that skill of problem finding is apparently much rarer than problem solving.
there are standard ways to go at it however, such as changing the hypoheses in a known theorem, or changing the conclusion.
i.e. if something is true in dimension one, ask what happens in dimension two. or if a differentiable function has a certain property, ask what happens for only continuous functions.try to get the habit of noticing what is not said, as well as what is said. and any time someone says "we do not nkow what happens when..." there is an open question.
it took me my whole life to notice it, but there is a standard theory of finitely generated classification of modules over pid's, but not more general rings, as far as i knew.
on the other hand, i knew that a dedekind domain is basically q ring that is locally a pid. thus there should be some weakker classification theorem for finitekly generated modules over a dedekind domain.
guess what? there is, and it appears at the end of books like dummitt and foote. but asking this question should have been obvious. notice many oif these books do not make it clear at all why such a questiion is an obvious extension of a well kmnown theory.
we are not very good at teaching how to ask questions in math, and it does not appear in many places.
one exception is basic algebraic geometry by shafarevich. when he explains a theorem, he ioften asks an open question extending what ahs been explained. but because a question, once asked is easier to answer, many of his questions have now been answered.
e.g. in the section on cycles, he posed the problem of fiun ite generation, but clemens answered it about 20 years ago, and in the section on uniformization, he posed a question on the structure of universal covers of algebraic varieties, and now a lot of work has been done on it by kollar and others.
I was so nose to the ground when ireqad shafarevich, i was only interested in the theorems there and ignored or did not even observe the open questions.
so to find open problems, you have to learn to be looking for them. As briefly put above, you have to spend some time reflecting, i.e. thinking, about what you have learned.
there is a famous book about this, called :"how to solve it" by george polya. read that. one tip irecall from that book, is that soklutions to problems are like grapes, they come in bunches. so when you have an idea that solves one problem, look around for another that it solves.
i used to not do this, and when i published a paper with an idea, I would notice later someone else had proved much more than i did with the same idea. I was occuopied with my problem, and having worked hard and long to find the key to its solution I was satisfied. It did not occur to me to maximize the productivity of my idea.
Ideas are few and far between. To expect to generate a new one for every problem is hard. So push them as far as they will go. But not forever. Try to think of a new one occasionally too. I also have had trouble doing this,a s we all have.
