I find value in reading and doing exercises in books about areas of mathematics that I didn't get to explore while in school. I try to do each exercise in each chapter with only the material contained in each book, but I sometimes can't think of a solution in a reasonable amount of time without using outside resources. In short, here's my main question:

When is it appropriate to use outside resources to complete book exercises during self-study?

As background, I am a software developer with a bachelors degree in math.

In terms of technology, I consider using outside resources to solve technical problems as a short-cut. It is more efficient to use a proven statistical library to do computations on a data set, but that won't necessarily teach me what the results actually mean. When I decide to work through a book, I work under the assumption that the book is self-contained, except where it says otherwise in the book itself. I realize that outside resources can accelerate my understanding, but I fear that I may miss the subtle points the exercise intends to emphasize by doing so.

In other words, how can someone ensure that eir learning of a new subject isn't harmed by turning to Google for help?

  • $\begingroup$ How long does it take till you turn to google? I'm in a similar situation and what hurt me the most was only attempting problems a couple of times and then turning to other sources. I learned to wait a couple hours, if you have time, days, and then come back to the problem. Re-read the chapters before the problems. Then if you're still stuck, I'd say there's no harm in looking elsewhere for help. This is what helped me, but it might not be the magic potion for you. $\endgroup$ – Dando18 Jun 5 '17 at 14:55
  • $\begingroup$ I usually wait a few days at least before turning to google. Self-study has the benefit of a lack of deadlines! However, if I spend too long without moving on, I find that I start getting off-track. Thanks for your input. :) $\endgroup$ – Jeff Freeman Jun 5 '17 at 15:04
  • $\begingroup$ If you've been thinking about a problem for a few days and not making progress, then it's certainly okay to look up an answer. The alternative is to put the problem on the back burner for a while and return to it later. $\endgroup$ – user49640 Jun 6 '17 at 21:38

Not every mathematics book is self-contained, and in particular, many include exercises or problems that require either a good deal of cleverness or knowledge of techniques that are not clearly explained in the book.

On the other hand, the point of the exercises is to gain technical strength and conceptual grasp by attempting to solve them on your own. After trying enough that you are convinced that nothing you are ever likely to come up with on your own, then that is the time to look to outside resources for the critical hint or solution technique. This site is a good resource for those hints or solutions.

Outside material can be helpful in other ways as well.

For example, if you are working your way through Munkries Topology, then the book "Counterexamples in Topology" only provides direct answers to a smattering of exercises, but it does provide examples that help you understand, for example, the sense in which topological spaces obeying different separation axioms are different.

  • $\begingroup$ Thank you for your input! $\endgroup$ – Jeff Freeman Jun 12 '17 at 14:05

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