*By the **Editorial Board**, based on an interview at the 2017 Joint Mathematics Meeting with REU students David Burton, Kelly Emmrich, Micah Henson, Andres Mejia, and Nina Pande.*

*Editor’s note: The editors thank David, Kelly, Micah, Andres, and Nina for taking the time to share their thoughts and insights with us. Biographical information for each of these students is included at the end of this article.*

“Something that I’ll remember the most is there were a couple epiphany moments where we just all of sudden we seemed like we just stumbled into this, you know magical place of thinking of something we never would have thought or come up with that was really important for our project and the reason I think that I’ll also remember that for a long time is that it gave me a lot of confidence that I could do research because being able to come up with a creative way forward is sometimes I think one of those important parts.”

— REU student

Why should students participate in a summer Research Experience for Undergraduates (REU)? What do undergrads gain from such programs? What has driven their growth and popularity over the past several decades? In this post, we share highlights of a conversation that the editors had with five undergraduates at the 2017 Joint Mathematics Meetings about their experiences at five different REUs (described in the final section). If you are a faculty member, we hope this inspires you to share information about REUs with your students. If you are an undergraduate student, we hope this inspires you to apply for an REU! (Lists of REUs can be found here and here.)

In our conversation, five major themes emerged regarding the students’ REU experiences:

*Collaboration:*the importance of collaboration, friendships, and networking*The Nature of Mathematics:*an appreciation or gained understanding of the nature of mathematics and mathematical research*Self-Beliefs and Agency:*heightened awareness and/or insight about oneself as a learner or person in general*Back to the Classroom:*the positive impact of REUs on subsequent coursework*Graduate School:*an increased or decreased interest in graduate school or insight into graduate school

While there were some additional comments off these themes, which we include below, in this article we hope to tell a story of the impact of REUs on undergraduates through the students’ own words. Note that all student quotes in this article have been lightly edited for clarity.

**Collaboration**

In the interview, a repeating theme was the value of collaboration for students faced with the task of engaging with challenging mathematics at a new level of depth, starting from the first day of the REU.

- The first two days were just lecture and these were all new ideas to me, words I… hadn’t heard before. Much less know how to compute things using them and that kind of thing… The comfort came in the fact that the other students felt the same way.
- There’re really solid friendships that come out of that struggle.
- What I think I’ll remember the most is the friendships.

The students’ friendships with each other were valuable because they found other people from different places and different backgrounds, in some cases finding a mathematical community for the first time.

- I appreciate the REU because it introduced me to a lot of people really serious about math and that’s something I don’t get too much at my college. There are a couple of people interested in grad school but not nearly as many as being at an REU and… now when I come to conferences I know a lot more people and it’s, we’ve kind of like supported each other through the application process of applying to grad school. So I guess, I mean like a lot of people I see myself being friends with pretty much for the rest of my life, being that we plan on being in the same field and I really, I really do appreciate that.
- It’s very easy to make good friends because you drop the pretense in some sense so it’s just a lot of fun. But yeah, definitely walking in ‘cause the different backgrounds you’re kind of able to collaborate in a different way from class … So you’re trusting someone else’s background, their knowledge, their expertise in these kinds of things and it allows you to work better as a group but also simultaneously it makes it easier to pull back.
- I think meeting different people from different places is the other thing you get out of REU and that should stick with you.

Eventually, in some cases, students formed efficient teams, creating “magical” moments.

- A lot of what I learned really happened outside of the hours in which we did research back at someone’s apartment asking, “Hey what does this mean again?” or “Can you explain this to me?” or “This came up today and at the time I didn’t know what it really meant,” and so I think that’s just a rewarding experience that I’m sure I’ll be able to take with me for the rest of my life.
- Working on the actual problem… there was little bit more of that give and take that [another student] spoke about. One was an applied math major I believe and the other a statistics major and I’m just a regular math major. So I guess I kind of contributed helping put stuff in “math speak”, and I guess that they contributed in other ways: ‘cause we were looking for patterns so one of the people had taken a number theory course and so he helped me and the other person in other areas.
- Attending the REU really kind of opened my eyes into the magic that can happen when you have a whole group of people who are interested in the same topic… I saw that change happen this past semester coming back from the REU, just my ability and my willingness to kind of invite that collaboration and to search for it in a greater way than I ever had before
- There are seven other people on this planet that have seen me at my most frustrated moment and at my happiest moment. You know you come to care about these people a lot… it’s just such a rewarding thing to know there are other people out there who have had the same passions as you and who will always kind of be that support team for you and who are going through the same, the same sort of I guess struggles that happen when you’re an undergrad looking toward a career in mathematics

**The Nature of Mathematics**

At their REUs, students learned a lot of new mathematics, delved deeply into the process of doing mathematics, and discovered how research is different than classwork.

- I think because these programs are well-organized [by] people who are really interested in helping and working with students, it [was] a little bit easier… I think for the most part trying to understand this background and the context with… the end goal of genuine research is much different than, I don’t know, someone sitting in a class and proving theorems that have been proved, I don’t know, hundreds of years before or something.
- If you’re just used to taking math classes the problems that you have experience solving are more straightforward and you have an idea of what tools you’re supposed to be using and what you’re drawing on. But then once you get into math research it’s like, you have everything anyone’s ever done to draw on and that can feel very overwhelming but also very exciting… Progress is not linear necessarily so it might feel like you’re making no progress at all for like several weeks and then suddenly you have a breakthrough and everything comes together.
- Our group had a few people who were much more experienced and then a couple of juniors and we just hadn’t seen a lot of things. So there were some things that came up all the time and we just didn’t know what they were and so getting to finally learn that stuff was so exciting.

Many students discovered that the world of mathematics is bigger than they had realized.

- Getting introduced to the math world and what math research is, is like almost bigger than learning the math than you need for your project.
- One thing I’ll definitely remember is a trip that we took to MathFest that summer and that was my first math conference and that was such an eye opener, because the REU was actually also at my college. So I had only, you know, really experienced math through this very small community at my school. And to suddenly be opened up to this world of thousands of mathematicians and all these different areas of math research that was all super exciting and something I’ll definitely remember.
- I felt, after my REU, that a lot about math not just as a subject but also culturally as a community in a lot of ways became demystified… for example, like, reading math research or something like that or reading math papers and this kind of thing just feels like something far off on the horizon or something like this and then you just sit down and do it.
- There’s so much math out there so it’s kind of, you know the saying “you don’t know what you don’t know”… I just learned that there’s so much more out there involving math that I had no idea existed. Which is really exciting for me so I guess what will affect me the most is it introduced me [to] something that I think will motivate me for the rest of my life.

**Self-Beliefs and Agency**

A key feature of these REUs was that students gained confidence and independence, became more comfortable admitting what they don’t know, and learned new ideas.

**You go to an REU and now you’re working with people from all universities. Everyone has a sort of different knowledge base, some people are more interested in things that you’ve never heard of and so what I found was this past semester these terms coming up, “oh hey I remember talking about that at the REU and I didn’t know what it meant then but now I have this motivation to learn what it means” because those conversations kind of made you want to learn these things that these other students knew.**- You find this sense of independence and you start learning these things and it is a very rewarding experience to look back after those two months time. You know, I didn’t even know what this word meant two months ago and now I like to talk to people about it kind of thing.
- I wish I had been a little more comfortable not knowing every last bit of the topic we were working on… trusting that you’re not going to spearhead every aspect of your project. There are other people in the group who are going to understand some things better than you.
- You can’t really have an ego, it’s very obvious what you know and what you don’t know.

In particular, students had to learn when and how to ask for help.

- We had very different levels of experience [with] the area that we were working in and so I think one of the difficulties was trying not [to be] discouraged by that. Because a lot of the time it can feel like you’re not contributing equally but just because you’re not, you know, the one like coming up with something new every single day, that doesn’t mean that you don’t have a role to play. And one other thing that I sort of learned over that summer was how to be assertive and make sure I knew what was going on and be willing to ask my group members to explain something in more detail so that we could all be on the same page. And that’s something they’re always willing to do but you sometimes have to ask for it, ‘cause they’re not always going to realize that they’re moving really fast and not everyone’s following.
- The REU experience, first off, is that it’s a very humbling experience, right. You find out what you know and what you don’t know very quickly… the most progress I think is when you finally admit that to yourself … It can be an uncomfortable thing but I think once you allow yourself to admit what you know and what you don’t know it opens the door, it opens the door to learn a lot more.
- It’s ok to ask people for help and there’s always going to be someone who knows something you don’t know. And you don’t have to look at that as any sort of discouragement towards yourself. It’s just an opportunity to learn something new. So I think that’s a very valuable group dynamic.

**Back to the Classroom**

When students returned to their home institutions following the REU, their summer experience influenced their work in subsequent mathematics courses.

- It was just very motivating and I found a lot of tie-ins to the course work I did the semester after the REU.
- [The REU] improve[d] my mathematical maturity so when I went to take analysis… it was much easier to read the analysis book than it had been to read the math stats book last year.
- I think it was from the first time I ever had to read some very difficult books. The books we had to read from were like really really tough to crack and they were using kind of a language I wasn’t very comfortable in… this made it easier I think to go back into the classroom and maybe read books.
- For me it kind of made it harder to go take classes because I enjoyed doing research so much.

**Graduate School**

The REU experience motivated students to consider whether or not to go to graduate school.

- It got me more motivated to go to graduate school
- Some of us have already decided “I don’t want to do research”… I’m really glad I did that [REU] so I found that out.
- Going into my REU I was sort of thinking of it as something, like, tell me whether or not I would like math research and whether or not I should think about going into [it]… and I think now, a year and a half later, I don’t think [this is] at all how you should approach your REU. I mean if you really really like it that’s definitely a good sign (chuckles) but if there are some things that you don’t like, that doesn’t necessarily mean that research is not for you.
- There are just so many factors going on in REU that I think even if it doesn’t go exactly how you would of wanted it to, you should still think about giving research another chance.

**Overall Observations**

Students agreed that one should not judge an REU by the first few days, and that the total REU experience was worthwhile and rewarding.

- One thing I wish I had known is that just as much as the details of your project, one thing that can determine like how the REU goes is your advisor’s advising style.
**Your experience the first two days is not representative of the whole experience.**- I don’t think you’re going to get the chance to work with professors who are so patient, so knowledgeable and so interested in their fields, I mean the amount of patience that it takes on the part of REU by the advisors is just entirely astounding to me. (laughing) It’s, to get someone from not knowing the definition of the basic object that you’re working on and getting them to produce a result I just I don’t even know what that takes.
- You know what everyone is facing and so it’s a really, it’s just a rewarding experience and I think that persevering past those initial discomforts is key to success.
- The graph of the REU experience is like the graph of the sine function: you’re going to have highs and lows and I promise you those positive parts are going to stick with you at the end.

**Interview Participant Bios **

David Burton completed his undergraduate degree at East Tennessee State University during the fall of 2016 and has just started as a graduate student there. He participated in an REU at the University of Connecticut Health Center for Quantitative Medicine during the summer of 2016, where he worked on an ongoing project entitled “Functional data analysis of copy number alterations in bladder cancer tumor chromosomes.” Next year, he hopes to enter a Ph.D. program in statistics.

Kelly Emmrich is a junior at the University of Wisconsin, La Crosse. She participated in the REU at California State University, Fresno, during the summer of 2016, where she worked on a project in complex analysis called “Sufficient conditions for a linear operator on R[x] to be monotone.” After graduation, she plans to enter a Ph.D. program in mathematics.

Micah Henson is a senior at Spelman College. She participated in the Mathematical Sciences Research Institute Undergraduate Program during the summer of 2016, where she worked on a problem in algebra called “The sandpile group of thick cycle graphs.” After graduation, she plans to pursue a Ph.D. in mathematics.

Andres Mejia is a junior at Bard College. He participated in the WADE Into Research REU at Wake Forest University during the summer of 2016, where he worked on a project in number theory called “Classically integral quadratic forms excepting at most two values.” After graduation, he plans to enter a graduate program in mathematics.

Nina Pande is a senior at Williams College. She participated in the SMALL REU at Williams College during the summer of 2015 and an REU at the University of Michigan during the summer of 2016. At Williams she worked on a problem in commutative algebra called “Controlling the dimensions of formal fibers,” and at Michigan she worked on another problem in commutative algebra called “First Koszul homology over a local Artinian ring.” After graduation she will teach high school mathematics in North Carolina before entering a graduate program in mathematics.