soft question - Making sure that you have comprehended a concept

There is a famous quote from John Von Neumann:

“In mathematics you don't understand things. You just get used to them.”

I don't know for sure what this means but, perhaps it means that asking if you understand something is a question that does not really make sense, in fact your question is really asking: What is the definition of understanding. Such a question may be doomed from the start. Even if we could agree on some definition and you were to apply it to some mathematical concept I am sure that at some later time, after you have learned other things, your new perspective will make you feel like you never understood that concept anyway. So my advice is to take your concept and get used to it. Here are some ideas on how.

1)If it's a definition try to think up some examples and non-examples. The non-examples are especially useful when your concept is adding an adjective to something you already know; like adding "prime" to "number"

2) I it's a theorem try to identify exactly where the hypotheses are used in the proof, and try to think up counterexamples to the statement as you remove those hypotheses.

3) If there are exercises, ie if you are reading a textbook, try them. Even if you don't get to them all at least read the statements.

4) After you think you are used to your idea try to explain it to someone. This is a really good way to see if you have overlooked something.

5) As to the remark about moving on to the next few pages: Don't expect to go through a book line by line and not have to go back. The stuff later in the book may help.

In short if you work a lot with your concept it will cease to intimidate you. Don't worry about achieving understanding (equivalently seeing the true meaning), This will never happen, which is fine since it's the trying that is important anyway. So keep trying, and have fun doing so!