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I am working toward a PhD degree and my major involves lot of mathematics. The problem is that my basis of math is not very strong but after putting a lot of time I can understand the theorems, lemmas, corollaries in the published papers. Sometime I wonder should I start learning the basics again or should I read the research papers only to make my math strong. If I start from the basics then it may consume lot of time however it may help me understanding things better and in applying accepted ideas to more general problems. On the other hand if I learn through research papers then my knowledge will be limited to only the problem I am working on and new ideas of solving the problem may not fluently come into my mind.

In essence which of the following strategy is better in your opinion

  1. Start learning from basics.
  2. Learn through published research papers during the literature review.
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    Can you be more specific as to the field of research and type of math it involves? Those with a background in that field may be able to provide specifics, while those with a more definitive math background, external to the research area, may have another perspective that is also helpful.
    – DMML
    Jun 17 '16 at 3:57
  • @DMlash Its wireless communication. The research is not defined yet however currently I am reading papers related to queuing theory and optimization. Jun 17 '16 at 4:33
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    Are you just going to use the results, or do you need to develop your own?
    – Davidmh
    Jun 17 '16 at 6:30
  • @Davidmh ideally I would like to develop my own. But its quite impossible for me right now. I just want to use the already existing results in some way but I want to learn their basics before I can apply them Jun 17 '16 at 7:24
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Why not both?

If you aren't explicitly a mathematician but there is a lot of math used as a tool in your field to solve problems (this is often the case with my field), it is understood that some people are stronger in the math than others. The whole point of having a field where people of different educational backgrounds work together is that you don't have to be an expert at every sub-skill and sub-task - you can have your own unique strengths without starting over at the fundamentals for every subject in play.

A strategy that I've found works very well is read some literature to see what techniques are used by others in their related research. If a technique comes up a few times (like, say, applying Dynamic Time Warping to analyze a time series), read into that bit to get a better high-level understanding of it - how it is used, it's strength and weakness, when it's not appropriate, etc.

At this point you may find that the technique just isn't relevant to your research - so you can probably skip it for now and move on. Sure, you don't understand it from top to bottom - but you just don't have time to learn everything in infinite detail! However, what if the technique seems really useful? Well, some people don't bother to understand it at all and just blindly apply it, because well other researchers have so it's probably fine. I'm not at all fond of this, and I would humbly suggest it leads to bad science, unreliable findings, and missed opportunities.

So if a technique seems useful to you, learn some more about it. Try to delve a little more deeply and find out just what the technique seems to actually do. What do some of the variables mean? How is the calculation performed overall - how does it behave based on some inputs compared to others, and why? Again, you probably have limited time, so don't feel you have to prove everything from first principles.

Next, you still have some time to advance your understanding of the fundamentals. So especially as you read through the literature and useful techniques, what area is really hard for you? Is probability stumping you, or calculus, or dynamic programming, or are the notations and implicit variable meanings alien to you? Reserve a little of your time - perhaps a few hours a week at most - to strengthen yourself on the most fundamental issues. Perform a trivial calculation by hand, or read a textbook explanation of the notations used.

I've found the biggest pitfall is thinking you have to understand everything from square one right at the very beginning, and I have to fight the tendency to get sucked in to minutiae. But then often as little as a few days pass and suddenly I realize I have a far better understanding of something than I thought I did, and it wasn't really as hard as it seemed at first.

Now this works for me, and for plenty of other people I know, but I cannot say it will be the best for you, in your field, with your personality and own unique traits and talents. As always YMMV - find what works best for you!

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  • Thanks a lot Brian. This is answer that I expected. I wanted to do it exact way you mentioned but I was not sure if its the thing done by other researchers also. As you mentioned you also does the same thing therefore I have more confidence in this approach now. Thanks again for clearly elaborating on it. Jun 18 '16 at 1:26
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It will not be very easy to learn a broad subject in mathematics by reading research papers. Books which cover the subject starting from the basics will be more cohesive (for instance, regarding notation and terminology) and will leave you with more approaches to solve problems. If your major requires a lot of mathematics, hopefully you can identify a specific area (like analysis, geometry, probability, etc.) that you need, and find a good text that synthesizes a lot of the research.

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The way I see it, the answer to this question really depends on your particular circumstance... To find out where you stand, you could seek someone to assess your mathematical ability.

Could you approach your advisor or a colleague about this? Ask someone with a strong mathematical background who understands your line of work to probe you. They could point you in the right direction for what foundational theory is necessary for your success, leading to book and even chapter recommendations.

I totally support building up the basics. I'd probably be driven towards that route, but realistically, your priority ought to be to further your work in the most effective way possible.

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