Background: I am a lecturer in computer science, but my research is mostly theoretic and mathematic, so I ask here.

I want to encourage my undergraduate students to become research students after their graduation (as master or Ph.D. students). To do this, I have to explain to them what it means to be a research student. As undergraduate students, they are used to just doing what we tell them to: we give them homework, they do the homework as requested, and get a grade. However, as research students they will have to be much more creative, bring more ideas of their own, take the initiative, try new directions, etc. I started to write a short document describing to undergraduates how the life of a research student looks like, but then I figured that someone must have done this before.

Do you know of a short text that explains to undergraduate students the meaning of doing research (as graduate students) in a way that is both accurate and attractive?

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    $\begingroup$ Maybe explain what you mean by "research student". As for mathematics (and in the U.S.), graduate students usually don't begin research until after passing their Ph.D. qualifying exams, which typically takes one to two years, and after this they are nearly always fairly closely supervised by someone who provides suggested topics/problems as promising avenues of research, taking into account the student's background and ability. Also, probably fewer than 10% (maybe 5%) of math majors (in the U.S.) enroll in a Ph.D. program, so it would also help if you explain why you think this is important. $\endgroup$ Aug 30, 2020 at 17:05
  • $\begingroup$ @DaveLRenfro: See the Joseph Gallian article to which I link for a different opinion. $\endgroup$ Aug 30, 2020 at 18:36
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    $\begingroup$ @Joseph O'Rourke: My puzzlement was with the "become research students after their graduation" aspect, not whether it's a good idea while a student. Every math major at one place I've taught was required to complete a "capstone" project, and at a couple of other places I've taught doing this was an option encouraged for stronger students. $\endgroup$ Aug 30, 2020 at 20:08
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    $\begingroup$ @Joseph O'Rourke: In looking at my earlier comment, I think the phrase "after their graduation" so stood out in my initial reading of the question that I overlooked mentioning it. I suspect the OP's intention is more along the lines of How do I show students the Beauty of Mathematics? than in trying to engage in what is described in this book (although simply reading about research, in that book, might be an example of what's asked for). (Better books, though, would be Ulam's book or E. T. Bell's.) $\endgroup$ Aug 30, 2020 at 20:35
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    $\begingroup$ I think it's a noble idea, but please consider this a complex (non math meaning) issue. Important to realize many students will not go on to grad work. It's a pyramid. And has to be. We don't need unlimited amounts of math researchers. There is not market demand for them. Also important to convey the downsides as well as upsides. Things like the job situation, advisor power level, etc. (Go read the million sob stories on academia SE. And they were normal well before SE in my experience.) $\endgroup$
    – guest
    Aug 31, 2020 at 0:13

3 Answers 3


Here's a useful article.

Gareth E. Roberts. "Conducting Mathematical Research with Undergraduates." 2008. PDF download

Abstract (extract). This article discusses some of the key issues concerning undergraduate mathematical research: selecting good research students, finding appropriate research questions, mentoring versus collaboration, and presenting and publishing student work. Some useful professional and financial resources supporting undergraduate research are also highlighted.

Along the way, Gareth Roberts mentions that,

Joe Gallian, who has been running a highly successful REU1 program at the University of Minnesota Duluth for over thirty years, spends the entire year leading up to his summer program searching for good research problems [15].

You might be interested in this recent history by Gallian:

Joseph A. Gallian. "A History of Undergraduate Research in Mathematics." 2018. PDF download.

Abstract (extract). In this article we identify the key events that have led to the current widespread acceptance of the importance of opportunities for undergraduates to engage in research in mathematics.

1REU = Research Experiences for Undergraduates, a US National Science Foundation summer program.

How to explain to undergraduate students what research means?

A nice, short and general explanation is given in The illustrated guide to a Ph.D, by Matt Might. As a complement for the other suggestions, you could use it as an introduction for the discussion.

  • $\begingroup$ Very cool, thanks (his website has other great posts too) $\endgroup$ Sep 5, 2020 at 21:51

Research: To boldly go to places no man has gone before...

I work in IT and it is only after working for 13 or so years, one day FINALLY, PC became personal to me. I felt as if it is watching me through it's camera, watching my actions, trying to know me, learning me. I said to my laptop one day, if you are trying to get inside me, I'll get inside you. And that's when I became interested in Machine Learning and Artificial Intelligence, I wanted to see what a computer sees when He(it) looks at me.

So Research has to be a personal quest. The subject becomes personal to you, you live it and it's fun. For example all my school life I struggled with Proofs in Mathematics. There are direct proofs, indirect proofs, proofs by contradiction, then there is logic, axioms, lemmas, corollaries and what not. So I decided to understand these mathematical proofs one day. I started studying and was so engrossed in the Logic part of it that I started living it. I was watching a sitcom and on listening to their dialogues I was getting irritated, I COULD NOT UNDERSTAND WHY?! Upon thinking I realised that since one can draw more than one meaning of what that person said, I was finding it irritating since logic says that unambiguously people MUST arrive at a single conclusion from premises. So I was living Logic during those times. I have the output here.

So research is devotion, it's a way of life. You don't don a "Researcher's hat" when you go to work, it's an attitude. One has to be an explorer at heart....

Ohh and by the way I found out how to make "Conscious Machines" NOT "Intelligent Machines" but Conscious machines, how to give them intuition...

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    $\begingroup$ Would you please change that (famous) quote to "... where no (one / person / something not gendered)"? I know it has meaning as is to lots of people (Star Trek, right?), but it is gendered, and that sort of language makes women feel less welcome. Thanks for considering it. $\endgroup$
    – Sue VanHattum
    Sep 1, 2020 at 18:48
  • $\begingroup$ "where no one has gone before" : en.wikipedia.org/wiki/Where_no_man_has_gone_before $\endgroup$
    – JRN
    Sep 5, 2020 at 16:11

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