What is a recommended strategy on exercises in a mathematical textbook at graduate level? [duplicate]

Question

When I read a mathematical textbook at graduate level to learn and master a particular subject that I am unfamiliar with, I always struggle with the strategy on the exercises. Sometimes I spent one hour on one exercise problem without making any progress. Sometimes I skip exercise problems with guilt and worried that I may not have learnt the material well. I can think of the following possible strategies on exercises:

1. Ignore all of them. Just focus on reading the main body of the textbook.
2. Set a time limit. Say for every hour I spent on reading the main body, I spent twice much time on solving exercise problems. I try to solve whatever I can within two hours and skip the rest.
3. Never move on to a new session unless I have solved all exercise problems in the previous session. If I get stuck, spend hours working on it and if still no success, look it up or ask somebody until I understand the solution.
4. Spend a reasonable amount of time on each problem and if I could not solve it, read the solution from a solution manual. This strategy seldom works as for most books on graduate level, solution manuals are unavailable.

What strategy do you recommend? I personal feel that Strategy 3 is the most time-consuming but it is also the only one that does not bother my conscience. But because it consumes a huge amount of time, I am not sure if it is wise to adopt it.

Answer 1

Disclaimer: My thoughts about this are somewhat unconventional, so keep that in mind.

I don't have a high opinion of the sausage-link structure of (advanced) mathematics courses, where you get a chunk of "material," and then exercises on that material, and then another chunk of material, and so on until the term ends. Since nearly all courses (and books) use this structure, I think the students who do well with it tend to be overrepresented among grad students due to survivorship bias, but I don't think it's well-correlated with any meaningful measure of mathematical talent. So my first observation is: Don't sweat it.

My second observation is: Some people just have innately less sequential, more integrated, and higher-context learning processes, which creates a mismatch with standard instruction. The bad news is that very few mathematics professors seem to be clued into this and/or have no intention of modifying their course structure to support a variety of learning styles.

However, if you're working from a book, you can reverse-engineer and restructure as you see fit. So skip around, use multiple references, work on what makes sense to you at any given time. Above all, don't feel constrained by the structure of the book and use the exercises as a check on your understanding rather than a goal in itself. You don't have to do the exercises from section 4 right after reading section 4. You might get more out of it if you come back to them after section 7.

This is going to take some experimentation and self-observation and not all books will be equally amenable to deconstruction. But we now live in an age where even fairly obscure topics will have multiple presentations that can be located with a few keystrokes. You just need to find what works best for you.

Answer 2

Treat exercises as optional adventures, not as boss monsters you need to beat before you can get to the next level. Use your own interests and common sense in deciding how hard you want to attack any specific exercise. You can still return to previously unsolved exercises from the next chapter.

Every once in a while, you'll encounter an exercise that is used later in a proof. That's the only case where you strictly need to solve it, and thus spending extra time, getting help etc. are good ideas. Note that such exercises are usually among the easier ones, unless the author is doing their writing wrong.

My strategy is generally to keep track of what I have solved by putting checkmarks near solved exercises and sometimes scribbling the main ideas of the solutions in. I don't do this with proper books, but I don't hesitate to do it with printed-out chapters, so I have several books or sets of lecture notes printed out and stapled together chapter by chapter, with lots of marginalia documenting my way through the text. (A sufficiently heavy-duty stapler -- say, able to staple 50 sheets -- plus some good tape for the "spine" of your stapled brochures, and some extra paper to use as cover are sufficient to make such printouts no less usable than books, but with the additional affordance that I can scribble around in them without feeling like a book-mutilating barbarian.)

Answer 3

Unfortunately, there is a tendency for the pedagogy to get worse as the topic gets harder. Probably because less people are learning the material (less investment in methods). Also more of a sink/swim Darwinian attitude.

However, the fundamentals of good pedagogy remain. If you are finding the material too difficult, you need to come up with methods to create progression (easier problems first) and assist yourself.

My advice:

A. Do a combination of 3&4. Spend a reasonable amount of time and then go get assistance (since the solution manuals don't exist). If they do, use the guides.

B. Try to use the minimum help first (again, within reason in terms of time management). I.e. See if just the answer is enough (not a full solution). Just look at the first few lines of a solution. Ask for a hint. Etc. The reason for doing this is that you are still learning the material, via some struggle. This helps it stay in your mind. If you just look at the solution and say "Oh...I get it", than you won't remember or benefit. Even if you get the crucial insight, you can still internalize it by working the entire problem.

C. Similarly to (C), after getting any assistance (limited or full solution), DO THE WHOLE PROBLEM over again. Act as if you haven't gotten the help and just pretend you are solving the problem from scratch (write it all out, too). This sounds hokey but it actually works. Helps to engrain patterns and remembering into your brain. Remember learning and pedagogy are issues of practical human psychology. We are NOT silicon. We are flesh, with imperfect memories and processors. As Aristotle said, man learns by imitation and practice. Of course if you can nuke it out, best. But you can still create brain grooves, by "imitation and practice".

D. Look for easier texts and easier homeworks. And then actually work those problems. It will give you confidence and help groove some things into your brain, that can then bubble up and help you on the harder problems. Also, if you request assistance, it's nice to have some basis to demonstrate.

E. Don't be proud. Use the TA. Use the instructor. Yes, they probably want to concentrate on research. Yes, they may prefer the students that are brilliant and don't need help. But if you are respectful and give evidence of working hard, you can win them over.

F. If they are "dicks" and you can't win them over, still don't sweat it. Keep getting help--be persistent. Don't react to brushoffs by anger OR by giving up. (Or by complaining commentish "questions" on this site.) Just smile and keep doing what matters to you--getting the help and learning the material.

Answer 4

The nature of grad-level "textbooks/monographs" varies wildly, so there's no single answer that can apply to all.

In the best case, which is not at all universally attained, the "exercises" are iconic examples illustrating the theorems of the chapter, with examples already occurring in the chapter. Great! But/and then there's not much work to be done, after reflecting on things enough to see that, yes, truly, these examples are "more of the same".

In one sort of "worst case", the chapter gives definitions, and all the theorems of interest are given as exercises. This is horrible and ridiculous.

A more typical middle-ground scenario is where the exercises do mention iconic examples, but with quite inadequate preparation in the preceding chapter, etc.

And in almost all cases, to take the book literally, everything is lined-up.

The seldom-noted point is that many things become vastly clearer with hindsight. So sitting on an exercise until solved is possibly the most foolish strategy... if, at least, it's a good exercise, oop, and if it's not (tho' how to judge, with insufficient info?) why do it at all?

Conceiving as mathematics as "a school subject", with orderly development, exams, homework, clearly specified prerequisites, etc., makes it much harder to understand, in fact.

EDIT: I forgot to say... that, really, why need there be "exercises"? Are there exercises at the end of chapters in novels? At the end of movements of symphonies? In coffee shops or bars? Math-as-filter has enormously corrupted the sense of what math is, and how to understand it.

Answer 5

TL;DR: Relax! Mathematics takes time!

I thought I would provide my perspective as a current pure mathematics student who is fairly successful at his studies and work. I read and collect a lot of mathematics textbooks in my spare time and I too have similar issues when learning new mathematics. In my earlier years as a mathematics student, I would always rush through reading textbooks and get flustered when I couldn't solve a problem quickly or understand the material. The moment I realised that it takes time to solve problems (often weeks or months) and it takes much longer to read, 'digest' and understand mathematics then other subjects, was the moment when my mathematics ability took a big leep forward.

When reading a mathematics textbook I will usually 'read' it 3 times over. Here what I usually aim to do on each read:

• On the first attempt, I read through the material very casually, not concerning myself too much on whether I understand every little detail or not. Here, I just want to try and pick up the definitions and the main concepts involved with the topic. I will usually try and remember how the concepts are developed and the results that are developed along the way if possible (e.g. lemmas, theorems etc.). I don't usually concern myself with the details of proofs too much. I might attempt some exercises if I think I can do them, however, sometimes I can't even do any at this point.

• On the second read through, I usually try and fill in the gaps of my knowledge from the first attempt. Here I will try and understand the details of proofs more which is much easier now that I have a bit more of a broad overview of the topic and have a bit more of a 'feel' for how things work. I usually try and get through as many exercises as I can on the second read but if there are exercises that I can't figure out within a reasonable time frame, I will just mark them out and come back at a later date when I have better skills in the subject.

• On the third read I don't really do much reading. Here I will just skim over all the content and mainly concentrate on understanding the proofs I still couldn't understand on the second read through and fill in any more details I couldn't quite work out before. Usually by this point I am fairly proficient in the subject and can figure out most of the exercises and try and complete some of the ones I marked out earlier that I couldn't do. Of course there are exercises I can't often do, and this just the nature of mathematics, sometimes you will just need to give it more time (possibly months or years) for you to develop the problem-solving skills required to answers these questions.

Reading and learning mathematics takes a lot, I repeat a lot of time to master. It might take me 6-12 months before I am on my final read-through of a textbook, and again, that's just the nature of mathematics. It takes time for concepts to sink in, for your skills to develop, and it takes even longer and longer as you progress to higher levels of mathematics.

Of course you should seek out help, either online (e.g. math stackexchange) or in person if you think it will benefit your learning, however, the take home message from my answer that I wanted to provide is that learning mathematics takes time. Don't overly concern yourself if you can't do problems or don't understand everything on your first attempt at reading through a textbook, and as has been suggested in other answers, try and stay away from seeking the solutions to problems you can't do. Keep reviewing and coming back as your skills develop over time. Patience and persistence are keys to succeeding at mathematics.

Answer 6

This depends on the course and what you want to get out of it. Some courses you take because you are deeply committed to that topic and want to know everything and attain deep insight. Others seem less "fundamental" to you. For me, Analysis and General Topology were in the first category and Algebra and Statistics were in the second. Over time, I was able to attain deep insight into my key subjects but only a general familiarity with the others. There is a lot of mathematics and we long ago passed the point where any single person could "know" all of it.

So, in the key subjects, your strategy 3 may be the best. In particular, in those courses and for those books you should be able in theory to solve any of the problems given. There are, however, a few books that are exceptional in that they deliberately hide research problems in the "exercises". Those books generally advertise the fact, or sometimes mark the questions that are, at time of printing, unsolved.

At the other extreme, your strategy 1 is likely to ever be successful if you actually want to learn anything. Reading is a passive activity. The exercises are there to get you engaged with the material and that is where learning happens.

But the other two strategies are probably enough for those courses that aren't key building blocks in your math psyche. Learn enough so that if it becomes necessary in future to go deeper you have the basic foundation in place on which to build.

But there is one key caveat. Mathematics can be subtle. You may think you have a solution, but don't. So, as with other learning you need a way to get feedback on what you produce. Perhaps the professor is willing to give you that (and hopefully will, at least on the assigned problems). Another way is to form a study group among students to examine and reflect on one another's work.

And don't forget that reading a solution to an exercise is worth very little in actual learning. Again, it is a passive activity that is unlikely to form deep associations that lead to insight. So, back to strategy 3 for the important stuff.