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I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.
I have a couple of pieces of advice:
Do not be intimidated by the introduction. This is where the whole paper is laid out,so it typically won't be easy to follow. The author will expand sentences into whole pages later on.
Take notes. I usually get a legal pad out and start copying down the paper, and fill in things, and make notes to myself or work out small examples.
Talk to people about the material. If you have an adviser or someone you are going through the paper with it is often helpful to present portions of the paper if not all of it.
Skip confusing bits. Feel free to black box certain parts. If you get stuck on something, don't worry, one sentence should not keep you from progressing through the paper. Box it off, pretend you understand the conclusion and make a note of it. Take that piece as a fact and move on to the next part of the paper. The goal is to get something out of a paper not understand it line for line the first time through.
Come back to the paper. You may want to go over the paper multiple times, take several passes at it. At least this is what I have to do when I am reading something. The amount of time it takes me to assimilate different levels/types of arguments gradually decreases, but it does take time.
I don't spend a lot of time reading papers right now, but this is what I try to do if I really want to understand something.
Edit: It has become easier and easier for me to read and digest papers over time. I am still slow, but I can get through a lot more. The above process is sort of a set out routine to help you digest it, eventually it will become obsolete.
How I read a paper really depends on why I'm reading the paper.
A lot of papers I go to because I have a specific goal. Maybe they have been cited elsewhere as containing a proof of something I want to understand. Or a different proof of something I already know how to prove. Or maybe someone refers to the paper as having a particularly lucid explanation of something. If that's my goal, I go straight to whatever is in the paper that I want, assuming the organization of the paper makes this possible (not always the case, but very common, at least with articles written after say 1950). In doing my close reading of the particular thing I want, I often take notes. Depending on the results of my reading the thing I set out to read, I might go on to read other parts of the paper. But I generally don't.
If I'm reading a paper without a specific goal, I generally start at the beginning and read the introduction. Ideally this contains the statements of the main results, but if it doesn't, I skim the paper for those, and also the statements of any lemmas, corollaries, etc. 19 times out of 20 my reading stops there, because I find that, beyond the statements of the theorems adding to my general awareness of the world, I'm not all that interested in the paper. That 1 time out of 20, I will probably try to come up with the proofs on my own, using the paper's introduction as a really big hint. Whether I succeed or not, I usually end up giving the paper a pretty thorough reading.
So most of the time, for me, "reading" a paper does not mean reading the entire paper, or even reading a large contiguous chunk of the paper. I could probably fit all of the papers I have ever read beginning to end in a regular sized binder with tons of room to spare. I think I am far from alone in this. In fact I think it is probably counterproductive to insist on reading things beginning to end, and I would advise that people who are new to the mathematical literature (e.g. students) suppress the desire to do this. Generally speaking, if you are new to the literature in a given subject, you will waste a lot of time if you read this way.
I generally have a "top down approach" to reading papers. More often than not, I am driven to a particular paper looking for a particular result. So I would start by looking at the statement of the result and try to see what background is needed in order to parse the statements which appear there. Once I have understood the philosophy of the result, I think about/look at the proof. Again, if I come across something unfamiliar, I try to acquaint myself with the relevant material on an as needed basis. I find this approach works better for me if I am interested in a paper for isolated results. Reading such papers from introduction to conclusion might not be of immediate (or even future) value.
The other kinds of papers I read would include survey papers (exposition of a particular new idea) and papers which essentially develop a new idea/concept/structure for the first time. A lot of these papers, I find, can (and eventually should) be read from front matter to bibliography. So I would essentially read these papers as I would read a textbook.
One thing which I find especially important while reading papers is to look at the cross references. This can be useful for motivation, alternate proofs and related ideas. It also gets me acquainted with who else is working in the area and the exposure to ideas from different mathematicians achieved thus is a nice side effect.
Finally, I feel it is important to ask a lot of questions while I read papers. Why is hypothesis X necessary. Can condition Y be weakened? Can this result be generalized to Z structure? This is eventually useful when I write my own results, to conceive the most general form that I can.
