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I am an undergraduate student in mathematics, and I will be starting my third year of university in the fall. I recently attended an NSF "Research Experiences for Undergraduates" (REU) program.

Unfortunately, the program did not go well for me. I can't engage with the material at the proper level, and I've accomplished very little while I've been here. For reference, the others here are writing multipage papers about things they've done or at the very least, learned, and some plan to submit theirs journals or present their findings at undergraduate-friendly conferences. I could probably write a paragraph of what I "learned" but in my opinion, I don't have enough background to even coherently think about what I was supposed to accomplish.

Here's a rundown of other possible reasons that I performed poorly:

  • As I said before I don't have much background. I've taken all of the pure math courses my school offers, but the others here came in knowing things like category theory, algebraic geometry, etc. that I simply haven't been able to learn well in my short time here. Notably, the program website never indicated that I would expected to know that stuff. I certainly exceeded the "minimum" requirements to apply.

  • My research advisor is advising three students and we're sharing projects. Since the others knew so much more than me, I feel like he was unwilling to help me with things that was probably trivial to others. (I know that in graduate school that attitude from professors might be common, but the other students had semester long classes on things I was trying to pick up in a few days.) This was also his first experience involving undergraduates with research.

  • While my program is considered competitive in the sense that they apparently get 300+ applications only accept 9 - 12 students each summer, I'm pretty certain that I only got in because I'm a member of an "under represented group" in mathematics and the program tries to favor such individuals. I was clearly not accepted on merit based on the caliber of the other students.

I am concerned about my performance because these programs are "designed to expose undergraduate math students to research to the extent that they are encouraged to attend graduate school in the mathematical sciences". Considering this experience makes me want to do anything but graduate school in math, I'm worried. Should I let this experience be a wake up call?

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    There are more qualified people here to give you feedback, but I can't help but feel that things aren't as bad as they might appear to you. 1) The "caliber" of students is not determined by which classes they have taken that you haven't. 2) Your program targets particularly the groups you say you are from (I checked 3 descriptions) in order to teach you; failure to do so is not yours as you disclosed your background beforehand. 3) Program targets junior/senior level. You are a sophomore. 4) Old testimonials included "...rewarding but frustrating." Best of luck! – gnometorule Jun 25 '16 at 2:22
  • Wait and answer this question at the final year at MSc. Some students need time to understand the theory, others treat it superficially. You need to build a solid background and this is not achieved easily. However, with proper guidance, you can publish a solid paper even undergraduate. The best actions would be to contact a professor at your home institution(or not) that you like and start collaborating. Then you will know what are your strength/weaknesses. Good luck! – Mikey Mike Jun 25 '16 at 11:21
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    I'd have to disagree with @MikeyMike's advice, insofar as "collaboration" between undergrads in math and faculty in math is at best contrived, and normally does not genuinely exist. To believe or insist otherwise usually merely reinforces a sensible (!) student's feeling that they're inadequate. I gather that mathematics differs from other subjects in this regard... Anyway, "expose to research" is one thing, "actively participate" or even "comprehend" is another. See Kimball's answer below. – paul garrett Jun 25 '16 at 22:48
  • @ paul garrett I know some undergrads in theoretical physics who actively participated in research projects with professors and published papers in somehow good journals(not the best impact factors, but respectable). – Mikey Mike Jun 26 '16 at 12:16
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To address the titular question, no, being a "poor undergraduate researcher" does not mean you are not cut out for graduate school. First, many people who go to grad school in math have no research experience in advance. Second, having one bad research experience does not mean you are a poor researcher--you're not even in your third year yet.

Having bad research experiences is pretty common, and I know several professional mathematicians who've had bad experiences. I was part of a summer program and a workshop where I felt I was in over my head and everyone else seemed to know the topic much more than me. Fortunately they weren't my first research experiences, and they didn't last as long as an REU, so they weren't too discouraging as a result. These experiences also provided no gauge as to whether or not I could be a successful researcher.

Research takes a lot of time, usually more than you expect (even when you expect it's more than you expect), as well as adequate preparation. It's also hard to find good problems that are suitable for undergraduates to do in 6-10 weeks. Part of the problem is, before you work on a project, you don't know how hard it will be or exactly what it will involve. Therefore lots of REU projects don't amount what was hoped for. It sounds a least like this project wasn't appropriate for you at that time. Whether that was due to the supervisor being too ambitious or you not having the expected background, I can't say. (There could be other reasons for your admission besides your "minority status.") What I can say is that is easy to get the impression that everyone around you understands or knows more than you just because they're more vocal or they know different things from you.

To address what seems to be your principal concern now, the primary reason to go to grad school in math should be because you want to learn and do more math. Getting a master's or a PhD in math does not mean you will be a math researcher, and you often don't even need to do any research to get a master's. It's unfortunate if this experience turned you off from math, but I suggest you wait a little while to see how you feel about math. Maybe after this experience fades from your memory, you'll get more excited about math again. Or maybe you'll find something else that excites you, which is good too.

By the way, if you've already taken all the math classes offered at your school by your second year, then possibly your school doesn't have an environment as conducive to developing mathematical maturity as other schools. If you do want to pursue math further, you might look into transferring, a semester abroad, or taking classes at other nearby schools.

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No. In mathematics all undergraduates are poor researchers.

You might object that if an undergrad did research that was strong enough, it really would serve as evidence that they will succeed in graduate school. However this simply does not happen: I am a math professor in pure math who has served on graduate admissions committees, and in the past 5 years I know of 2 undergraduates in the US who I'd say did research at this level. Others might argue the number is higher, perhaps on the order of 1 or 2 per year; but in any case it is extremely rare.

Along the same lines, Andy Putman said in this answer:

I am the director of graduate admissions in the pure math department of my university.

In my experience, we basically ignore "research" conducted by undergraduate and masters students ... It just isn't a good predictor of whether or not you will be successful in a PhD program.

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    Seconding this: indeed, by some (unreasonable but popular) standards, essentially all math undergrads are "bad researchers", in the sense that they don't have much idea what's going on, in the first place. The nearly universal "timetable" for "getting up to the starting line for (meaningful) research" is apparently much different from other subjects (... or, perhaps, those are misrepresented, too, by people who don't know any better...). REUs in math are at best encouraging about social interactions, inspirational about active math rather than compliance-oriented classroom-math, etc. – paul garrett Jun 25 '16 at 23:20
  • In mathematics all undergraduates are poor researchers — Well, almost all. There are rare exceptions like Daniel Kane. – JeffE Jun 26 '16 at 23:49
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    @JeffE: When I read Tom Church's first sentence, my first thought was Jacob Lurie. My second thought was that his sentence is still true almost everywhere. – Dave L Renfro Jun 27 '16 at 16:51
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"Should I let this experience be a wake up call?" Absolutely.

"Does being a poor undergraduate researcher mean that I have a poor chance of being successful in graduate school?" Not necessarily.

Everyone starts out as a poor researcher. Becoming a good researcher requires hard work, preparation, and some chance. It is pretty clear you do not have the preparation yet. If you have taken all the pure math classes at your home institution, this suggests that your home institution may not be able to prepare you for math research.

It's a personal decision to pursue research. If you decide to do that, you should talk to your advisors at both the REU and your home institution about what you need to change to be better prepared.

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    It is pretty clear you do not have the preparation yet. I wouldn't go that far. How much preparation you need depends almost entirely on your project. – Kimball Jun 25 '16 at 11:26
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You'd think it might - but a counter example.

I was an appalling undergraduate researcher. I had a project I chose - but it turned out I wasn't that interested in it. I scraped a pass.

I got back into research about 15 years later - 5 years ago. In those 5 years I've obtained around £10m of funding, about half of that as lead applicant. I've published in multiple high impact journals and impacted national policy. People from across the country seek me out as a collaborator.

The difference - The first project was wet lab immunology, and I learned I detest lab work. The last 5 years has all been near patient pragmatic research.

If you can find an interest and a passion, you'll find the background you need will be little effort to acquire.

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