Different people learn in different ways. Some people can pick things up in the lecture, others need time to go home after and reconstruct everything for themselves. While it's certainly impressive that some people can "get it" in the moment, it's by no means required for success in STEM, especially as a grad student. You should try to avoid comparing yourself to others, since you may have different areas where you excel. You should also keep in mind that you only see what other people project about themselves to the outside world and you don't have access to their inner monologue; because of this, I think many grad students tend to overestimate the abilities of their peers and underestimate their own. In the end, there's a certain rate at which you can absorb technical information, and there's no point in simply wishing the rate was higher. While you can increase this rate with lots of hard work and experience, that's not something you should expect to happen magically or quickly.
Having said that, if your goal is to be able to ask questions in lecture, here are some thoughts on various strategies to try:
- most lectures start relatively easy and then ramp up in difficulty. Try to listen carefully and follow the logical steps of the lecture, see if you can identify the first spot where you start feeling confused. Take a moment and try to pin down exactly what you are confused about as precisely as possible (is it an undefined word; a new concept; an old concept being used in an unfamiliar way...) and ask a question about this.
- don't be afraid to ask about ambiguous or undefined symbols; it could well be that simply getting over this "mathematical language barrier" will make the content a lot clearer to you and you can spot tricky steps more easily.
- review the material at your pace beforehand so you know what the confusing parts are in advance.
- do a lot of problems in this subject so you become more familiar with standard notation and the ways of thinking about it, which in turn will increase the rate at which you can absorb information in this area.
- don't try to follow and absorb every detail in the moment, but only try to follow the logical outline of the lecture, since you will be able to fill in details later. If you don't see how the lecture logically flows from one part to the next at a high level, it's probably work asking a question to clarify the logic.
- if the lecture is covering a general case of something, try reducing various statements to a special case you understand well and see if you can follow the details there. For example, if a derivation is being done in three dimensions, see if you can follow how the arguments work if you project out one of the coordinates.
- try to explain tricky concepts out loud to yourself and to other people. While doing this, try to think of ways you could explain the material that weren't the way that was done in the lecture. This may either lead you to a better explanation, or to a realization of why the material was structured a certain way. Getting a feel for how your lecturer likes to think about and explain things, can give you some intuition for which aspects of what they are talking about are the most important.
To summarize: (a) definitely speak up to clarify unambiguous or undefined notation, since you will have no chance of following anything if the symbols are not defined, (b) working hard outside of class can help you follow things more quickly in class, and (c) try to identify the most important points and focus on those, leaving more detailed calculations or proofs to understand later. This frees your brain up to think about the material at a higher level; it is much easier to ask questions if you are following the structure of the lecture rather than trying to follow every step line-by-line. Note: "higher level" here is not a value judgment that some material is "better" or "more intrinsically interesting," but essentially a kind of coarse-graining of the material where you assume certain calculations / proofs work out and see if you can follow the logical implications of the results that are derived. You want to aim to be able to (a) identify the key results in the lecture and (b) for each result, understand what all the inputs are (and why they are needed), what all the outputs are (and what outputs you'd like to get that you can't get), and what you can do with the result. You do need to understand the details eventually, but you don't need to understand them in the moment, especially if you feel it is hurting your ability to understand the overall lecture. I suspect that many questions you perceive as being very intricate or detailed, are really coming from someone following at this kind of higher level finding that two results don't seem to fit together or making a connection with another subject they know well (again keep in mind you don't know the internal monologue of other people).
