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?
"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.