How to deal with math problems if there is no solution book and no helpful teachers?

Question

Since a short time ago, I refuse to try solving math questions of which there is no solution available, unless I'm working on an own/open problem in my thesis. In subjects however, I refuse, because it is frustrating to get stuck for days and knowing that a solution exists. So knowing that I miss some important way of thinking.

Now you might think Why? If you can't solve it, just look at the course material and if that doesn't help then ask the teacher. Yes I tried that. By both mouth and mail. Unfortunately a lot of them don't like to get disturbed by students, because they are actually not real teachers, but very busy researchers who hardly have time for anything else. I also tried google, mathstack and asking students, but without a solution book it seems like it is always possible that nothing helps and you thus stay where you are: stuck.

Having said this, is there a way to learn without solution books but with the guarantee that you can always get access to the solution for the problem you work on? Maybe I do something wrong with the ways already tried?

Answer 1

You should find someone who is willing to spend enough time helping you get unstuck. The generic answer is: asking classmates, asking professors, asking on MathStackexchange and MathOverflow. Of course when you ask a question you should have thought about what you really don't understand. Do you know the definitions? Are there similar examples in the book? Can you ask a simpler question that you don't know the answer to? If you spend some time with your book and you're not making progress then you should ask for help if there's no solutions manual.

is there a way to learn without solution books but with the guarantee that you can always get access to the solution for the problem you work on?

No. Ask people for help and hope that things will get resolved.

Answer 2

frustrating to get stuck for days and knowing that a solution exists. So knowing that I miss some important way of thinking.

So your solution to missing some important way of thinking is ... to not think at all?

If you try hard to solve a problem, you are learning. You learn what doesn't work and why. That's a valuable lesson, and it's how you get better. In fact, I'd venture to say if it takes you 4 days to solve a problem, then even if you don't end up solving it, you probably learned more than the person who solved it in 1 hour.

In mathematics, knowing the exact solution of some made-up problem usually isn't the key. After all, the exercise question tells you the conclusion of the solution. You already know the result from the onset. You just have to prove it, and it is the things you learn from proving it that are important, and that's my point: even if you don't manage to prove it, you still did worked with the details and learned similar concepts to the person who did manage to prove it. You just learned it slighty differently, e.g. somebody who proved it learned "okay, this technique works in this case", while you learned "okay, this technique doesn't work in this case". Both are equally valuable.

Answer 3

Actually, you will learn more without the hints provided by a solution, provided that you can find a way to get some feedback on your attempts. Perhaps your professor or someone else can provide that.

But you should investigate the Moore Method of teaching mathematics, created by Robert Lee Moore and used by some of his successors. It is, roughly, a discovery method of learning higher math with few hints (hmmm, no hints). Some of Moore's students have become outstanding mathematicians and educators.

The method sounds a bit brutal, of course.

However, if you are in a graduate program, you need to understand that being able to re-prove things that are already known is a lower level skill than true mathematics. Even being able to prove things never proven before, but having been given a good statement of the problem requires only a lower level of insight. But you have to achieve at least that before you can reach the highest level of insight - having a good idea of what might be proven and thus worthy of exploration.

I suggest you press on. But also that you find some way to get feedback. If you go wrong somewhere it is good to know where and why so that you don't develop misconceptions.

However, to avoid frustration, it is also useful to put hard problems aside for a while as you work on other things - or even just take a break. Trying to run your brain on "nitro" all the time leads to burn out and poor results. Give yourself a break in general, and specifically a break from any problem that is eluding you at the moment. Insight will likely come, but it can't be scheduled or forced.

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For what it's worth, I don't think I had the "higher levels" of insight into mathematics until I'd completed by doctorate.