you have to worry about spending unlimited time because you don’t know when you’re done, it’s going to be very difficult to work on the assignments in order without shifting between one and the other constantly so you don’t spend all your effort on increasing one by 5% when that same effort could have increased the other one by 10%.
the overall effect is just that everyone spent an extra two hours for little benefit.
Woah! I sure hope not! The two or three times I had challenging assignments in school (my school encouraged undergraduates to take graduate classes if interested) they were tremendously valuable. If thinking about difficult problems and solving them has no marginal benefit, I can’t imagine what part of schooling does! (perhaps the diploma mill would be ideal in that scenario? I’m having a hard time simulating this hypothetical student).
It’s not that it has no marginal benefit, it’s that it has diminishing marginal benefit for the effort spent. At some point, you’re better off working on another class’s assignment instead. At some point you’re better off taking leisure time, or even sleeping. If even people with a good understanding are only expected to get 50% correct, how are you supposed to know when you’re better off going to sleep, only knowing that you’ve completed 50% and not knowing whether the extra 5% from sacrificing some sleep is worth it?
The returns diminish when it comes to impact on your grade, yes, and I certainly enjoyed transparency about how the grades I got would be impacted by my work.
The distribution of value for learning, though, goes up with difficulty until it drops to zero (the point at which you cannot solve the puzzle at all). My only point is that we should strongly prefer systems that allow us to soak up all that high-intensity high-value work—modern universities aren’t that for many students, though, but independently reading textbooks could/should be.
Yeah, you don’t want this sort of thing graded on a curve—though the “race to the bottom” issue isn’t substantially worse here than it would be with a more conventional problem set. Curves are generally set by the amount of effort the average student is willing to spend rather than the amount of effort an arbitrarily ambitious student is willing to, meaning that if you’re in a position to be making decisions about how to allocate your limited study time you probably don’t need to be doing so.
The teacher in my example didn’t grade on a curve, he assigned grade boundaries based on his idea of an acceptable level of effort and understanding. He also included estimates of each problem’s difficulty, which was helpful for time management.
As it happens, Art of Problem Solving questions actually fit the bill of being really difficult. Most chapters have about 15 or so “challenge” problems, of which 5 or so are really hard.
The scoring system I used was to (a) give myself double points for answering the really hard challenge problems, and (b) only require a total score of 10 for all the challenge problems combined. So if I got all the regular problems correct + 10 points on the challenge problems (1 point per “standard” challenge problem and 2 points per “hard” challenge problem) then I’d score 100. One side effect of this was that on chapters with a particularly large number of challenge problems it was possible to get something like a 140. But in any case, getting a 100 actually wasn’t that easy.
Well, yes, but in my judgement, OP has at least reached the point where marginal skill at algebra won’t translate into marginal gains in things OP’s interested in.
In particular, learning algebra is a means to an end, and that end is calculus. That requires being good enough to not be limited by sketchy knowledge of algebra when you get to calculus, not being good enough to ace the AMC10 (warning: nerd snipe). I’ve updated my comment to reflect that getting all the way to AMC10-acing levels is a poor use of time, and were I running the class, I wouldn’t encourage it. (On the other hand, if you’re under the age of 16 and have talent for math, then getting to AMC-acing levels of algebra is a fine use of your time. If you plan on attending university in America, I’m given to understand that the elite universities you’re going to want to gain admissions to take AMC scores because they have so many indistinguishable SAT-math scores. Even if not, getting that good at algebra may be a good use of your time if it means you can have high scores in math competitions to your name. If that’s not true, then your time is probably better spent reading, say, Spivak or something like 6.042J (for anyone interested, read the first 4 chapters of the 6.042J text, then Spivak, then the rest of 6.042J. And (this should go without saying), watch the lectures and do the problem sets. One does not simply get good at math without a ridiculous amount of practice.))
I would argue that math skills correlates with many facets of general intelligence, and that getting all the way up to “AMC10-acing levels” might not prepare you for calculus, but you’ll probably pick it up much faster due to your increased ability to think flexibly and pattern-match, which is really useful early on when first picking up on tricks for differentiation/integration, etc. Learning new material and genuinely learning to think are two different tasks, and I myself value the latter over the former, especially seeing as getting better at the latter makes it easier to do the former. At AMC/AIME levels of math, I would say raw intelligence is a lot more important than real math knowledge, and I value being intelligent extremely highly, so it was easy for me to see the allure of AMC/AIME. Naturally, if that’s not your thing and you just value math for the sake of real-world applications (which is an entirely defensible position), then yes, I agree that the AMC/AIME are not that useful. In terms of increasing fluid intelligence, though, I’d recommend it wholeheartedly.
(Note: The final statement in the above paragraph about increasing fluid intelligence is not supported in the academic literature as far as I’m aware, and is purely anecdotal advice to which the usual disclaimers must be applied, etc.)
I’ve had the same experience with studying math intensely feeling like it buffs fluid intelligence, and then I hear from many sources about how skill training is domain-specific and doesn’t transfer well (1), which would mean that’s impossible, and I’m stuck hoping that intensely studying math for several hours a day is something that’s fringe enough that it wouldn’t be picked up in research.
But, if the fluid-intelligence buff comes from studying consistently and at the edge of your skill level, then going through the two AoPS algebra books, but not getting to be able to solve 100% of their “challenge” problems, is going to do just as much as getting through one of them and getting to the point of being able to solve all the challenge problems. There’s no real shortage of math-at-skill-level to study, but not knowing calculus is absolutely crippling; you can’t learn physics, chemistry, economics, programming, or really any higher math, because at a certain point, every just kind of assumes you understand calculus.
(1) According to my memory textbook, two things that enhance transfer-of-learning are learning things deeply and learning them concretely in one context or abstractly and then applying it to a wide variety of contexts. Going through math textbooks, proving every theorem before reading the book’s proof, and then applying the theorems to a wide range of context seems to fit both bills, so take +1 plausibility. However, this is one of the less well-supported areas, and the book didn’t point to any original studies that I could find (2), so grain of salt.
(2) Which isn’t to say the research hasn’t been done. Just, the “do things in one context or abstractly and then in multiple contexts” was done by a grad student and then published in a book by his adviser, and the “learn things deeply” was presented as a generic experiment without reference to primary literature.
I agree that not learning calculus is absolutely crippling. If the OP has not yet learned calculus, then I would say learning calculus should take precedence. On the other hand, if the OP has already learned calculus, then depending on the circumstances, studying an AoPS algebra book might not be a bad use of his/her time. (Or really just any AoPS book, really; from my experience with AoPS, they tend to write very good math books—or at least very good practice problems.)
Unfortunately, I have very little to say on the topic of increasing fluid intelligence. Outside of some basic introductory texts, my experience in the fields of cognitive science is close to nil, so anything I say about this can and probably should be taken with a generous helping of salt. I will point out that in my experience, getting better at math seems to have helped my general reasoning skills in ways largely unrelated to math; however, seeing as I would strongly like this to be true, it’s possible that I am allowing myself to notice a trend where there is none. Additionally, it seems plausible to me that learning new math might not help intelligence due to domain-specificity, but that using already-learned math in extremely difficult situations might make a difference because in the latter case, the actual skill domain itself is less important than the mental gymnastics—and the increased ability to do those gymnastics might be transferable. Going through every theorem in the book and proving it, then, would probably fall into the latter category rather than the former.
So with all of that said, the question is: when you say you “[study] math intensely”, do you mean covering new material, or doing inventive things with old material?
Going through every theorem in the book and proving it
This is how I study math. A result I’ve come across a few times is that trying a problem before learning how to do it, even if you fail, will result in you learning how to solve the problem better (this study (pdf), for instance). AoPS pedagogy reflects this; they encourage the reader to try each problem before reading how to solve it, even if they don’t succeed. (I’ve been doing this long enough that I forget that this is a minority approach rather than common sense. Reminds my of this paper. tl;dr, most students study by rereading the book, even though this is completely ineffective, and very few practice retrieval, which is wildly effective.)
In this way, I don’t see the dichotomy between learning new math and using already-learned math in new situations. Going off stuff I read here (written for popular audience, grain of salt, although it’s used as a text for an upper-level psych course at Harvard), pushing your mind to its limit learning new math is going to increase your ability to learn new math and pushing your brain to its limit using known math in novel situations will increase your ability to use known math in novel situations (which feels kinda, sorta like buffing fluid intelligence)—but there’s no reason you can’t learn math by using it in novel situations because that’s exactly what I do when learning math. So, when I talk about “[studying] math intensely”, I’m referring to doing inventive things with old material to learn new material.
One more thing: there’s no widely-accepted method for increasing fluid intelligence that I’m aware of. This is all based on fairly new research and personal experience. Fortunately, between the age distribution of the US population and how many diseases we can treat or cure, there’s a lot of money going into research neuroscience because Alzheimer’s, so hopefully this is changing quickly relative to the counterfactual.
use known math in novel situations (which feels kinda, sorta like buffing fluid intelligence)
If we temporarily disregard the academic literature here (partly due to my unfamiliarity with it and partly due to there not being much on fluid intelligence increase in the first place) and just go by personal experience, do you feel that there are any differences between “[using] known math in novel situations” and “buffing fluid intelligence” that may be of significance?
From personal experience, “using known math in (appropriately difficult) novel situations” implies “fluid intelligence buff”, but “not using known math in novel situations” doesn’t imply “no fluid intelligence buff.” For instance, I think I could get the same result replacing math with certain parts of CS.
This seems plausible to me. Note that I consider trying to do things like proving theorems to be “novel situations”, whereas simply learning what the theorem says doesn’t feel like it would be as effective at increasing fluid intelligence, if at all. Since CS (I assume you mean programming here, since I consider theoretical CS part of mathematics) often requires you to put your thinking cap on, so to speak, I can very easily see that buffing fluid intelligence as well. Which, of course, requires that fluid intelligence increase be possible in the first place. As you said, it certainly feels like doing math/CS buffs fluid intelligence, but...
According to my notes from the above Doidge book, things we can change include “memory”, “processing speed”, “intelligence”, and “ability to learn.”
While I’m uncertain whether the rest are well-accepted, being able to improve memory is well-accepted among cognitive psychologists. They trained this one subject to a digit span of 80. Also, one feature of mnemonic techniques is “speed up”, or the ability to improve with practice, hence memory competitions. Method of loci is just ridiculously powerful (for the narrow and typically unhelpful types of information that can be method-of-loci’d).
Memory certainly is improvable from what I’ve heard/read, and it doesn’t seem too much of a stretch to accept that processing speed might be improvable as well. That being said, I’m a bit more skeptical of improving intelligence (maybe studying math does it, and maybe not), and I’m not quite sure what “ability to learn” entails—if we’re using the phrase conventionally, I’d assume that “ability to learn” would simply be a combination of intelligence and memory—the former for understanding something in the first place, and the latter for retaining that understanding. If there’s something about “ability to learn” outside of this, I’d be interested to hear about it.
In order of how much each aspect matters to me, personally, I’d probably rank intelligence as most important, followed by memory, then processing speed. (I’m not confident enough in my current understanding of “ability to learn” to attempt to rank it yet—see the above paragraph.) If you asked me why I care so much about intelligence… I honestly don’t know, actually. To paraphrase HJPEV, it’s such a fundamental theorem in my values system that it’s hard to go about describing the actual proof steps. (Actually, to continue the analogy, this seems more like an axiom than a theorem, which is weird because I don’t remember ever consciously deciding to terminally value intelligence.) This is a somewhat concerning realization for me, and suggests that I should perhaps consider my reasons for caring about stuff a lot more closely than I’ve been doing as of late.
Of course, this is very unfortunate for me, seeing as “intelligence” has always been considered fuzzy and hard to quantify… maybe that’s why there’s not a lot of academic material out there on it? I’m really good at math for a normal person, but a fair number of the more “mathy” discussions here on LW somehow manage to completely lose me (think “decision theory”, and so forth), so I’m thinking that maybe “really good” is a bit of a premature designation. I probably wouldn’t be the only one here in saying that LW is the first online discussion board in which I’ve felt completely outclassed intellectually—I’m generally used to be the smartest person in the proverbial “room”, so being here is a simultaneously humbling and exciting experience for me.
Back onto the topic of whether math improves fluid intelligence: it has been my experience that people who are really good with math also tend to be really smart, but 1. that could easily be a selection effect, and 2. correlation does not imply causation. At this point, I’m basically just left hoping that studying math does have appreciable effects on fluid intelligence, but given the dearth of research on the subject, I have very little basis to say anything one way or another, which is rather frustrating for me personally. Alas...
(And now, reading through this again, I see I somehow managed to turn a discussion on improving mental capability into a dissertation on my own perceived problems… yeah, I think I should probably stop typing now.)
Regarding “ability to learn”: Ebbinghaus (famous for discovering forgetting curves which lead to Anki) showed that amount you learn is proportional to the amount of time you spend learning it; L(t) = at . Increasing “ability to learn” could be thought of like the constant of proportionality, a.
Obviously, there’s values of time, t, for which this breaks down. I don’t have the academic citations, but a professor I once had (who I trust) said that your brain’s ability to learn can be saturated and marginal time spent learning won’t help until you sleep.
However, I am aware of Walker et al. (2002) (pdf), which shows that you don’t show improvements from practice until you sleep. In hierarchical situations, where every new thing you learn depends on something you’ve already learned, this implies you should sleep in between every new thing you learn. This effect is separate from distributed practice, which you should do anyway.
I agree that these things are also important, but I’m not sure they should be classified as “basic” traits the way memory, processing speed, and intelligence are. Then again, I could be mistaken.
If you plan on attending university in America, I’m given to understand that the elite universities you’re going to want to gain admissions to take AMC scores because they have so many indistinguishable SAT-math scores.
No, just take the Advanced Placement calculus classes.
Based on the commentary I’ve heard from grad students, getting truly good at math is on par with bringing the ring to Mt. Doom in terms of difficulty, if not danger.
Oh, I at least somewhat disagree with that. The form of mathematics is extremely difficult and even somewhat obscurantist. The content is, while not easy, no harder than any of the other difficult intellectual skills we regularly teach to intelligent people, and gets easier with practice just like everything else.
No, the miserable thing is when someone starts out into “Let there be a Blah from the set Herp, then define its Bibbity to be the Derp such that De Hurr...” when what they really mean is, “I will prove a theorem quantified over all Herp’s by pretending I have an arbitrary Herp called Blah, and then using Blah to build Bibbity, which is a Derp, and due to the way I built it, it has this property De Hurr that I wanted.” While all that English is obviously quite verbose for everyday work, computer theorem-proving languages are usually about as terse as “real” mathematics while being much clearer and easier to understand, especially because in programming, giving things descriptive names is considered good form, whereas in mathematics the custom is to use single letters, ideally Greek ones or in funny fonts, for absolutely everything, the better to prove the theorem without ever explaining what it means or why you built it that way.
If we bothered to treat mathematics as a form of communication (specifically: communication of rigidly defined, highly specific computational structures) rather than as a personal exploration of Platonic Higher Realms, we could definitely all get much better at it.
(Why yes I am a massive fanboy of Bob Harper… Why did you ask?)
Thanks for the thanks, but I’ll also comment on iarwain1′s discipline getting this done. This journey began about 7 months ago, and through all the difficulties life has thrown, the progress has been consistent.
I’ll also comment that the grading system above is very difficult, and were I running this particular course, it would’ve been upwards 100%.
Perhaps in any ordinary class at a North American institution. But I don’t think such grading schemes are reasonable—there’s more to reach for (and more humility, and much finer discrimination) when 80% is difficult to achieve.
Yeah, bearing in mind that the mapping from percentage scores to letter grades or the equivalent is pretty much arbitrary, I much prefer systems where some of the problems presented are really hard and the grade boundaries placed correspondingly lower. It allows for more ambition and more flexibility, and perhaps more importantly it’s just more interesting than a system where you get a perfect score if you don’t screw up each of twenty virtually identical basic exercises. I still have fond memories of a high-school physics class where I once earned an A on a test with a score of 57%. (The median was somewhere in the 30s.)
That presumes it’s real difficulty rather than busywork or pointless procedural stuff, though, which is harder to design and in some fields harder to grade: in mathematics you can grade only on the final answer (with partial credit if you e.g. obviously lost a sign somewhere), but that’s not true for something like e.g. physics lab notebooks.
One problem with this is that the amount of effort you can spend on a set of problems of this sort is nearly unbounded. If the problems are simple enough that a decent understanding of the subject leads you to get them all correct, you will only have to spend as much time as it takes to finish the assignment, and then you’re done. If a decent understanding of the subject only leads you to get 50% correct, then you’ll probably be in a position where you can spend another hour and raise that to 55%, and another hour for 57%, etc. You don’t know (until after the fact) how much you need to get correct to actually get a good grade, so you’re stuck not knowing how much effort is reasonable.
Furthermore, if it’s graded on a curve, this will result in a race to the bottom where everyone spends an extra two hours for that 7% advantage over everyone else and since everyone’s spent it, the overall effect is just that everyone spent an extra two hours for little benefit.
And woe be it if you have two such assignments at the same time. Not only do you have to worry about spending unlimited time because you don’t know when you’re done, it’s going to be very difficult to work on the assignments in order without shifting between one and the other constantly so you don’t spend all your effort on increasing one by 5% when that same effort could have increased the other one by 10%.
Woah! I sure hope not! The two or three times I had challenging assignments in school (my school encouraged undergraduates to take graduate classes if interested) they were tremendously valuable. If thinking about difficult problems and solving them has no marginal benefit, I can’t imagine what part of schooling does! (perhaps the diploma mill would be ideal in that scenario? I’m having a hard time simulating this hypothetical student).
It’s not that it has no marginal benefit, it’s that it has diminishing marginal benefit for the effort spent. At some point, you’re better off working on another class’s assignment instead. At some point you’re better off taking leisure time, or even sleeping. If even people with a good understanding are only expected to get 50% correct, how are you supposed to know when you’re better off going to sleep, only knowing that you’ve completed 50% and not knowing whether the extra 5% from sacrificing some sleep is worth it?
The returns diminish when it comes to impact on your grade, yes, and I certainly enjoyed transparency about how the grades I got would be impacted by my work.
The distribution of value for learning, though, goes up with difficulty until it drops to zero (the point at which you cannot solve the puzzle at all). My only point is that we should strongly prefer systems that allow us to soak up all that high-intensity high-value work—modern universities aren’t that for many students, though, but independently reading textbooks could/should be.
Yeah, you don’t want this sort of thing graded on a curve—though the “race to the bottom” issue isn’t substantially worse here than it would be with a more conventional problem set. Curves are generally set by the amount of effort the average student is willing to spend rather than the amount of effort an arbitrarily ambitious student is willing to, meaning that if you’re in a position to be making decisions about how to allocate your limited study time you probably don’t need to be doing so.
The teacher in my example didn’t grade on a curve, he assigned grade boundaries based on his idea of an acceptable level of effort and understanding. He also included estimates of each problem’s difficulty, which was helpful for time management.
As it happens, Art of Problem Solving questions actually fit the bill of being really difficult. Most chapters have about 15 or so “challenge” problems, of which 5 or so are really hard.
The scoring system I used was to (a) give myself double points for answering the really hard challenge problems, and (b) only require a total score of 10 for all the challenge problems combined. So if I got all the regular problems correct + 10 points on the challenge problems (1 point per “standard” challenge problem and 2 points per “hard” challenge problem) then I’d score 100. One side effect of this was that on chapters with a particularly large number of challenge problems it was possible to get something like a 140. But in any case, getting a 100 actually wasn’t that easy.
Well, yes, but in my judgement, OP has at least reached the point where marginal skill at algebra won’t translate into marginal gains in things OP’s interested in.
In particular, learning algebra is a means to an end, and that end is calculus. That requires being good enough to not be limited by sketchy knowledge of algebra when you get to calculus, not being good enough to ace the AMC10 (warning: nerd snipe). I’ve updated my comment to reflect that getting all the way to AMC10-acing levels is a poor use of time, and were I running the class, I wouldn’t encourage it. (On the other hand, if you’re under the age of 16 and have talent for math, then getting to AMC-acing levels of algebra is a fine use of your time. If you plan on attending university in America, I’m given to understand that the elite universities you’re going to want to gain admissions to take AMC scores because they have so many indistinguishable SAT-math scores. Even if not, getting that good at algebra may be a good use of your time if it means you can have high scores in math competitions to your name. If that’s not true, then your time is probably better spent reading, say, Spivak or something like 6.042J (for anyone interested, read the first 4 chapters of the 6.042J text, then Spivak, then the rest of 6.042J. And (this should go without saying), watch the lectures and do the problem sets. One does not simply get good at math without a ridiculous amount of practice.))
I would argue that math skills correlates with many facets of general intelligence, and that getting all the way up to “AMC10-acing levels” might not prepare you for calculus, but you’ll probably pick it up much faster due to your increased ability to think flexibly and pattern-match, which is really useful early on when first picking up on tricks for differentiation/integration, etc. Learning new material and genuinely learning to think are two different tasks, and I myself value the latter over the former, especially seeing as getting better at the latter makes it easier to do the former. At AMC/AIME levels of math, I would say raw intelligence is a lot more important than real math knowledge, and I value being intelligent extremely highly, so it was easy for me to see the allure of AMC/AIME. Naturally, if that’s not your thing and you just value math for the sake of real-world applications (which is an entirely defensible position), then yes, I agree that the AMC/AIME are not that useful. In terms of increasing fluid intelligence, though, I’d recommend it wholeheartedly.
(Note: The final statement in the above paragraph about increasing fluid intelligence is not supported in the academic literature as far as I’m aware, and is purely anecdotal advice to which the usual disclaimers must be applied, etc.)
I’ve had the same experience with studying math intensely feeling like it buffs fluid intelligence, and then I hear from many sources about how skill training is domain-specific and doesn’t transfer well (1), which would mean that’s impossible, and I’m stuck hoping that intensely studying math for several hours a day is something that’s fringe enough that it wouldn’t be picked up in research.
But, if the fluid-intelligence buff comes from studying consistently and at the edge of your skill level, then going through the two AoPS algebra books, but not getting to be able to solve 100% of their “challenge” problems, is going to do just as much as getting through one of them and getting to the point of being able to solve all the challenge problems. There’s no real shortage of math-at-skill-level to study, but not knowing calculus is absolutely crippling; you can’t learn physics, chemistry, economics, programming, or really any higher math, because at a certain point, every just kind of assumes you understand calculus.
(1) According to my memory textbook, two things that enhance transfer-of-learning are learning things deeply and learning them concretely in one context or abstractly and then applying it to a wide variety of contexts. Going through math textbooks, proving every theorem before reading the book’s proof, and then applying the theorems to a wide range of context seems to fit both bills, so take +1 plausibility. However, this is one of the less well-supported areas, and the book didn’t point to any original studies that I could find (2), so grain of salt.
(2) Which isn’t to say the research hasn’t been done. Just, the “do things in one context or abstractly and then in multiple contexts” was done by a grad student and then published in a book by his adviser, and the “learn things deeply” was presented as a generic experiment without reference to primary literature.
I agree that not learning calculus is absolutely crippling. If the OP has not yet learned calculus, then I would say learning calculus should take precedence. On the other hand, if the OP has already learned calculus, then depending on the circumstances, studying an AoPS algebra book might not be a bad use of his/her time. (Or really just any AoPS book, really; from my experience with AoPS, they tend to write very good math books—or at least very good practice problems.)
Unfortunately, I have very little to say on the topic of increasing fluid intelligence. Outside of some basic introductory texts, my experience in the fields of cognitive science is close to nil, so anything I say about this can and probably should be taken with a generous helping of salt. I will point out that in my experience, getting better at math seems to have helped my general reasoning skills in ways largely unrelated to math; however, seeing as I would strongly like this to be true, it’s possible that I am allowing myself to notice a trend where there is none. Additionally, it seems plausible to me that learning new math might not help intelligence due to domain-specificity, but that using already-learned math in extremely difficult situations might make a difference because in the latter case, the actual skill domain itself is less important than the mental gymnastics—and the increased ability to do those gymnastics might be transferable. Going through every theorem in the book and proving it, then, would probably fall into the latter category rather than the former.
So with all of that said, the question is: when you say you “[study] math intensely”, do you mean covering new material, or doing inventive things with old material?
This is how I study math. A result I’ve come across a few times is that trying a problem before learning how to do it, even if you fail, will result in you learning how to solve the problem better (this study (pdf), for instance). AoPS pedagogy reflects this; they encourage the reader to try each problem before reading how to solve it, even if they don’t succeed. (I’ve been doing this long enough that I forget that this is a minority approach rather than common sense. Reminds my of this paper. tl;dr, most students study by rereading the book, even though this is completely ineffective, and very few practice retrieval, which is wildly effective.)
In this way, I don’t see the dichotomy between learning new math and using already-learned math in new situations. Going off stuff I read here (written for popular audience, grain of salt, although it’s used as a text for an upper-level psych course at Harvard), pushing your mind to its limit learning new math is going to increase your ability to learn new math and pushing your brain to its limit using known math in novel situations will increase your ability to use known math in novel situations (which feels kinda, sorta like buffing fluid intelligence)—but there’s no reason you can’t learn math by using it in novel situations because that’s exactly what I do when learning math. So, when I talk about “[studying] math intensely”, I’m referring to doing inventive things with old material to learn new material.
One more thing: there’s no widely-accepted method for increasing fluid intelligence that I’m aware of. This is all based on fairly new research and personal experience. Fortunately, between the age distribution of the US population and how many diseases we can treat or cure, there’s a lot of money going into research neuroscience because Alzheimer’s, so hopefully this is changing quickly relative to the counterfactual.
One more thing: OP doesn’t know calculus.
If we temporarily disregard the academic literature here (partly due to my unfamiliarity with it and partly due to there not being much on fluid intelligence increase in the first place) and just go by personal experience, do you feel that there are any differences between “[using] known math in novel situations” and “buffing fluid intelligence” that may be of significance?
From personal experience, “using known math in (appropriately difficult) novel situations” implies “fluid intelligence buff”, but “not using known math in novel situations” doesn’t imply “no fluid intelligence buff.” For instance, I think I could get the same result replacing math with certain parts of CS.
This seems plausible to me. Note that I consider trying to do things like proving theorems to be “novel situations”, whereas simply learning what the theorem says doesn’t feel like it would be as effective at increasing fluid intelligence, if at all. Since CS (I assume you mean programming here, since I consider theoretical CS part of mathematics) often requires you to put your thinking cap on, so to speak, I can very easily see that buffing fluid intelligence as well. Which, of course, requires that fluid intelligence increase be possible in the first place. As you said, it certainly feels like doing math/CS buffs fluid intelligence, but...
According to my notes from the above Doidge book, things we can change include “memory”, “processing speed”, “intelligence”, and “ability to learn.”
While I’m uncertain whether the rest are well-accepted, being able to improve memory is well-accepted among cognitive psychologists. They trained this one subject to a digit span of 80. Also, one feature of mnemonic techniques is “speed up”, or the ability to improve with practice, hence memory competitions. Method of loci is just ridiculously powerful (for the narrow and typically unhelpful types of information that can be method-of-loci’d).
Memory certainly is improvable from what I’ve heard/read, and it doesn’t seem too much of a stretch to accept that processing speed might be improvable as well. That being said, I’m a bit more skeptical of improving intelligence (maybe studying math does it, and maybe not), and I’m not quite sure what “ability to learn” entails—if we’re using the phrase conventionally, I’d assume that “ability to learn” would simply be a combination of intelligence and memory—the former for understanding something in the first place, and the latter for retaining that understanding. If there’s something about “ability to learn” outside of this, I’d be interested to hear about it.
In order of how much each aspect matters to me, personally, I’d probably rank intelligence as most important, followed by memory, then processing speed. (I’m not confident enough in my current understanding of “ability to learn” to attempt to rank it yet—see the above paragraph.) If you asked me why I care so much about intelligence… I honestly don’t know, actually. To paraphrase HJPEV, it’s such a fundamental theorem in my values system that it’s hard to go about describing the actual proof steps. (Actually, to continue the analogy, this seems more like an axiom than a theorem, which is weird because I don’t remember ever consciously deciding to terminally value intelligence.) This is a somewhat concerning realization for me, and suggests that I should perhaps consider my reasons for caring about stuff a lot more closely than I’ve been doing as of late.
Of course, this is very unfortunate for me, seeing as “intelligence” has always been considered fuzzy and hard to quantify… maybe that’s why there’s not a lot of academic material out there on it? I’m really good at math for a normal person, but a fair number of the more “mathy” discussions here on LW somehow manage to completely lose me (think “decision theory”, and so forth), so I’m thinking that maybe “really good” is a bit of a premature designation. I probably wouldn’t be the only one here in saying that LW is the first online discussion board in which I’ve felt completely outclassed intellectually—I’m generally used to be the smartest person in the proverbial “room”, so being here is a simultaneously humbling and exciting experience for me.
Back onto the topic of whether math improves fluid intelligence: it has been my experience that people who are really good with math also tend to be really smart, but 1. that could easily be a selection effect, and 2. correlation does not imply causation. At this point, I’m basically just left hoping that studying math does have appreciable effects on fluid intelligence, but given the dearth of research on the subject, I have very little basis to say anything one way or another, which is rather frustrating for me personally. Alas...
(And now, reading through this again, I see I somehow managed to turn a discussion on improving mental capability into a dissertation on my own perceived problems… yeah, I think I should probably stop typing now.)
I found the Doidge book therapeutic.
Regarding “ability to learn”: Ebbinghaus (famous for discovering forgetting curves which lead to Anki) showed that amount you learn is proportional to the amount of time you spend learning it; L(t) = at . Increasing “ability to learn” could be thought of like the constant of proportionality, a.
Learning = acquiring and retaining knowledge or skills. This definition comes from a book written by cognitive scientists coming off a decade of researching optimal learning recommended by Robin Hanson, which is also worth reading.
Obviously, there’s values of time, t, for which this breaks down. I don’t have the academic citations, but a professor I once had (who I trust) said that your brain’s ability to learn can be saturated and marginal time spent learning won’t help until you sleep.
However, I am aware of Walker et al. (2002) (pdf), which shows that you don’t show improvements from practice until you sleep. In hierarchical situations, where every new thing you learn depends on something you’ve already learned, this implies you should sleep in between every new thing you learn. This effect is separate from distributed practice, which you should do anyway.
Skills, techniques and habits are also rather important.
I agree that these things are also important, but I’m not sure they should be classified as “basic” traits the way memory, processing speed, and intelligence are. Then again, I could be mistaken.
No, just take the Advanced Placement calculus classes.
So, it’s like walking into Mordor, then? X-D
Based on the commentary I’ve heard from grad students, getting truly good at math is on par with bringing the ring to Mt. Doom in terms of difficulty, if not danger.
Oh, I at least somewhat disagree with that. The form of mathematics is extremely difficult and even somewhat obscurantist. The content is, while not easy, no harder than any of the other difficult intellectual skills we regularly teach to intelligent people, and gets easier with practice just like everything else.
No, the miserable thing is when someone starts out into “Let there be a Blah from the set Herp, then define its Bibbity to be the Derp such that De Hurr...” when what they really mean is, “I will prove a theorem quantified over all Herp’s by pretending I have an arbitrary Herp called Blah, and then using Blah to build Bibbity, which is a Derp, and due to the way I built it, it has this property De Hurr that I wanted.” While all that English is obviously quite verbose for everyday work, computer theorem-proving languages are usually about as terse as “real” mathematics while being much clearer and easier to understand, especially because in programming, giving things descriptive names is considered good form, whereas in mathematics the custom is to use single letters, ideally Greek ones or in funny fonts, for absolutely everything, the better to prove the theorem without ever explaining what it means or why you built it that way.
If we bothered to treat mathematics as a form of communication (specifically: communication of rigidly defined, highly specific computational structures) rather than as a personal exploration of Platonic Higher Realms, we could definitely all get much better at it.
(Why yes I am a massive fanboy of Bob Harper… Why did you ask?)
And you get to keep the ring afterwards! :-)