A Quick and Dirty Survey: Textbook Learning

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'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.)

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...

• what is the goal of people when reading textbooks? being able to solve real-world problems? passing exams? solving all the excercises? getting the warm fuzzy feeling of having eaten a huge book with lots of formulas while getting that "yes I understand" feeling that may or may not be the same as really understanding stuff?
• what is the goal of people who write textbooks? is the fact that they are hard to read an unavoidable thing, a way-too-common flaw or... are there people who read math like I read MLP fanfics? Is it possible to fix this?
• and also, the statistics you mention. About the average WPM when reading math books...

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)

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.)