I say everything I’m about to say as a person who is more certain than not that you have something valuable to contribute through this sequence, and who eagerly awaits more.
All of your posts in this sequence have purportedly been written to motivate your main thesis, but it’s not clear to me what that is. I think you should stop motivating and very clearly reveal your Big Secret. What can I do right now to improve my mathematical ability? That’s what I want to know.
Consider these points:
Eliezer tried to explain his metaethics the first time and failed, but it was okay because he was able to write his epistemology sequence after that and clarify. His posts pretty much always motivate people to continue reading because they usually have such high insight density and people know that they do; but your posts in this sequence, as far as I can tell, have just been anecdotes, quotes, nonstandard definitions, references detailing your nonstandard definitions, and promises of elucidation in future posts, and in your case, there isn’t common knowledge of high insight density to make people trust you even when they don’t understand where you’re going. Your second post had less karma than your first post, and I predict that this post will have even less. I think that people are getting antsy. If no one understands your Big Secret and it turns out that all of this motivation really was necessary, then you can just do what Eliezer did and try again.
It seems to me that you’re getting strawmanned a lot. I want to argue against those things but I don’t have a real man to point to, and if you wrote a clear thesis statement, then I would. Maybe someone would say that I’m defending a side rather than seeking the truth if I want to argue against another position when I have no clear position, but I think that amounts to claiming that your predicate’s extension is empty rather than that it’s vague. There are things that people have claimed about your posts that I can’t consider an accurate statement of your (vague) position no matter how charitable my interpretation. The most obvious is that you’re claiming that innate ability is irrelevant; you’ve explicitly claimed the opposite. Other possibilities in order of increasing plausibility include:
(Almost) anyone can be a famous mathematician.
(Almost) anyone can do mathematical research.
(Almost) anyone can be what an average person considers ‘good at math.’
Current mathematical pedagogy sucks.
You’ve explicitly stated the fourth item, and it’s the one I’m most sympathetic to, and it would be useful to me and many others if this were elucidated regardless of whether or not it means I can be a famous mathematician. If you’re claiming more than one of the above, then maybe you could just refine your points about that particular point into public form, and substantiate your more radical assertions afterwards. I really want to see what you have to say about pedagogical techniques more than anything else. I want to be better at math. I’m open to the possibility that I’m misinterpreting everyone’s attitudes about this and I’m the only one who wants to know what you have to say about that more than anything else.
I also think it’s weird that I see a lot of people throwing around ‘good at math’ and ‘bad at math’ as if those terms mean the same thing to everyone. Some people mean Calculus III, some people mean Fields Medal or thereabouts, and some people mean somewhere in between. Whether someone is good or bad will depend on what people mean. It also surprises me because that’s an amateur mistake here. It also doesn’t help when you quote modest mathematicians but describe people who are characterized as ‘bad at math’ in your anecdotes. It makes it too easy to assume that you’re claiming those people are secret Grothendiecks, and I don’t think you are. If you are, then I want to know.
(a) A single number model of intelligence is toxic and silly. IQ is a single number proxy for a complex multidimensional space.
(b) Effective test taking has very little to do with math ability. Many excellent mathematicians are bad test takers (e.g. do not think quickly on their feet): this means basically nothing.
(c) Brains are complicated, and there is a huge amount of heterogeneity in how people process information and think about mathematics (and indeed all topics, but it is clearer in mathematics perhaps). Some are very visual, some are big on calculation.
(d) There is no separate magisterium called “math,” there is a gently sloping continuum from common sense to “novel math work.” When someone says “I am bad at math,” I am not sure if they mean “I can’t think carefully at all,” “math notation scares me,” “I can’t think abstractly,” [something else].
(e) If you haven’t engaged with math beyond high school, you probably don’t have enough information to evaluate the counterfactual “would a hypothetical me that pursued a math education make a good mathematician?” A lot of people’s current trajectories in life are based on contingent things like what sort of teachers you had, social situation, etc.
(f) There is a skill component to math that needs practice and repetition, as with anything else. I call it “taking the time to insert metal struts into your brain.”
(g) I think a non-trivial % of college-educated population can have a non-trivial engagement with mathematical topics, if they get over their panic. A smaller % can do novel work, if they wanted to put in the time.
(h) There is an enormous overlap between doing math properly and doing rationality properly. Math gives you quite a bit of good rationality habits for free (e.g. “I am notice I am confused..”), and (?conversely?) you can’t engage with rationality without a bit of math.
When someone says “I am bad at math,” I am not sure if they mean “I can’t think carefully at all,” “math notation scares me,” “I can’t think abstractly,” [something else].
A data point for you: I am not particularly good at math. What this means is that at certain levels going forwards suddenly becomes much more difficult. I can continue, but slowly and only with a lot of effort. It’s a slog. By comparison, I’m much better at logic/patterns and going deeper there is just easier. I do NOT mean that I can’t think carefully or abstractly or that notation scares me.
Note that I’m using a fairly narrow definition of math here. In particular, I distinguish math and statistics and believe that they require two different propensities. People good at math are rarely good at statistics; people good at statistics are rarely good at math.
I am not sure what exactly going deeper at logic/patterns means if not getting into mathematical logic. It is incredibly easy to read mathematics you know and incredibly difficult to read mathematics that you don’t due to how dense it is. It might be the case that your impression is due to comparing these two.
I am training to become a mathematician and I do not know of a single person for whom learning mathematics is not slowly and with a lot of effort, I do not think you are particularly exceptional in that but I know very little about your particular scenario.
I am not sure what exactly going deeper at logic/patterns means if not getting into mathematical logic.
I am not sure how to properly express the difference, but it has to do with math being more, um, hard-edged and rigid (in the sense of a rigid mechanical construction as opposed to a reed bending with the wind).
To use a quote from Conan Doyle, a logic/patterns ability would allow one to “from a drop of water … infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other.” Mathematical logic cannot do that.
Definitely less rigorous formalization and more gestalt pattern recognition.
In general, I think of math as dealing with well-defined “things”—you may not know the shape/properties/characteristics at the moment, but they exist, they are precisely defined, and they are not going anywhere. In contrast to math, statistics deals with fuzzy amorphous “things” that you will likely never know in precise detail, that mutate as more data becomes available, and that usually require interpretation and/or some guessing to fill in the gaps.
Cutting edge math is actually mostly about converting fuzzy stuff, at least the parts of math I am interested in(Algebraic Geometry—Grothendieck/Weil for example). Both the mentioned mathematicians worked in a field where people had some stuff that worked but no foundations.
Also, the foundations of math have been changing for quite a long time and continue to do so. I think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced. However, I don’t see an easy way to fix it.
To teach mathematics well would require a high amount of mastery and we don’t have enough people like that around.
I think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced.
I doubt it—I generally teach myself things and just ignore bad instruction. The underlying cause is likely to be the curse of the gifted—I’m lazy and when I run into walls I usually go around instead of starting a wall disassembly project. And I was never attracted to math sufficiently to apply a lot of effort.
Can you give an example of the level where things suddenly become more difficult?
As I said in another post, I struggled quite a bit with early calculus classes, but breezed through later “more difficult” classes that built on them.
Also, I disagree with the math and stats thing. Many of the best statisticians I know have strong grounding in mathematics, as do many of the best data scientists I know.
Can you give an example of the level where things suddenly become more difficult?
I hit a wall in my string theory course, after having to apply a lot more effort than expected in a QFT course the year before. Didn’t have that issue with GR at all. Well, maybe with some finer points involving algebraic topology, but nothing insurmountable.
Brains are complicated, and there is a huge amount of heterogeneity in how people process information and think about mathematics (and indeed all topics, but it is clearer in mathematics perhaps). Some are very visual, some are big on calculation.
And importantly, brains are more heterogeneous at the extremes than at the means. A person of average intelligence might not have much difference between their verbal and calculation abilities; a person of great verbal or calculative intelligence may have a large gap between their stronger and (comparatively) weaker cognitive abilities.
I say everything I’m about to say as a person who is more certain than not that you have something valuable to contribute through this sequence, and who eagerly awaits more. All of your posts in this sequence have purportedly been written to motivate your main thesis, but it’s not clear to me what that is. I think you should stop motivating and very clearly reveal your Big Secret.
I very much appreciate your interest. I’m sympathetic to the points that you raise. The trouble is that it’s hard to even state my main thesis without presenting a lot of background information (!!). It’s as though someone wanted to know about monstrous moonshine, and I started explaining what a modular function is, and the person said “ok, rather than giving so much motivation, I’d prefer it if you just told me what monstrous moonshine is.” Then I state the theorem, and the person says “wait, what’s the modular j-function?”
But it’s much easier to speak to an individual’s situation than it is to speak to the general question of how people can get better at math. What’s your background and what are your goals? I may not be able to respond at length individually, but I’ll try to offer quick thoughts at least, and your comments will inform what I write about subsequently.
Yeah, I know people who have seen his calculus lectures, and if I understand correctly he sticks to the book when he teaches undergraduate courses and doesn’t share his special insights. You would have had to go to office hours :P.
I very much appreciate your interest. I’m sympathetic to the points that you raise. The trouble is that it’s hard to even state my main thesis without presenting a lot of background information (!!). It’s as though someone wanted to know about monstrous moonshine, and I started explaining what a modular function is, and the person said “ok, rather than giving so much motivation, I’d prefer it if you just told me what monstrous moonshine is.” Then I state the theorem, and the person says “wait, what’s the modular j-function?”
I did consider the possibility when I wrote that that I was grossly underestimating the amount of background information necessary to understand your main thesis. After all, you’ve been tutoring for, 15 years, I believe you said? Presumably you’ve collected a ton of anecdata, besides the information you’ve gathered from actual research. That’s a lot to distill. But do you really think it’s comparable to having to understand advanced abstract algebra/group theory? If you’re really confident that stating your main thesis in your next post will only discredit you, then I don’t think you should do it. But I do think you should be open to the possibility that that’s not the case, and I think there’s a non-negligible risk that you’ll alienate the portion of your audience that strongly believes that mathematical ability is almost completely, if not completely, innate, if you take too much time to motivate.
But it’s much easier to speak to an individual’s situation than it is to speak to the general question of how people can get better at math. What’s your background and what are your goals? I may not be able to respond at length individually, but I’ll try to offer quick thoughts at least, and your comments will inform what I write about subsequently.
The rest of this comment is going to require considerable personal disclosure, so others should move on if they don’t like that or care. I hope that my comment doesn’t decrease my credibility with you or others; I think I’ve made valid points. My background is weak, far below what I expect to be the LessWrong norm. I’m going to explain my background, innate ability, interests, and the provisional goals that I’ve set based on my preliminary research. I understand that your time is limited, so I should say that any links are for elaboration and do not necessarily need to be read. I do apologize if this is too long, and I won’t be offended if you don’t offer a response.
I only have a high school diploma and I’m not a college student. I was trapped in the bowels of anti-epistemology until the end of last year. I have strong ugh fields around the traditional math curriculum because I always memorized passwords and my math classes took place very early in the newest part of my school, in which the floors were ceramic, the walls concrete, and the air conditioning diabolically effective; I literally associate the traditional curriculum with being exhausted and confused in a cold, hard, white room that is beset with motivational posters and otherwise featureless. A pressing need for mathematical logic (as you will see) segued nicely into my need for a subject in formal science that I had not been conditioned to hate and that required little or no prerequisite knowledge from the traditional curriculum. I started reading about logic in February, and I got up to soundness and completeness proofs a few days ago. My first book used semantic tableaux, and now I’m working my way from the bottom up again with Fitch-style natural deduction. I have other books for Gentzen-style, Hilbert-style, and sequent calculi.
As for innate ability, that timeline should give you an idea of my learning rate, although I should mention that my mother died on February 24th and it decreased my productivity considerably. I identified the pattern in the Raven matrix in Innate Mathematical Ability. I took the Stanford-Binet IV when I was 8 and scored 135, although I know that one test before age 16 is not a very reliable indication of my adult IQ. Presumably my writing and analysis on this post thus far should have given you an idea of my verbal reasoning ability. I detail my innate ability because I expect there’s a possibility that the benefits of your main thesis are diminishing as innate ability increases, just as rationality training is usually a monumental improvement for those with little initial rationality and not-so-great for those with considerable existing rationality.
As for interests, I’m curious about AGI (I won’t go so far as to say that I expect to become a researcher), but I’m lacking in tools to seriously understand the field. I’ve already read the popular introduction.
As for goals, I’ve been trying to construct a model of what I need to study because the MIRI Research Guide assumes more background knowledge than one would infer at a glance. Khan Academy is enough to solidify my traditional foundations once I get over the ugh fields. Peter Smith’s Teach Yourself Logic guide is basically all I need to give me an idea of how to work through first-order logic, model theory, proof theory, etc. I’ve found Halmos accessible for some set theory. The early relevant chapters in Jaynes require at least calculus, and I have Apostol and Spivak (to compare one another) for that. Axler seems good enough for linear algebra; I checked out the first chapter and it wasn’t too frightening. I’m aware of the Decision Theory FAQ, and have also found that accessible.
I infer that more reading recommendations, especially on subjects that I may have missed, and general learning techniques of the sort that I expect you will eventually detail and that I might not have picked up elsewhere, would be most useful to me, although I defer to your pedagogical experience in determining what advice to give me.
I am a bachelor’s in mathematics and estimate my current knowledge to be around a second year graduate student’s if my mathematical knowledge is useful. I am interested in getting better at doing math as well as teaching it.
I also think it’s weird that I see a lot of people throwing around ‘good at math’ and ‘bad at math’ as if those terms mean the same thing to everyone. Some people mean Calculus III, some people mean Fields Medal or thereabouts, and some people mean somewhere in between. Whether someone is good or bad will depend on what people mean.
Well, I certainly find it plausible that most reasonably intelligent (ie: at the mean to one standard deviation above) people can learn math to the level of, say, a first-year undergrad, or an advanced high-schooler. For one thing, some countries have more advanced and rigorous math curricula for high-schoolers than others (“engineering-major calculus” and linear algebra are reasonably common in high school curricula), and yet their class-failure rate, to my very limited knowledge, seems to vary with the quality of the pedagogy rather than being uniformly higher.
Could most people of such an intelligence level also learn an entire undergraduate math major? I don’t know: nobody is trying that experiment. I do think most of the people who already complete engineering, natural science, or computer-science degrees could probably complete undergraduate math—but nobody is trying the experiment of controlling the “double major” variable, either. Instead we discourage physics, engineering, and comp sci majors who don’t voluntarily double-major from doing additional theoretical math courses in favor of the applied math they need for their own domain (ie: algebra for the quantum physicist, differential equations and optimization for the engineer, logic and computability for the computer scientist). And that’s when they take a theoretical track, instead of just cutting out to the applications ASAP!
Could most people who do PhD-level research work in other natural and formal sciences do work in mathematics? At the research level, the distinction has collapsed: they partly already do! I’ve actually heard it said that you’re not really well-prepared for PhD-level CS or physics if you didn’t double-major in math, or for PhD-level biology if you didn’t take an undergrad major or minor in statistics, anyway. You certainly can’t work in type theory or machine learning (to bang on my own interests) these days without using and doing research-level work in “math” as an inherent part of your own research field.
It’s very hard to make the relevant inferences when we lack data on what happens when we try to teach people math instead of using math as a weed-out subject, and then jumping out from behind a bush at young researchers in the other sciences yelling, “SHOULDA LEARNED MORE MATH!”.
I’m finding this discussion very interesting because of my personal background. The general population would describe me as “good at maths”. I would describe myself (because of context) as “bad at maths”. I was one of the best all the way through high school and then started an undergraduate maths course known for being challenging. After a few weeks I completely hit a wall and couldn’t progress any further with it. (I changed course to music.)
My sister, father and brother-in-law all completed a whole undergraduate course in maths—I couldn’t finish the first year. So I think I am bad at maths.
Following on: I think I have a much deeper aesthetic understanding of music than my father and sister. They, the “better” mathematicians, are excellent musicians, but in a functional sense. I’d say that I, the “worse” mathematician, have a much more profound insight into music than they do.
And I failed my first go at Machine Learning, and nearly failed my first go at Intro to Statistics!
(Because I hadn’t taken continuous probability first, and was depressed.)
What was it like from the inside, hitting that wall? Math is a particularly easy subject to hit a wall in, because if you’re lacking a prerequisite of some sort, all of a sudden everything you’re seeing turns to apparent nonsense.
Actually I found it very difficult to understand how it had happened. At school I was one of the best, I enjoyed maths, I understood the concepts and mostly found it easy. At University that all reversed: I was one of the worst, I couldn’t do the assignments, I found the lectures boring, and I thoroughly disliked it. I found it very hard to comprehend how such a complete reversal had happened. And more than 15 years later, I still don’t really get it… It’s rather destabilising when you can’t do the thing you expected to devote three years of your life to!
I say everything I’m about to say as a person who is more certain than not that you have something valuable to contribute through this sequence, and who eagerly awaits more.
All of your posts in this sequence have purportedly been written to motivate your main thesis, but it’s not clear to me what that is. I think you should stop motivating and very clearly reveal your Big Secret. What can I do right now to improve my mathematical ability? That’s what I want to know.
Consider these points:
Eliezer tried to explain his metaethics the first time and failed, but it was okay because he was able to write his epistemology sequence after that and clarify. His posts pretty much always motivate people to continue reading because they usually have such high insight density and people know that they do; but your posts in this sequence, as far as I can tell, have just been anecdotes, quotes, nonstandard definitions, references detailing your nonstandard definitions, and promises of elucidation in future posts, and in your case, there isn’t common knowledge of high insight density to make people trust you even when they don’t understand where you’re going. Your second post had less karma than your first post, and I predict that this post will have even less. I think that people are getting antsy. If no one understands your Big Secret and it turns out that all of this motivation really was necessary, then you can just do what Eliezer did and try again.
It seems to me that you’re getting strawmanned a lot. I want to argue against those things but I don’t have a real man to point to, and if you wrote a clear thesis statement, then I would. Maybe someone would say that I’m defending a side rather than seeking the truth if I want to argue against another position when I have no clear position, but I think that amounts to claiming that your predicate’s extension is empty rather than that it’s vague. There are things that people have claimed about your posts that I can’t consider an accurate statement of your (vague) position no matter how charitable my interpretation. The most obvious is that you’re claiming that innate ability is irrelevant; you’ve explicitly claimed the opposite. Other possibilities in order of increasing plausibility include:
(Almost) anyone can be a famous mathematician.
(Almost) anyone can do mathematical research.
(Almost) anyone can be what an average person considers ‘good at math.’
Current mathematical pedagogy sucks.
You’ve explicitly stated the fourth item, and it’s the one I’m most sympathetic to, and it would be useful to me and many others if this were elucidated regardless of whether or not it means I can be a famous mathematician. If you’re claiming more than one of the above, then maybe you could just refine your points about that particular point into public form, and substantiate your more radical assertions afterwards. I really want to see what you have to say about pedagogical techniques more than anything else. I want to be better at math. I’m open to the possibility that I’m misinterpreting everyone’s attitudes about this and I’m the only one who wants to know what you have to say about that more than anything else.
I also think it’s weird that I see a lot of people throwing around ‘good at math’ and ‘bad at math’ as if those terms mean the same thing to everyone. Some people mean Calculus III, some people mean Fields Medal or thereabouts, and some people mean somewhere in between. Whether someone is good or bad will depend on what people mean. It also surprises me because that’s an amateur mistake here. It also doesn’t help when you quote modest mathematicians but describe people who are characterized as ‘bad at math’ in your anecdotes. It makes it too easy to assume that you’re claiming those people are secret Grothendiecks, and I don’t think you are. If you are, then I want to know.
Let me take a stab at my (not OPs) views on math:
(a) A single number model of intelligence is toxic and silly. IQ is a single number proxy for a complex multidimensional space.
(b) Effective test taking has very little to do with math ability. Many excellent mathematicians are bad test takers (e.g. do not think quickly on their feet): this means basically nothing.
(c) Brains are complicated, and there is a huge amount of heterogeneity in how people process information and think about mathematics (and indeed all topics, but it is clearer in mathematics perhaps). Some are very visual, some are big on calculation.
(d) There is no separate magisterium called “math,” there is a gently sloping continuum from common sense to “novel math work.” When someone says “I am bad at math,” I am not sure if they mean “I can’t think carefully at all,” “math notation scares me,” “I can’t think abstractly,” [something else].
(e) If you haven’t engaged with math beyond high school, you probably don’t have enough information to evaluate the counterfactual “would a hypothetical me that pursued a math education make a good mathematician?” A lot of people’s current trajectories in life are based on contingent things like what sort of teachers you had, social situation, etc.
(f) There is a skill component to math that needs practice and repetition, as with anything else. I call it “taking the time to insert metal struts into your brain.”
(g) I think a non-trivial % of college-educated population can have a non-trivial engagement with mathematical topics, if they get over their panic. A smaller % can do novel work, if they wanted to put in the time.
(h) There is an enormous overlap between doing math properly and doing rationality properly. Math gives you quite a bit of good rationality habits for free (e.g. “I am notice I am confused..”), and (?conversely?) you can’t engage with rationality without a bit of math.
A data point for you: I am not particularly good at math. What this means is that at certain levels going forwards suddenly becomes much more difficult. I can continue, but slowly and only with a lot of effort. It’s a slog. By comparison, I’m much better at logic/patterns and going deeper there is just easier. I do NOT mean that I can’t think carefully or abstractly or that notation scares me.
Note that I’m using a fairly narrow definition of math here. In particular, I distinguish math and statistics and believe that they require two different propensities. People good at math are rarely good at statistics; people good at statistics are rarely good at math.
I am not sure what exactly going deeper at logic/patterns means if not getting into mathematical logic. It is incredibly easy to read mathematics you know and incredibly difficult to read mathematics that you don’t due to how dense it is. It might be the case that your impression is due to comparing these two.
I am training to become a mathematician and I do not know of a single person for whom learning mathematics is not slowly and with a lot of effort, I do not think you are particularly exceptional in that but I know very little about your particular scenario.
I am not sure how to properly express the difference, but it has to do with math being more, um, hard-edged and rigid (in the sense of a rigid mechanical construction as opposed to a reed bending with the wind).
To use a quote from Conan Doyle, a logic/patterns ability would allow one to “from a drop of water … infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other.” Mathematical logic cannot do that.
So something more like intuitive pattern-recognition/completion than rigorous formalization?
Definitely less rigorous formalization and more gestalt pattern recognition.
In general, I think of math as dealing with well-defined “things”—you may not know the shape/properties/characteristics at the moment, but they exist, they are precisely defined, and they are not going anywhere. In contrast to math, statistics deals with fuzzy amorphous “things” that you will likely never know in precise detail, that mutate as more data becomes available, and that usually require interpretation and/or some guessing to fill in the gaps.
Cutting edge math is actually mostly about converting fuzzy stuff, at least the parts of math I am interested in(Algebraic Geometry—Grothendieck/Weil for example). Both the mentioned mathematicians worked in a field where people had some stuff that worked but no foundations.
Also, the foundations of math have been changing for quite a long time and continue to do so. I think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced. However, I don’t see an easy way to fix it.
To teach mathematics well would require a high amount of mastery and we don’t have enough people like that around.
I doubt it—I generally teach myself things and just ignore bad instruction. The underlying cause is likely to be the curse of the gifted—I’m lazy and when I run into walls I usually go around instead of starting a wall disassembly project. And I was never attracted to math sufficiently to apply a lot of effort.
Can you give an example of the level where things suddenly become more difficult?
As I said in another post, I struggled quite a bit with early calculus classes, but breezed through later “more difficult” classes that built on them.
Also, I disagree with the math and stats thing. Many of the best statisticians I know have strong grounding in mathematics, as do many of the best data scientists I know.
I hit a wall in my string theory course, after having to apply a lot more effort than expected in a QFT course the year before. Didn’t have that issue with GR at all. Well, maybe with some finer points involving algebraic topology, but nothing insurmountable.
And importantly, brains are more heterogeneous at the extremes than at the means. A person of average intelligence might not have much difference between their verbal and calculation abilities; a person of great verbal or calculative intelligence may have a large gap between their stronger and (comparatively) weaker cognitive abilities.
I very much appreciate your interest. I’m sympathetic to the points that you raise. The trouble is that it’s hard to even state my main thesis without presenting a lot of background information (!!). It’s as though someone wanted to know about monstrous moonshine, and I started explaining what a modular function is, and the person said “ok, rather than giving so much motivation, I’d prefer it if you just told me what monstrous moonshine is.” Then I state the theorem, and the person says “wait, what’s the modular j-function?”
But it’s much easier to speak to an individual’s situation than it is to speak to the general question of how people can get better at math. What’s your background and what are your goals? I may not be able to respond at length individually, but I’ll try to offer quick thoughts at least, and your comments will inform what I write about subsequently.
Completely unrelated: long ago, I took a lower division math class from Borcherds. I don’t think anything rubbed off, though :(.
Yeah, I know people who have seen his calculus lectures, and if I understand correctly he sticks to the book when he teaches undergraduate courses and doesn’t share his special insights. You would have had to go to office hours :P.
I did consider the possibility when I wrote that that I was grossly underestimating the amount of background information necessary to understand your main thesis. After all, you’ve been tutoring for, 15 years, I believe you said? Presumably you’ve collected a ton of anecdata, besides the information you’ve gathered from actual research. That’s a lot to distill. But do you really think it’s comparable to having to understand advanced abstract algebra/group theory? If you’re really confident that stating your main thesis in your next post will only discredit you, then I don’t think you should do it. But I do think you should be open to the possibility that that’s not the case, and I think there’s a non-negligible risk that you’ll alienate the portion of your audience that strongly believes that mathematical ability is almost completely, if not completely, innate, if you take too much time to motivate.
The rest of this comment is going to require considerable personal disclosure, so others should move on if they don’t like that or care. I hope that my comment doesn’t decrease my credibility with you or others; I think I’ve made valid points. My background is weak, far below what I expect to be the LessWrong norm. I’m going to explain my background, innate ability, interests, and the provisional goals that I’ve set based on my preliminary research. I understand that your time is limited, so I should say that any links are for elaboration and do not necessarily need to be read. I do apologize if this is too long, and I won’t be offended if you don’t offer a response.
I only have a high school diploma and I’m not a college student. I was trapped in the bowels of anti-epistemology until the end of last year. I have strong ugh fields around the traditional math curriculum because I always memorized passwords and my math classes took place very early in the newest part of my school, in which the floors were ceramic, the walls concrete, and the air conditioning diabolically effective; I literally associate the traditional curriculum with being exhausted and confused in a cold, hard, white room that is beset with motivational posters and otherwise featureless. A pressing need for mathematical logic (as you will see) segued nicely into my need for a subject in formal science that I had not been conditioned to hate and that required little or no prerequisite knowledge from the traditional curriculum. I started reading about logic in February, and I got up to soundness and completeness proofs a few days ago. My first book used semantic tableaux, and now I’m working my way from the bottom up again with Fitch-style natural deduction. I have other books for Gentzen-style, Hilbert-style, and sequent calculi.
As for innate ability, that timeline should give you an idea of my learning rate, although I should mention that my mother died on February 24th and it decreased my productivity considerably. I identified the pattern in the Raven matrix in Innate Mathematical Ability. I took the Stanford-Binet IV when I was 8 and scored 135, although I know that one test before age 16 is not a very reliable indication of my adult IQ. Presumably my writing and analysis on this post thus far should have given you an idea of my verbal reasoning ability. I detail my innate ability because I expect there’s a possibility that the benefits of your main thesis are diminishing as innate ability increases, just as rationality training is usually a monumental improvement for those with little initial rationality and not-so-great for those with considerable existing rationality.
As for interests, I’m curious about AGI (I won’t go so far as to say that I expect to become a researcher), but I’m lacking in tools to seriously understand the field. I’ve already read the popular introduction.
As for goals, I’ve been trying to construct a model of what I need to study because the MIRI Research Guide assumes more background knowledge than one would infer at a glance. Khan Academy is enough to solidify my traditional foundations once I get over the ugh fields. Peter Smith’s Teach Yourself Logic guide is basically all I need to give me an idea of how to work through first-order logic, model theory, proof theory, etc. I’ve found Halmos accessible for some set theory. The early relevant chapters in Jaynes require at least calculus, and I have Apostol and Spivak (to compare one another) for that. Axler seems good enough for linear algebra; I checked out the first chapter and it wasn’t too frightening. I’m aware of the Decision Theory FAQ, and have also found that accessible.
I infer that more reading recommendations, especially on subjects that I may have missed, and general learning techniques of the sort that I expect you will eventually detail and that I might not have picked up elsewhere, would be most useful to me, although I defer to your pedagogical experience in determining what advice to give me.
I am a bachelor’s in mathematics and estimate my current knowledge to be around a second year graduate student’s if my mathematical knowledge is useful. I am interested in getting better at doing math as well as teaching it.
Note: I am not the person you replied to.
Well, I certainly find it plausible that most reasonably intelligent (ie: at the mean to one standard deviation above) people can learn math to the level of, say, a first-year undergrad, or an advanced high-schooler. For one thing, some countries have more advanced and rigorous math curricula for high-schoolers than others (“engineering-major calculus” and linear algebra are reasonably common in high school curricula), and yet their class-failure rate, to my very limited knowledge, seems to vary with the quality of the pedagogy rather than being uniformly higher.
Could most people of such an intelligence level also learn an entire undergraduate math major? I don’t know: nobody is trying that experiment. I do think most of the people who already complete engineering, natural science, or computer-science degrees could probably complete undergraduate math—but nobody is trying the experiment of controlling the “double major” variable, either. Instead we discourage physics, engineering, and comp sci majors who don’t voluntarily double-major from doing additional theoretical math courses in favor of the applied math they need for their own domain (ie: algebra for the quantum physicist, differential equations and optimization for the engineer, logic and computability for the computer scientist). And that’s when they take a theoretical track, instead of just cutting out to the applications ASAP!
Could most people who do PhD-level research work in other natural and formal sciences do work in mathematics? At the research level, the distinction has collapsed: they partly already do! I’ve actually heard it said that you’re not really well-prepared for PhD-level CS or physics if you didn’t double-major in math, or for PhD-level biology if you didn’t take an undergrad major or minor in statistics, anyway. You certainly can’t work in type theory or machine learning (to bang on my own interests) these days without using and doing research-level work in “math” as an inherent part of your own research field.
It’s very hard to make the relevant inferences when we lack data on what happens when we try to teach people math instead of using math as a weed-out subject, and then jumping out from behind a bush at young researchers in the other sciences yelling, “SHOULDA LEARNED MORE MATH!”.
I’m finding this discussion very interesting because of my personal background. The general population would describe me as “good at maths”. I would describe myself (because of context) as “bad at maths”. I was one of the best all the way through high school and then started an undergraduate maths course known for being challenging. After a few weeks I completely hit a wall and couldn’t progress any further with it. (I changed course to music.) My sister, father and brother-in-law all completed a whole undergraduate course in maths—I couldn’t finish the first year. So I think I am bad at maths.
Following on: I think I have a much deeper aesthetic understanding of music than my father and sister. They, the “better” mathematicians, are excellent musicians, but in a functional sense. I’d say that I, the “worse” mathematician, have a much more profound insight into music than they do.
And I failed my first go at Machine Learning, and nearly failed my first go at Intro to Statistics!
(Because I hadn’t taken continuous probability first, and was depressed.)
What was it like from the inside, hitting that wall? Math is a particularly easy subject to hit a wall in, because if you’re lacking a prerequisite of some sort, all of a sudden everything you’re seeing turns to apparent nonsense.
Actually I found it very difficult to understand how it had happened. At school I was one of the best, I enjoyed maths, I understood the concepts and mostly found it easy. At University that all reversed: I was one of the worst, I couldn’t do the assignments, I found the lectures boring, and I thoroughly disliked it. I found it very hard to comprehend how such a complete reversal had happened. And more than 15 years later, I still don’t really get it… It’s rather destabilising when you can’t do the thing you expected to devote three years of your life to!
This is a gentle reminder for everyone reading these comments that if you enjoyed this post, please go back up top and upvote it real quick.
I myself forgot at first. I think when I see a post that looks especially interesting to me, I tear right into it and don’t look back.