How can I learn to recognize the obvious questions? I have heard and seen several references to "obvious questions", "obvious axioms" and other "obvious" things (I am not referring to obvious results!). Now, in a seminar I am taking, at the end of each talk, the lecturer ask us for questions, and then tells us some questions we should be asking ourselves. Once he states the questions, they feel very natural to ask. He does not refer to them as obvious, but it seems like the term perfectly fits.
In category theory, you gain a feeling for the obvious axioms after spending some time with the subject. I understand that it takes a while, because if the subject is new to you, you do not have an overview and need to settle in first. But this seminar is on much simpler topics and the questions the lecturer in this seminar is presenting are not complicated at all.
An example of such a question: We derived an inequality, and he said we should be asking ourselves how sharp the inequality is. This seems glaringly obvious to me, but only in hindsight. I have proven sharpness of an inequality so many times, it is regularly part of any discussion that involves an inequality, yet it didn't come to me to ask that.
After some unsuccessful research on Google and here, my question is this:

How can I learn to ask the obvious questions?

To be more concrete about the advice that I am seeking: What are specific learning techniques I can apply to develop a stronger foresight for obvious questions? Alternatively, are there tools that I can use to systematically enumerate these sorts of questions?

I think the best way to discover "obvious" questions is to act as though you're going to present the material to others. 
Presuming one of a few pedagogical approaches, then preparing the material in this way will force you to anticipate questions in such a way that you'll tend to see these "obvious" questions materialize.
