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My question is about how to best read mathematics papers for research.

I just passed the last qualifying exam as a Ph.D. student in mathematics (studying statistics) and I'm reaching the point where I will need to do more research, which involves reading papers. I used to write, in my own words in a journal, pre-reading thoughts (why I'm reading this paper, what I already know, etc.), a summary of the paper's contents while reading, post-reading thoughts (what I think about the ideas, how they connect to others, how this could be useful, opinions, etc.), and a list of potentially interesting bibliography entries for later reading. When reading I would essentially read linearly, making margin notes, highlighting interesting sections, and so on. Reading a single paper could take a day, going through this process.

This did well enough for my undergraduate thesis on a topic in economics. It also worked well for the paper that my adviser and I are wrapping up and submitting to journals. But I get the impression that this is not an optimal way to read papers, especially in subjects as technically intense as mathematics. (My role in the paper I co-authored with my adviser was to do simulation studies for a test statistic he and another co-author developed, which is more computational and programming intensive than mathematically intensive, and paper reading was basically to get background on the subject. This is fine for me as a starting researcher, but I don't think I will get by on this alone for a Ph.D. thesis, let alone in my later career.)

I'm shy about asking my adviser the best way to read papers, so I'd like to hear others' approach. Is my approach to paper reading a good one, or is there a better way?

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    "I'm shy about asking my adviser the best way to read papers, so I'd like to hear others' approach." It's great to hear a variety of approaches, but you should not be shy about asking your advisor this. That's what they're there for, and they are better positioned to give you a useful answer than anyone else. – Pete L. Clark Aug 31 '17 at 14:58
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    I suggest you join or start a journal club with some fellow students. This won't completely answer your question but it would help. – aparente001 Sep 1 '17 at 4:02
  • I tell my students: have a question in mind when reading a paper. Be strategic. For example, you might have idea or problem X. Then you can go looking for idea/problem X in papers or compare and contrast the Y in papers with X. – Prof. Santa Claus Sep 2 '17 at 22:01
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My PhD was in statistics, and I have also supervised PhD students in this field, so hopefully I can offer some useful advice. Here are some thoughts:

  • Talk to your supervisor: There is no room for intellectual shyness in that relationship. Your supervisor is not going to be upset with you for asking questions, evening if it is something you think you should know how to do already. The fact that you even think to think about this is a good sign, and your process is quite impressive. Don't be shy.

  • Streamline your process: Your process sounds quite comprehensive already. Personally, I find that writing a short summary in your own words/notation is probably the most useful exercise to understand math papers. Some of the other stuff you're doing might be a bit excessive, especially if you're only getting through one paper a day.

  • Learn the maths, expand your tool-kit: In a mathematical field, in additional to understanding the specifics of the paper you're reading, you also want to use this as an opportunity to gradually expand your mathematical tool-kit. To do this, make sure you understand the derivation/proof of each result, and if you see an unfamiliar mathematical technique, talk to your supervisor about why it works, and when you might deploy it.

  • Learn to "see the matrix": Think about the intuition of each method and and make sure you can describe it intuitively. When reading mathematical theorems, ask yourself lots of questions: Do I understand this result and its relevance to the field? How would this be applied in practice? Do I understand the proof(s)? Are there other ways to prove this? If there are necessary conditions for a result, do I understand why they're necessary (i.e., why does the result fail without them)? If there are sufficient conditions for a result, do I understand why these are sufficient (i.e., why does the result hold in this case)?

  • Jot down your own ideas: As you read the literature, jot down your own ideas for possible extensions and interesting questions that come out of this. At first these might be quite ill-posed, but you can gradually refine them as you learn more.

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I think your approach is good. Here are some suggestions:

  • I doubt that writing what you already know is ever likely to be useful to you in the future, so you could save time by not doing that.
  • At the end, I would write two or three sentences to summarize the paper, at the top of your notes. I mean a kind of replacement for the abstract, but easier to read, stripped of the self-promoting aspects, and including your overall opinion of the paper's quality and usefulness.
  • I usually make a note of how easy the paper was to read, as I don't want to re-read a paper in the future if I found it painful to read or understand the first time. Or at least I want to be forewarned.
  • Include keywords or other important words and phrases. If the paper contains an important piece of information, then you will want to be able to find that in the future, if you have forgotten where you saw it. You have to anticipate what words or phrases you will be likely to search for.
  • A day is rather long. You will probably need to become faster, but that may happen naturally. I usually do the same kinds of things as you, and I don't think I am slower than anyone else at reading papers. I probably pick up more details.
  • As Pete L. Clark said, there is no reason to be shy about asking your adviser.

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