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I started a PhD in applied mathematics and things seem to be going well after an initially terrible start. My supervisor and I have devised a clear set of objectives for the next three years and I’m slowly starting to learn the ropes of independent researcher life.

I recently hit a snag when my supervisor decided to take a more hands off approach with my learning insofar as they told me to “continue with my research”. Naturally this left me feeling a bit overwhelmed having only just started a few months ago. How exactly does one “do research”? Am I simply expected to keep reading articles and analysing their findings?

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    This isn't long enough to be an answer but it may help you. My own advisor, later committee chair, told me to read papers and look for gaps. I found a tiny gap in the area of pattern recognition, and now they call me Doc.
    – Bob Brown
    Apr 1, 2021 at 22:59
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    I'm curious what "clear set of objectives" means to you. "Solve world hunger" is maybe a clear objective, but it isn't a particularly useful one to organize around. You haven't operationally defined the thing you'd like to solve in a way that you can measure any progress, haven't indicated any specific proximate steps, etc. Can you see some path to your clear objectives, or are they too abstract to be useful?
    – Bryan Krause ♦
    Apr 1, 2021 at 23:46
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    "How exactly does one “do research”?" -- I really don't think this question is answerable here; learning how to do research is the entire purpose of graduate school; we cannot boil it down to a 500-word answer. Moreover, you are already working with an advisor and have formulated objectives, so any answers without that context will not be very useful to you.
    – cag51 ♦
    Apr 2, 2021 at 1:00
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    Agree with @cag51 - I was going to say the same thing before reading the other comments. Terry Tao’s blog has several dozen very good pages with advice about doing research. But even that doesn’t begin to really be enough to enable someone to learn how to be a successful researcher without proper mentorship (unless you are many standard deviations away from normal abilities, similarly to someone who’s a top research mathematician).
    – Dan Romik
    Apr 2, 2021 at 2:51
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    You should talk to your supervisor again and explain that you need more guidance.
    – astronat supports the strike
    Apr 2, 2021 at 7:16

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This might be simplified but I hope it's useful. You can think of the research process as consisting of 4 steps.

Step 1. Setting your topic

Here you will define your topic, understand the research landscape (the tops and valleys, the things that are important and those ideas that have been discarded).

A key activity here includes formulating your research questions. It's important to note that the research questions shouldn't answer themselves, why else would you undertake the research? :D Also, do not assume the very thing that you are trying to answer ("Why is X a better solution for developing Y" assumes that X is better to begin with).

There is no shortcut to understanding your topic, you need to read, and read a lot. If you're merely interested in understanding the research field, focusing on systematic literature reviews is the best way to go. These are papers that summarize the state of the literature for a given field.

Step 2. Locate information about your topic

In order to "do research" you will need to have access to research databases. Most likely, your university will have access to some of the larger databases, such as Web of Science, Scopus etc. Google Scholar aggregates research papers from a wide array of sources and is a good starting point (there are some concerns with it which I shouldn't skip over, but it's nonetheless a useful tool).

Sometime a book written for laymen could be a good place to begin, even though that book may not be authoritative, they are usually easier to digest than your average journal paper. It may seem silly, but Youtube and Podcasts are a good source for understanding a specific research field, but not particularly useful for citation purposes.

Step 3. Analyze/Evaluate information Once you have scoped through the research literature, you will need to evaluate your sources and distinguish between primary and secondary sources.

Step 4. Write and take notes Write! Always write even when you're not suppose to write. Writing a lot will help you. My advise is to set a writing quote (e.g. 300 words/day) and then meet that quota everyday. The text might not be the best starting out, but eventually you should be able to edit the text and make it better (or discard the parts that are not needed). Also: take a lot of notes! Mental notes. Notes on paper. Notes on the screen. Notes while walking.

Step 5. Keep and organize sources Cite your sources and be sure to use a good reference management software for that purpose. Zotero is Open Source and easy to use but there are many others to choose from as well.

The five steps above will be true for virtually any research field. The remainder will differ a lot depending on your field (quantitative/qualitative research, empirical data vs other forms of data collection etc...)

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