I've done a substantial amount of independent learning in math, but almost none of it was through textbooks. (I am very bad at forcing myself to do textbook exercises.) Most of what I've learned outside of classrooms has been from a combination of Wikipedia, reading math blog posts, and writing math blog posts. My main substitute for doing exercises is trying to work through the details of a result I find interesting on my blog; I try to prove everything that leads to or fits in the context of the result along the way, and those are my exercises. (Edit: my secondary substitute is solving problems on sites like math.stackexchange.)
It would be very helpful if you were more specific. What's an example of an exercise in Spivak that took you a long time, and what did you do during that time?
There wasn't a single problem in particular I found took me a long time. It was more... when I finished a set of exercises, I looked at the clock and realized a couple of hours have gone by, and that seemed way too long (though it really wasn't).
For example, the first chapter deals with properties of numbers (multiplicative/additive inverses and identities, associative laws, etc.) and using these properties in proofs. Some of the exercises, say, #4, are split up into several other exercises (i, ii, iii, iv, etc.). "Prove: if a > 1, then a^2 > a" -- Which seemed fairly trivial to me at first, but proving them using only the properties of numbers/inequalities provided made the task more difficult.
During the time it took for me to solve them, I just... well... sat down and solved them. Played around with what I could and couldn't do, referred to some proofs he mentioned earlier, etc. :)
There's an old saying among mathematicians that no one ever understands calculus until they teach it. I'm living proof that this isn't strictly true, since I do think I understand calculus at some reasonable level and I've never taught it (though I have tutored folks in calculus once or twice).
Nonetheless, one really good way to learn any subject is to teach it, and calculus is no exception. I've forced myself to learn many subjects by committing to give presentations on the topic at a fixed time, and then preparing the relevant lecture materials. Usually when I come back to the subject a few years later I'm embarrassed by the naivete of my initial efforts, but nonetheless I do learn the material and most importantly get over the initial hump of complete ignorance.
So try signing up to teach or tutor calculus, and see if that helps.
taken me hours upon hours upon hours of combined work.
There's a reason that good college programs are said to require many hours of homework. Real skills, as in math and language, can only be gained by practice, although you can acquire "background-knowledge" level of a field with mere reading.
I spend more time than I should learning about learning, instead of learning the material itself.
Maybe you're doing this because it's more comfortable that actually learning the material. If I have six hours to cut down a tree and I spend the first five hours sharpening my axe, I'm probably not going about this the right way unless I'm very good at sharpening.
I have to carefully comb through all the concepts and equations and structure them intuitively in a way I see fit. I hate not having a very fundamental understanding of the things I'm working with.
It sounds like you're reading the chapter very closely before trying the exercises. Perhaps you'd learn much faster by trying the exercises before you feel ready. It might feel uncomfortable for you. But maybe it's worth it to solve the problem of being uncomfortable with not understanding.
On the other hand, maybe you really do need to read the chapter carefully before doing the exercises. But you should at least try not doing that.
If a text is too difficult, work through an easier text on the same topic (or prerequisite topics) first. For calculus, as preliminary training, I recommend Thompson's (original, 1914) "Calculus Made Easy" (basic intuition; you likely already have that if you're learning multivariable) and Courant's "What Is Mathematics" (partially as an introduction to proofs and various topics that won't be brought together in this manner in standard introductory courses). Spivak's text is great as a first exercise in meticulous proofs, but do take an actual analysis course after multivariable calculus (the current best book for this role seems to be Pugh's "Real Mathematical Analysis", which works as an improvement over baby Rudin). For multivariable calculus, I've heard good things about Hubbard's "Vector Calculus".
From Linear Algebra Done Right:
"If you zip through a page in less than an hour, you are probably going too fast."
It's a step higher than Spivak, but not a large one. 10 chapter, 250 pages, so about 25 hours a chapter. Plus exercises, which sometimes take upwards an hour each.
Math is hard. It takes a long time, especially if you don't immediately see how to do something, or try to solve a problem by conjecturing a lemma that turns out to not be true. If you're in flow when you study, then you're maximally utilizing your cognitive resources, and going as fast as your brain will let you. Don't try to go faster, you'll just start missing things.
... Are you me?
I've just completed the survey; I'd be very interested if you were willing to discuss the results you've received. I'm going through the exact same thing at the moment, so I was really glad to see that someone has launched up a discussion about this method of learning. Thank you!
I used to be like you, but over time I've gotten myself to follow the procedures described in this blog post more closely: http://steve-yegge.blogspot.com/2006/03/math-for-programmers.html
As a former perfectionist, it was easy for me to fall in to the trap of learning "brick by brick", where I would try to understand everything in great detail and know it once and for all. I think a better learning metaphor may be splattering paint on a canvas. You're going to forget lots of stuff, so don't worry if you're setting yourself up for going over the same stuff twice, it'll probably be helpful review anyway. (I think I read some LWer write something like "people tend to really grasp things once they've read their 3rd textbook on the subject".) You might as well follow your nose and learn in a way that interests you. Also, there may be unresolvable dependencies in what you're learning... e.g. maybe to understand A fully, you need to know B, but to fully know B, you need to know C, and to fully know C, you need to know A. You may also find that reading something without understanding it one day allows you to read with full understanding a few days later. And unlike many types of work, your benefit from learning is a smooth linear function of the amount of it you do... i.e. reading and understanding 10% of a textbook is probably 10% as good as reading and understanding 100%. So embrace the messiness of things.
An even better metaphor than paint splattering may be "just-in-time" learning, where instead of learning stuff, you figure out what you want to accomplish with what you're going to learn, then learn only what you need to complete that accomplishment, in order to accomplish it. One problem with "just-in-time" learning is that sometimes you don't know what would be useful to know. For example, maybe the problem you're working on is isomorphic to some problem in a field you haven't studied at all. But if you had studied the field at least a little, you would be reminded of it, and then you could study up on it more in order to see how it might apply to your problem. This is an argument for the style of learning in the above blog post: the more fields you have basic knowledge of, the better the odds that you'll have a vague idea of what might be brought to bear on your problem. I also suspect that learning stuff can develop useful skills even if you forget everything you learn, e.g. lots of MIT alumni profiles in their alumni magazine seem to say stuff like "MIT taught me to think analytically, which has been really helpful for [job that doesn't require science or math]". Figure you're overcoming your aversion to using System 2 or something.
I've experimented just a little for using Anki to retain technical knowledge, but it seems like the time investment for memorizing stuff is really high. And it's all a few clicks away on the internet, so I'd rather just look stuff up when I need it. But it's probably worth keeping in mind that if you want to get a good, semi-permanent grasp of something conceptually, it's optimal to space your study out, and try to answer questions for yourself instead of just imbibing information directly (IIRC a study showed that taking a quiz is a more effective way to review for a quiz than rereading was. Also, the Socratic method rocks, in my experience... kinda hoping that online education will eventually shift to that.).
(Note: I'm a novice as autodidacts/researchers go, so don't take my advice too seriously.)
It may be worthwhile for me to develop a sense as to when I'm able to fully understand something. I wonder how one would go about doing that?
Also, I've been using SuperMemo consistently for about six months now. I understand that rote memorization isn't really "learning", but it's helped actually learning in so many ways. At first, it took a while to set up the cards in a fashion I approved of, but now it only takes a few minutes to set up multiple cards, and 10 minutes at most to do the review every morning. I think the time I've spent in SuperMemo is well worth what I've gained from it.
Henceforth, I'm going to make a conscious effort to "embrace the messiness of things" when it comes to learning. I seem to do do that pretty well when it comes to my environment. ;)
At first, it took a while to set up the cards in a fashion I approved of, but now it only takes a few minutes to set up multiple cards, and 10 minutes at most to do the review every morning.
Could you try to summarize what made you faster at this? Maybe I'm missing something.
Do most of the exercises in a textbook, but seek help when you're stuck, and switch textbooks when you get stuck too often. Challenging exercises are where the deep learning happens. ETA: skip exercises when you "feel" how you would solve them, but randomly check whether this estimate is correct.
I went through an Optimization course last semester (CS, grad), so it doesn't really qualify as an "out of class experience", nevertheless reading it was quite optional, and, actually, the questions I asked myself were very similar to yours.
Especially in the light of those small remarks textbooks tend to make along the lines of "we don't have any more space here, so if you're interested, the excellent book by X and Y is a very nice read". As if they were referring to some light and entertaining book if the one you were holding weren't really enough to fill up your entire afternoon.
Instead, I spent hours on the part about Conjugate Gradients for example, coming up with different (mostly wrong) mental models, drawing various maps, and thinking about what's wrong with the way I try to study math. (I also ended up at #lesswrong, asking people how they study. Also brought home some ideas.)
So, in the second half of the semester, I upgraded my method to the proposed-by-some-people "ignore all the proofs, and generally, all of the textbook, try to complete the excercises that are likely to come up on the exam, and don't try to see everything". Which kind of worked: I understand most of the concepts, I can solve actual problems, and also, passed the exam. (As if that one counts as a proof of knowledge...)
But I'm still curious how studying textbooks is supposed to work. Like...
I used Spivak in my calc class in college, and it was a decent text. Having the answer key is extremely helpful, especially with the later chapters. There are some methods of proofs that are needed to solve the problem that aren't discussed at all in the text, and you are unlikely to be able to come up with them on your own.
Obviously, don't use the answer key unless you've spent a significant amount of time trying to find the answer yourself.
Spivak is not an easy book and is probably not a great way to learn calculus. It is, however, an excellent start to learning rigorous mathematical techniques and gives you great foothold for for later topics in math such as analysis. The calculus course I took used "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by Hubbard and Hubbard - it is essentially a slightly more modern and easier to read version of Spivak. The people I took that course with all had superb starting points for their later math courses compared to others (warning: selection bias! those who took the course were more interested in math, etc.), but we often lamented the fact that we weren't very good at actually doing much with multivariable calculus when it came to, say, physics courses. That's more because the differential forms introduced in Spivak are not the standard way of writing calculus in other domains than because what Spivak is doing is inherently harder.
In short: Spivak is hard. Taking a long time on it is not surprising in the least. It's probably not your most 'efficient' route to learning multivariable calculus, but it is a good way to learn to do proofs and think mathematically. Keep chugging away at it! Doing all the problems is probably a bit much and I, at least, would be more likely to not complete it if I were doing all the problems. Maybe you can find the list of used problems from a course and use that instead. Good luck!
(Edit: disclaimer, I've only used Spivak as a supplement to Hubbard & Hubbard)
You seem to be talking about Spivak's "Calculus on Manifolds", while the post is about "Calculus". Munkres's "Analysis on Manifolds" is a more didactically forgiving book than "Calculus on Manifolds" (and Hubbard's book works as preparation for both).
That's quite possible - I spent some time trying to figure out which was which but gave up and no longer have my copy. Which one is known as "baby Spivak"? (That's the one I was referring to.)
Hey I can input! I'm also a physics undergrad. Math textbooks are always tough. Go talk to a math professor and see if they recommend one for you. This is good because they know about where your knowledge level is and can suggest an appropriate book, plus you can come to them with questions. I do the same for physics textbooks too.
Do all the exercises, it should take a long time. I've done ~3/4 of the exercises in Griffiths E&M in the last four months, and that's a reasonable pace.
Most of the time when I'm reading a textbook, I just try to read everything, make sure I actually understand how everything works (i.e. derivations of results and formulas), and then skip all the exercises because of a lack of motivation. I can see how you'd maybe want to do lots of exercises if whatever you're learning is something you expect to use for itself, but a lot of the time (especially with something like calculus), I feel it's just a stepping stone, and I've found that when you're learning the thing you're actually interested in you'll be able to solidify your understanding of the basics at that stage.
Mind you, I write this as a data point, not as a recommendation. What works reasonably well for me may fail horribly for you.
(Also, I started filling out your survey, but then I gave up, 'cause I think my answers to some of the questions would've just been weird/not exactly answering the right questions.)
Hello, folks. I'm one of those long-time lurkers.
I've decided to conduct, as the title suggests, a quick and dirty survey in hopes of better understanding a problem I have (or rather, whether or not what I have is actually a problem).
Here's some context: I'm a Physics & Mathematics major, currently taking multi-variable. Lately, I've been unsatisfied with my understanding and usage of mathematics—mainly calculus. I've decided to go through what's been recommended as a much more rigorous Calculus textbook, Calculus by Michael Spivak. So far I'm really enjoying it, but it's taking me a long time to get through the exercises. I can be very meticulous about things like this and want to do every exercise through every chapter; I feel that there's benefit to actually doing them regardless of whether or not I look at the problem and think "Yeah, I can do this." Sometimes actually doing the problem is much more difficult than it seems, and I learn a lot from doing them. When flipping through the exercises, I also notice that—regardless of how well I think I know the material—there ends up being a section of exercises focused on something I've never heard of before; something very clever or, I think, mathematically enlightening, that's dependent on the exercises before it.
I'm somewhat embarrassed to admit that the exercises of the first chapter alone had taken me hours upon hours upon hours of combined work. I consider myself slow when it comes to reading mathematics and physics literature—I have to carefully comb through all the concepts and equations and structure them intuitively in a way I see fit. I hate not having a very fundamental understanding of the things I'm working with.
At the same time, I read/hear people who apparently are familiar with multiple textbooks on the same subject. Familiar enough to judge whether or not it is a good textbook. Familiar enough to place how they fit on a hierarchy of textbooks on the same subject. I think "At the rate I'm going, it will take me a very long time to get through this."
So...
Here's (what I think is) my issue: I don't know whether or not I'm taking too long. Am I doing things inefficiently? Is there a better way to choose which exercises I do and don't work through so that I learn a similar amount of material in less time? Or is it just fine that I'm taking this long? Am I slow and inefficient or am I just new to this process of working through a textbook cover-to-cover, which is supposed to take a very long time anyway?
I spend more time than I should learning about learning, instead of learning the material itself. I find myself using up lots of time trying to figure out how to learn more efficiently, how to think more efficiently, how to work more efficiently, and such things—as opposed to actually learning and actually thinking and actually working, which ends up being an inefficient use of my time. I think part of this problem stems from the fact that I don't have much of a comparison for when I can say "Ok, I'm satisfied and can stop focusing on improve how I do this act—and just do it already." I want to solve that issue now.
Which brings us to...
Here's my attempted solution: A survey! I assume many people here at LessWrong have worked through a science or mathematics textbook on their own. Mainly I'd like to gauge whether or not you thought you were taking a very long time, how long it took you, etc. I'd also like to know what your approach was: Did you perform every exercise, or skim through the book finding things you knew you didn't know? Did you skip around or go from the first chapter to the last? Do you have any advice on how one should approach a given textbook?
Here's the survey: https://docs.google.com/forms/d/1S4_-7_dxgmgprMbNhL1dNmX_0Zq9QrA9lpTl9ZHHxMI/viewform
I'm not sure how interested anyone but me is in this, but on a later date I could make another post showing the data. I considered checking "Publish and show a link to the results of this form", but I wasn't sure if that kept everyone anonymous or not. Also, feel more than free to post any criticism, shortcomings, improvements, etc. Have I left anything out? Is there anything you'd like to see me add? This is my first attempt at a survey like this and I'd appreciate any feedback (though I know it's not necessarily a rigorous survey, just a quick data-collection, I suppose).
I strongly encourage the posting of any textbook-reading tips or guidelines in the comments. I left that out of the survey so that anyone who's interested has immediate access to tips.
Here's an edit: Thanks for all the responses, everyone. Not only was my original question sufficiently answered (that is, it doesn't seem like I'm taking too long; there were only a few survey takers, but in between the comments and the survey answers, I'm not going at an extraordinarily slow rate). There's some very solid advice for different methods I might try to optimize my learning process. One that especially hit home was the suggestion that the large amounts of time spent "learning about learning" are such because it feels more comfortable than actually learning the material. In short, it's a safety blanket that makes me feel like I'm doing something productive when I'm really just avoiding what needs to be done. Some other useful pieces of advice are:
- Try being open to learning a broader range of materials without necessarily mastering each one. It might be the case that you need to know one thing in order to master the other, and need to know the other in order to master the one—trying to master either of them in isolation ends up being somewhat futile. Not everything needs to be "brick by brick" structured. (This was a lesson I found useful when I first learned that a number raised to the "one half" power was the square root of that number: Trying to master it in terms of the rules I already knew ended up in a thought like, "... Two to the third power is two times two times two. Two to the one-half power is two... times two one half times?"
- Though it may be uncomfortable at first, it could make learning easier to try the exercises before reading the chapter super-carefully; trying them before you feel ready to try them. You don't necessarily have to fully comprehend all of the proofs in the chapter to get through some exercises.
- Textbooks might just be the wrong way to go in the first place. Try resources like Wikipedia, math blogs, and math forums.
- "Don't use the answer key unless you've spent a significant amount of time trying to find the answer yourself!" (This may seem obvious, but a few years ago, I'd spend a couple of minutes on the problem, not understand it, look to the answer key, and wonder why I wasn't learning anything.)
- Skip exercises when you feel you could solve them, but randomly check whether this estimate is correct by doing the problem anyway. (I like this one a lot).
- Talk to a professor!
- It may be the case that you learn well via just reading, and not spending so much time on the exercises.
Here are some websites/blogs mentioned:
(Blog) Math for Programmers - http://steve-yegge.blogspot.com/2006/03/math-for-programmers.html
(Blog) Annoying Precision - http://qchu.wordpress.com/
(Math Forum) Mathematics - http://math.stackexchange.com/
Excellent, excellent stuff, though. Thank you. :) There's a lot of material and advice for me to work with—while simultaneously making sure I don't avoid my work by hiding under the guise of productivity.