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Is solving all of the exercises in a textbook a good idea? I'm particularly concerned with textbooks on mathematics. I have this obsession that I should solve all of the problems that a textbook has. It takes a lot of time and energy but usually I'm satisfied with the end result being me having a better understanding of that particular subject.

Any similar experience of this sort? How's this going to work in the long-run?

closed as primarily opinion-based by Bryan Krause, user3209815, Coder, J.R., user67075 Aug 21 '17 at 22:14

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  • 8
    If it's helping you learn and not taking up too much time, then I don't see any negatives. – astronat Aug 20 '17 at 22:32
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    At some point, the exercises help more with memorizing vs conceptualizing. I've forgotten almost all of my integral tables, but I know how integration works and can massage an equation into something I can look up. – Nick T Aug 21 '17 at 4:20
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    There is also the question of what you count as solving. If you are doing it only for yourself anyway, there are some shortcuts you can take. Say you are looking at the hundredth integral in a long list in some book and you immediately see a working substitution which reduces it to something you know how to integrate, without even picking up the pen. Then you should count it as solved and move on to the next, in the hope of finding something more interesting. Sure, you may not have inserted the boundary values, but after a hundred integrals, this part is purely mechanical anyway. – mlk Aug 21 '17 at 8:30
  • The Art of Computer Programming has a good number of unsolved research problems among its exercises. Fermat's Last Theorem was among them, but it was downgraded because it's not unsolved anymore. – gnasher729 Aug 21 '17 at 19:53
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It depends enormously on your personal objectives, apart from your personal predilections. For example, if a significant goal is to advance your understanding of mathematics, then obsessing over exercises (many of which are contrived busywork in undergrad textbooks, and sometimes in grad-level textbooks in the U.S.) is a dubious investment of your personal resources.

For one thing, apart from the articiality of some of the exercises, many of them will be semi-incomprehensible if you've just read the chapter they appear after... but obvious after you've read further! A significant reason for this is that mathematics has developed with various goals in mind, so that the most important enduring concepts and facts refer to important phenomena... not just to some artificial choice of linear logical development as is the common style in textbooks.

On another hand, if you do not aim to be a professional mathematician, or if somehow you have a lot of spare time, sure, why not do whatever you want? Indeed, another common trap of studying mathematics is being too obedient about following some syllabus or textbook, as opposed to following one's own curiosity and interests. It is subtler to parse the situation that your impulse is to do all the exercises... :)

Another practical point is that, at some point, probably soon, unless you severely restrict what books you look at, there's no way you'll have time to do all the exercises in detail, even if you are a whiz-kid. There are too many, and sometimes they are prankish. For example, the "exercises" in Atiyah-MacDonald's "Commutative Algebra" (a misleadingly slim volume) are mostly "theorems" one would find in other books on the same topic.

And, then, there's the point that novices' "solutions" to difficult exercises are often severely suboptimal, even if "successful". Sure, it's good to think about issues, but, at the same time, you'll be able to approach those questions far more wisely later (if you still care, and the things haven't become completely obvious anyway!).

But certainly no one is "required" to do all the exercises, despite some propaganda on the internet. For that matter, it is probably not optimal for most peoples' circumstances and goals. Still, taking an extreme stance, maybe the world will end tomorrow, and if you want to spend the evening doing exercises that you find enchanting, I'd be the last one to try to discourage you. :)

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There is no right answer here, although you need to think about the time commitment involved. For instance, I doubt you need to solve every problem in an introductory calculus book. However, in a textbook where the number of problems per section or chapter is limited, it may be instructive to you to solve the problems that aren't assigned.

However, ultimately, this is a function of how you as an individual learn best. For some people, additional repetition of skills can help; for others, they can pick it up much more quickly, in which case the additional problems may not yield many benefits.

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No, it's not. All questions are not equally good exercises. You'd be better served to purchase 2 books on the subject (ideally that take different approaches to the subject matter) and solve 1/2 the questions (picking out the better ones) in each text. This is of course contingent on wanting to learn a particular subject very thoroughly, and that solving questions helps you a lot. In many cases, it'd be a better use of your time to look at other related subject rather than spend so much time on one in particular, but I leave that to you.

Frankly, I can imagine few exceptions to this. The one that I can think of is that a text is very sparse in the questions is has and all the questions are good, but even then, you should evaluate a question's usefulness independently of others. A blanket policy like just doing all the problems in a text is very likely to to have a lot of redundancy in the problems you're solving. Textbooks aren't cheap, but there's a lot of material online nowadays as well if cost is an issue. If you're studying a field that's so small that there simply aren't many textbooks on the subject, chances are you'd be well served to read papers rather than textbooks.

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    This raises the question of how a novice can pick out the better problems. – SolveIt Sep 22 '17 at 17:15
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As a student of mathematics, I like to state that none can learn mathematics without solving mathematical problems in hand. Exercises in textbooks are a collection of mathematical problems. A good book contains beautiful problems. When you are learning a branch of mathematics you must select a good book to solve its exercises as much as you can. Solving problems is more important than reading the text.

  • Surely we can still learn the concepts without solving problems? We may not be as adept at applying them, but that seems secondary in some cases. – Weckar E. Aug 21 '17 at 9:41
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For your grades: doing as many exercises as you can is always a good idea, especially in math / physics / computer science.

For your mental health: obsessions are dangerous sometimes. Once you've done all the exercises in a book, pick another text and solve only a few problems ;-)

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