What should I be doing in a grad level math course? I have an undergrad degree in an engineering field and switched to applied math for a PhD. In engineering, I have had my share of good and bad professors but I knew what I had to be doing during a lecture (math or engineering). I would simply take notes and ask questions for what I did not understand. Graduate level lectures, however, seem to be an all new ball game.
They are more rigorous (with more of Theorem - Proof-  Corollary - Applications type structure) and its easy to get lost in case you don't remember something from undergrad math. I used to ask questions but I felt that I am slowing down the class with basic questions. My question is essentially, what should I be doing in such classes:
With Theorem-Proof-Corollary classes, what is the most effective thing for students to be doing?
 A: Prepare yourself in advance of class.
Read the relevant material - the next section not yet covered (in anticipation of new information), and review your notes from the previous lecture,  and do this ahead of class. This will give you a chance to read and take notes on definitions, theorems, and working through proofs. Write the proofs out in order to make them explicit enough to understand "down the road", later in the semester (and connect those proofs to the newly covered definitions and theorems). Personally, I had always rewritten my class notes. I find that concepts and information get "etched" in my "brain" through the act of writing: taking notes as I'm listening, rewriting them, taking notes AS I'm reading...moreso than when I do not much else than read and listen.
That way, you will be "primed" for the material covered in the lecture, not necessarily having mastered it, but you will have developed enough familiarity with the material covered in lecture to be able to really listen, having developed a keen sense of what you grasp and don't grasp. You will be prepared to answer questions left over from you pre-class preparation, and for anything you don't understand in lecture, you'll be in a position to ask intelligent, informed questions.  So ask away. My general impression has been that professors relish good questions, and graduate level classes are often "seminar format" anyway, where you are expected to interact intelligently (and if not yet, they soon will be!)  
So don't think of "homework" as something to do after class, but before class, as well. Between your self-study and your attendance at lectures, you'll be well prepared to complete problem sets, to start working them as soon as they are assigned. 
You'll be a "step ahead" of the game with such a strategy!
