Recently I’ve been trying to catch up in math, with a goal of trying to get to calculus as soon as possible. (I want to study Data Science, and calculus / linear algebra seems to be necessary for that kind of study.) I found someone on LW who agreed to provide me with some deadlines, minor incentives, and help if I need it (similar to this proposal), although I’m not sure how well such a setup will end up working.
The Art of Problem Solving books deliberately make you think a lot, and a lot of the problems are quite difficult. That’s great, but I’ve found that after 2-3 hours of heavy thinking my brain often feels completely shot and that ruins my studying for the rest of the day. It also doesn’t help that my available study time usually runs from about 10am-2pm, but I often only start to really wake up around noon. (Yes, I get enough sleep usually. I also use a light box. But I still often only wake up around noon.)
One solution I’ve been thinking of would be to take the studying slower: I’d study math only from 12-2, and before that I’d study something else, like programming. The only problem with that is that cutting my study time in half means it’ll take twice as long to get through the material. At that rate I estimate it’ll take approximately a year, perhaps a bit more, before I can even start Calculus. Maybe that’s what’s needed, but I was hoping to get on with studying data science sooner than that.
Another possible solution would be to try an easier course of study than the AoPS books. I’ve had some good experiences with MOOCs, so perhaps that might be a good route to take. To that end I’ve tentatively signed up to this math refresher course, although I don’t really know anything about it. Or perhaps I could just CliffNotes my way through Algebra II and Precalculus, and then take a Calculus MOOC. I wouldn’t get the material nearly as well, of course, but at least I’d be able to get to Calculus and move on with my data science studies from there. I could even do one of these alternatives while also doing the AoPS books at a slower pace. That way I could get to data science studying as soon as possible, and I’d also eventually get a more thorough familiarity with the material through the AoPS books.
Be very very careful of studying beyond the level you think is comfortable. My experience has been that you cannot push yourself to learn difficult things, especially math, faster than a certain pace. Sure, your limit may be 20% higher than what you think it is, but it’s not 200% higher. Spending more time on a task when you just don’t feel up to it is useless, because instead of thinking you’ll just be spending more time staring at the page and having your mind drift off.
I’ve found that the various methods of ‘productivity boosting’ (pomodoros, etc) are largely useless and do one of two things: Either decrease your productivity, or momentarily increase it at the expense of a huge decrease later on (anything from ‘feeling fuzzy for a couple of days’ to ‘total burnout for 3 weeks’). Unless you have a mental illness, your brain is already a finely-tuned machine for learning and doing. Don’t fool yourself into thinking you can improve it just by some clever schedule rearrangement.
The point to all of this is that you should refrain from ‘planning ahead’ when it comes to learning. Sure, you should have some general overall sketch of what you want to learn, but at each particular moment in time, the best strategy is to simply pick some topic and try to learn it as best you can, until you get tired. Then rest until you feel you can go at it again. And avoid internet distractions that use up your mental energy but don’t cause you to learn anything.
The general rule of thumb for raw intelligence probably applies, you can damage it with unwise actions (like eating lead paint or taking up boxing), but there aren’t really any good ways to boost it beyond its natural unimpeded baseline. Good instrumental rationality can help you look out for and avoid self-sabotaging behavior, like overworking your way into burnout.
Largely, but not entirely. There are cases where evolution optimises for something different from what you want. And there are cases where the environment has changed faster than evolution can track.
If some particular method of learning can be shown, through evidence, to be an improvement long-term, then by all means go for it. But until then, your prior belief has to be that it isn’t.
One of my professors once mentioned that there’s an upper limit to how much learning you can do in a sleep cycle [citation needed]. This is congruent with my experience, both before and after he mentioned that, so I tend to believe it. Personally, I tend to max out around 3-4 hours, so the times you’re talking about seem reasonable. If you can restructure your work times, napping is a good strategy; I’ve talked to a few people who report getting through grad school by napping once they’d saturated their brain’s capacity to learn new stuff.
Interleaved practice is good. This study had subjects practice finding the volume of unconventional geometric solids. One group clustered their practice; they found the volumes of a bunch of wedges, then a bunch of spheroids, etc. The other group had their practice problems mixed. On a final test, the former group got 20% right, and the latter group got 63% right. citation.
What this suggests is you should perhaps study programming and algebra at the same time, switching between the two fairly frequently. It feels like you’re going slower, but, as the authors of the book emphasize, you’re trading the illusion of learning for more durable learning.
The act of acquiring new, or modifying and reinforcing, existing knowledge, behaviors, skills, values, or preferences and may involve synthesizing different types of information
My experience is in math (and the prof in question taught math), which is fairly light on the memorization. Sure, you memorize definitions, but most of the effort is in internalizing new ways of thinking about things. Like, the derivation of the quadratic formula doesn’t contain any new information to memorize, and I definitely didn’t memorize the steps, but when I learned it, I spent a bunch of time looking at what the steps were, why they were legal, and why we decided to use those particular manipulations to solve the problem, and internalizing those things. And after doing enough stuff like that, I’d try to internalize some new stuff, and my brain would say “No!” And then I stopped.
ETA: I’m not sure if I’d call that memorization. I’m certainly talking about putting things in your head that weren’t there before, but it’s not the type of thing you could easily make into an Anki card.
I’m not sure if I’d call that memorization. I’m certainly talking about putting things in your head that weren’t there before
Well, that’s the thing. Learning can be quite different. Some of it is putting new things into your head. But some of it is rearranging your internal maps. And some of it is generating new connections between things inside your head. A whole bunch of it is all of the above.
I understand the idea of limited capacity per sleep cycle—I’m curious whether it works in different ways for different kinds of learning.
I understand the idea of limited capacity per sleep cycle—I’m curious whether it works in different ways for different kinds of learning.
Personally I’d be surprised if it did. The maximum amount of deliberate practice you can get in a day tops out at 3-4 hours, according to K. Anders Ericsson. I think that’s quite close to the limits of what the brain can do. I’ll honestly be surprised if napping tesets that clock or he or other psychologists woul have uncovered them.
Well, first of all “deliberate practice” is different from “learning”. The paper is concerned with ability to perform which is the goal of the deliberate practice, not with understanding which is the goal of learning.
Second, the paper is unwilling to commit to this number saying (emphasis mine) ”...raising the possibility of a more general limit on the maximal amount of deliberate practice that can be sustained over extended time without exhaustion.”
I certainly accept the idea that resources such as concentration, attention, etc. are limited (though they recover over time) and you can’t just be at your best all your waking time. But there doesn’t seem to be enough evidence to fix hard numbers (like 2-4 hours) for that. And, of course, I expect there to be fair amount of individual variation, as well as some dependency on what exactly is it that you’re learning or practicing.
Recently I’ve been trying to catch up in math, with a goal of trying to get to calculus as soon as possible. (I want to study Data Science, and calculus / linear algebra seems to be necessary for that kind of study.) I found someone on LW who agreed to provide me with some deadlines, minor incentives, and help if I need it (similar to this proposal), although I’m not sure how well such a setup will end up working.
Originally the plan was that I’d study the Art of Problem Solving Intermediate Algebra book, but I found that many of the concepts were a little advanced for me, so I switched to the middle of the Introduction to Algebra book instead.
The Art of Problem Solving books deliberately make you think a lot, and a lot of the problems are quite difficult. That’s great, but I’ve found that after 2-3 hours of heavy thinking my brain often feels completely shot and that ruins my studying for the rest of the day. It also doesn’t help that my available study time usually runs from about 10am-2pm, but I often only start to really wake up around noon. (Yes, I get enough sleep usually. I also use a light box. But I still often only wake up around noon.)
One solution I’ve been thinking of would be to take the studying slower: I’d study math only from 12-2, and before that I’d study something else, like programming. The only problem with that is that cutting my study time in half means it’ll take twice as long to get through the material. At that rate I estimate it’ll take approximately a year, perhaps a bit more, before I can even start Calculus. Maybe that’s what’s needed, but I was hoping to get on with studying data science sooner than that.
Another possible solution would be to try an easier course of study than the AoPS books. I’ve had some good experiences with MOOCs, so perhaps that might be a good route to take. To that end I’ve tentatively signed up to this math refresher course, although I don’t really know anything about it. Or perhaps I could just CliffNotes my way through Algebra II and Precalculus, and then take a Calculus MOOC. I wouldn’t get the material nearly as well, of course, but at least I’d be able to get to Calculus and move on with my data science studies from there. I could even do one of these alternatives while also doing the AoPS books at a slower pace. That way I could get to data science studying as soon as possible, and I’d also eventually get a more thorough familiarity with the material through the AoPS books.
What would you suggest?
Be very very careful of studying beyond the level you think is comfortable. My experience has been that you cannot push yourself to learn difficult things, especially math, faster than a certain pace. Sure, your limit may be 20% higher than what you think it is, but it’s not 200% higher. Spending more time on a task when you just don’t feel up to it is useless, because instead of thinking you’ll just be spending more time staring at the page and having your mind drift off.
I’ve found that the various methods of ‘productivity boosting’ (pomodoros, etc) are largely useless and do one of two things: Either decrease your productivity, or momentarily increase it at the expense of a huge decrease later on (anything from ‘feeling fuzzy for a couple of days’ to ‘total burnout for 3 weeks’). Unless you have a mental illness, your brain is already a finely-tuned machine for learning and doing. Don’t fool yourself into thinking you can improve it just by some clever schedule rearrangement.
The point to all of this is that you should refrain from ‘planning ahead’ when it comes to learning. Sure, you should have some general overall sketch of what you want to learn, but at each particular moment in time, the best strategy is to simply pick some topic and try to learn it as best you can, until you get tired. Then rest until you feel you can go at it again. And avoid internet distractions that use up your mental energy but don’t cause you to learn anything.
Does this by extension imply that the type of instrumental rationality training advocated by LW is useless? Why, why not?
The general rule of thumb for raw intelligence probably applies, you can damage it with unwise actions (like eating lead paint or taking up boxing), but there aren’t really any good ways to boost it beyond its natural unimpeded baseline. Good instrumental rationality can help you look out for and avoid self-sabotaging behavior, like overworking your way into burnout.
Decreasing work-load when you feel tired—the thing you naturally want to do—is also a reliable way to avoid burnout.
Largely, but not entirely. There are cases where evolution optimises for something different from what you want. And there are cases where the environment has changed faster than evolution can track.
Evolution always optimizes for the same thing :-/
If you want something different, that’s your problem :-D
Is it time to restart the “Read the Sequences” meme?
Specifically: The Tragedy of Group Selectionism
Well, at least read the wiki entry.
If some particular method of learning can be shown, through evidence, to be an improvement long-term, then by all means go for it. But until then, your prior belief has to be that it isn’t.
One of my professors once mentioned that there’s an upper limit to how much learning you can do in a sleep cycle [citation needed]. This is congruent with my experience, both before and after he mentioned that, so I tend to believe it. Personally, I tend to max out around 3-4 hours, so the times you’re talking about seem reasonable. If you can restructure your work times, napping is a good strategy; I’ve talked to a few people who report getting through grad school by napping once they’d saturated their brain’s capacity to learn new stuff.
Interleaved practice is good. This study had subjects practice finding the volume of unconventional geometric solids. One group clustered their practice; they found the volumes of a bunch of wedges, then a bunch of spheroids, etc. The other group had their practice problems mixed. On a final test, the former group got 20% right, and the latter group got 63% right. citation.
What this suggests is you should perhaps study programming and algebra at the same time, switching between the two fairly frequently. It feels like you’re going slower, but, as the authors of the book emphasize, you’re trading the illusion of learning for more durable learning.
The AoPS textbooks are really, really good. In fact, I’m pretty sure they’re the only good algebra textbooks you’re going to find, unless you count abstract or linear algebra; most textbooks at that level are mediocre. As luke_prog has mentioned, good textbooks are the usually the quickest and best way to learn new material. Quality learning takes time, and you’re doing yourself no favors by spending that time looking for faster alternatives.
By “learning” do you actually mean “memorization”?
By learning, I actually mean
My experience is in math (and the prof in question taught math), which is fairly light on the memorization. Sure, you memorize definitions, but most of the effort is in internalizing new ways of thinking about things. Like, the derivation of the quadratic formula doesn’t contain any new information to memorize, and I definitely didn’t memorize the steps, but when I learned it, I spent a bunch of time looking at what the steps were, why they were legal, and why we decided to use those particular manipulations to solve the problem, and internalizing those things. And after doing enough stuff like that, I’d try to internalize some new stuff, and my brain would say “No!” And then I stopped.
ETA: I’m not sure if I’d call that memorization. I’m certainly talking about putting things in your head that weren’t there before, but it’s not the type of thing you could easily make into an Anki card.
Well, that’s the thing. Learning can be quite different. Some of it is putting new things into your head. But some of it is rearranging your internal maps. And some of it is generating new connections between things inside your head. A whole bunch of it is all of the above.
I understand the idea of limited capacity per sleep cycle—I’m curious whether it works in different ways for different kinds of learning.
Personally I’d be surprised if it did. The maximum amount of deliberate practice you can get in a day tops out at 3-4 hours, according to K. Anders Ericsson. I think that’s quite close to the limits of what the brain can do. I’ll honestly be surprised if napping tesets that clock or he or other psychologists woul have uncovered them.
Do you have a link?
See pg. 391-392 of The Role of Deliberate Practice in the Acquisition of Expert Performance.pdf), the paper that kicked off the field. A better summary is that 2-4 hours is the maximum sustainable amount of deliberate practice in a day.
Ah, so that’s where you are coming from.
Well, first of all “deliberate practice” is different from “learning”. The paper is concerned with ability to perform which is the goal of the deliberate practice, not with understanding which is the goal of learning.
Second, the paper is unwilling to commit to this number saying (emphasis mine) ”...raising the possibility of a more general limit on the maximal amount of deliberate practice that can be sustained over extended time without exhaustion.”
I certainly accept the idea that resources such as concentration, attention, etc. are limited (though they recover over time) and you can’t just be at your best all your waking time. But there doesn’t seem to be enough evidence to fix hard numbers (like 2-4 hours) for that. And, of course, I expect there to be fair amount of individual variation, as well as some dependency on what exactly is it that you’re learning or practicing.