By Luis David García Puente, Contributing Editor, Sam Houston State University
Over the years I have been asked the questions: Why do you direct undergraduate research? How do you pick a research problem for your students? How do you manage a research group? In this blog post I would like to present my personal points of view regarding these questions.
I have been involved in research with undergraduates since 2001. I have worked with students as part of REU programs at large research universities, at mostly undergraduate state universities, and at programs in mathematics institutes. I have also worked with small groups of local students. In 2001, I was a graduate TA at the Summer Institute of Mathematics for Undergraduates (SIMU), an REU program hosted at the University of Puerto Rico – Humacao that received the first ever Mathematics Programs that Make a Difference award from the AMS in 2006. This program fundamentally shaped my view regarding working on research with undergraduate students.
There are several resources for students to learn about undergraduate research programs. There are webpages with lists of active REU programs, some maintained by the AMS, NSF, and the Math Alliance and also some other independent websites. The webpages of many of these programs also relate the points of view of previous students and what makes these programs successful and special. There is an MAA website that, while outdated, does a nice job at answering the question: Is an REU for You? There are also more recent articles and blogs describing the relevance of undergraduate research from the students’ own point of view.
In contrast, there is only a limited amount of information targeting faculty who are interested in leading an REU but lack the experience to work with undergraduate students on research. Currently, the American Institute of Mathematics (AIM) and The Institute of Computational and Experimental Research in Mathematics (ICERM) offer a one-week workshop on Research Experiences for Undergraduate Faculty (REUF). The goal of this workshop is to equip faculty at primarily undergraduate institutions with tools to engage in research with undergraduate students. In 2014, Leslie Hogben and Ulrica Wilson published an article in Involve detailing this program. The REUF program has been very successful and I encourage anyone who wants to start advising undergraduate students in research to apply. As another example, for several years the Center for Undergraduate Research in Mathematics (CURM) has promoted academic year undergraduate research in mathematics. This program, located at Brigham Young University, has provided training and funds to professors to establish undergraduate research groups. Some of the materials developed in this program are accessible online. For those who are interested in reading more about leading REUs, the journal PRIMUS published in 2013 a Special Issue on Undergraduate Research in the Mathematical Sciences. All the articles in this issue are great reads for anyone interested in the topic. In the rest of this article, I will provide some personal reflections about leading REUs.
Why do you direct undergraduate research?
In my opinion, the job of a mathematician consists in learning, discovering, and disseminating mathematical knowledge. More than 60% of undergraduate mathematics degrees are awarded by colleges and universities that do not have doctoral programs. At these institutions, students may not necessarily get enough training focused on discovering and presenting mathematical knowledge. So, working with students on research is an essential complement to their undergraduate education.
Some mathematicians feel that students interested in research should simply continue to graduate programs and do “real research” there. First, I think that undergraduate research can be real research, and I will talk more about this later. But as a general response to this perspective, I offer an analogy. When a child starts to learn how to ride a bike, they go in stages. First, they have training wheels, then the wheels are removed but the very concerned adults remain jogging right next to the kids to catch them in case they are about to fall, and finally the stage arrives when the kids ride free and unassisted. The second stage is very short but important in building confidence and self-assurance. Undergraduate research plays the role of this second stage. In our classes, students learn mathematics using training wheels. The problems are not too difficult and they have all the tools that they need to solve the problem. Undergraduate research is a short experience where the safety net is removed, students explore their capabilities, but the faculty is nearby to make sure that students do not fall or, if they do, to encourage them to get up and continue working.
I also believe that it is important to provide mentoring of students from a diverse range of backgrounds and demographics. Doing research with students that are ready and prepared to do research is very exciting. Doing research with students that will greatly benefit from having companionship, guidance, and mentoring is similarly extremely fulfilling. For this reason, I have focused my attention on first generation and other underrepresented groups in the STEM sciences. At the end of the day, these groups of students have produced mathematics that are just as beautiful and significant as any other “top students”. But this group usually lacks knowledge about graduate school and the diverse jobs that exist outside of academia which require advanced degrees, and they tend to be more aware of their perceived mathematical limitations. They also have to fight against some preconceived notions in their family and society about their future careers.
How do you pick a research problem?
A research problem must truly be carefully selected to simultaneously provide an honest “real research” experience for the student but also an experience that is meaningful and productive. I do not think that the ultimate product should necessarily be a research paper. But at the end of the program, a student should be able to pinpoint some specific contribution to the subject that is entirely their own. I have always been open to both concrete problems and open-ended investigations that start without a clear target. The main thing is to pick a problem that is flexible enough to get adjusted according to students’ needs so that there can be a successful outcome at the end of the experience. Partial results, conjectures, and databases of non-trivial computations, or even a detailed report regarding the pitfalls in a certain approach are great examples of positive outcomes.
Yet how does one find such problems? It is usually difficult to find problems that satisfy the above constraints and that are also in the faculty’s research program. So one has to be willing to expand the search. In my opinion, there are three main sources: articles, talks, and conversations. Read undergraduate research journals like Involve, Principia, Rose-Hulman undergraduate math journal, SIURO and the Minnesota Journal of Undergraduate Mathematics. CURM has a more complete list of undergraduate math journals. Travel to conferences like the Joint Math Meetings, Math Fest, Field of Dreams, SACNAS or the NSF Mathematics Institutes’ Modern Math Workshop (at SACNAS). Most of these conferences have poster and/or talk sessions devoted to undergraduate research. Talk to colleagues or presenters at conferences or workshops.
Through the years, I have done a combination of the three activities detailed above. I would usually write a note on the main area of research, a certain open problem, and some references. I would then read a couple of introductory papers, write a short introduction to the problem and perform some computations. At the end, I have a 4-5 page self-contained note that I can use to remember the problem or give to a group of students. Some of these notes get refined over the years as students work on some aspects of the problem or discover new avenues to pursuit.
How do you manage a research group?
First, I always start a collaboration with a crash course on the subject. My goal is to provide all the necessary information that the students need to understand the given problem and be able to do some experiments. I do this to cover the background material in the shortest possible time but also to establish a relationship with the students. Once students start working on their problem, I meet with them every day. In a short 15-20 minute meeting, each group presents the advances and challenges of the previous day. Only one student in the group presents on a given day and he or she must discuss the advances of the entire group. Students rotate through the week and at the end of the week they give a beamer presentation on their weekly advances. After this presentation, I discuss the goals for the weekend and the following week but also improvements on their presentation and their report. I have found that at the end of the program, it is much easier to compile the partial reports into the final report and the weekly presentations into a poster or a final talk.
As I have mentioned above, the experience must be a real research experience. So I listen to the students, point them to some useful references. I also give them suggestions. But mostly I act as a cheerleader. Using the bike analogy as before, I am not as close as to catch them when they fall, but I am always right there to cheer them up and continue looking at the road ahead.
Undergraduate research is usually a short time event. But faculty involvement is a long race. It takes time to find good problems. It also takes time to learn how to interact with students to improve their abilities and confidence while making sure that students retain ownership of their own work. It takes time to successfully find funding to support this activity. And despite all the time it takes, one only affects very few students every year. But even then, for me it has been one of the most rewarding activities that I have been involved in.