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I'm an incoming PhD student. I did some research during my undergrad and master. One of the things my advisors told me was to read papers but not too focused on "technical details". This did help me navigate literature faster by just skimming for the "main ideas" in the paper. But is this the right attitude when the purpose of reading papers is to learn techniques (e.g. how to prove certain bounds, how to analyze certain matrix operations)? I felt that understanding an overall "strategy" or "essential ideas" has not been very effective in advancing my own research. That is, I'm still not very good at doing those type of complicated analysis and my research effort remains "elementary". It's like taking a math class without doing any homework: I feel that I know what is what but tend to struggle when actually sitting for an exam and solving problems.

What is the most effective way to cultivate technical abilities as a researcher, without the aid of textbooks and problem sets?

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    My first reaction is that you're giving up too easily - you're finding the complicated analysis 'hard' because you're expecting to figure it out in a few hours, when in reality you might have to spend a few months thinking about it in order to figure it out. – Alexander Woo Feb 2 '20 at 23:36
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When you were an undergraduate, studying (I assume) math, the advice you got was good. Read a lot of papers, somewhat superficially, to get an idea of what it is possible to do and how to go about it generally.

But now that you are in a graduate research program you need to do more. The basis you get from textbooks isn't enough to get you to the research frontier in mathematics. It won't get you very close, actually. So, to approach the research boundary of what is known, you need to read (a few) papers fairly deeply. For advanced work, elementary, textbook, techniques may not work. So, you need to delve into the proofs of things to see if there are new and interesting techniques that you need in your own toolkit.

But, as you read a paper there are some things you need to ask yourself, beyond the question whether everything is correct. The big question is whether the paper is complete and is the last word on a topic, or can it, perhaps, be generalized in some interesting direction.

Another big question is whether the techniques used in the paper (not the results themselves) can be used and modified (extended) to solve other similar problems. In this sense the proofs in a math paper may be more important than the theorems. And it is why some new proofs of old theorems are extremely important. The proofs open up new areas of exploration.

Another area of research can be opened up by comparing two papers and examining whether there is a possibility to combine what the two say to come up with something new and interesting. This may or may not require a deep reading of the papers, of course.

To do math research requires insight. Basically, it means being able to find things that might be true but haven't been proven true. Various limitations of logic make this harder than it might seem. So, when reading a paper, another question to ask is "What led the author(s) to think this might be true?" Why did they think it something worth time studying? That can lead to insights.

But, math research, of its nature is narrow and deep. So, you often need to do deep dives into ideas to discover whether there is something hidden but worth exploring. What works for a textbook learner doesn't get the job done for a researcher.

Caveat: This applies to mathematics and some theoretical aspects of other technical fields. I'll make no claim about "technical fields" in general.

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In my experience (in the field of statistics - but I suspect it is of wider application) you need to switch between two modes. Sometimes, all you need is to skim a paper for its main ideas. But sometimes you will never understand a paper unless you try to reproduce its results. The reason for that is that published papers tend to be highly compressed, leaving out major steps in the argument. Unless you try to reproduce those arguments in detail I think that you will not fully understand the authors' work.

I have found that sometimes I cannot reproduce the work in a paper because I do not sufficiently understand the theory. There is nothing wrong with consulting a textbook - even a first year undergraduate textbook - just because you are now a PhD student. I even had to buy a second-hand copy of such a textbook, having long ago thrown out the copy of the same book that I bought in my first year of undergraduate study.

I have certainly worked through set exercises to help me understand basic theory. You should feel no shame in doing so.

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Textbooks and problem sets are the specialized tools to teach you technical abilities.

Research papers may describe a technique, but they're often more focused on proving that the technique is sound, or a promising direction for research. Teaching you how to use the technique is not their primary aim. That's probably why your professors told you not to get stuck on the technical parts.

The time-tested ways of learning technique are textbooks, exercises, and taking classes where a teacher guides you through the material. Nowadays of course there also online courses, which makes it easier to do during hours that suit you.

One of the key advantages of a textbook, or a course or problem set, is completeness. If the author knew what they were doing, the programme will give you a thorough and complete set of skills.

Compare this to trying to learn from the examples in individual papers, or learning by tackling each problem as it appears (the classic self-taught programmer). The self-taught method doesn't know what it doesn't know. If you missed a key insight while trying to learn techniques from a stack of papers, because it wasn't explicitly used in those papers, you might never notice.

That said, not all textbooks and courses are equally good, and some are good for some students but not others. So look for one that actually works for you.

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    Your suggestions here are good for general learning, but not sufficient to get a person to the research frontier in math. Math research is a specialists game. Textbooks are good for general knowledge. But, yes, the practice of doing exercises helps you develop some skill. – Buffy Feb 2 '20 at 23:20
  • Fair enough. It started out as a "textbooks aren't bad" that grew into a longer answer. – ObscureOwl Feb 2 '20 at 23:36

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