This is a post written with the goal of reaching undergraduate students who have participated in a research experience for undergraduates (REU) or are in the process of completing one. I have been thinking of the lessons that students can learn during a research experience, especially how to make the positive effects of the REU last well beyond the summer, perhaps forever. Like most things in life, this requires effort from the student as well as the research mentor, so I urge faculty mentors to consider these suggestions too. What follows is a result of conversations with faculty members Lei Cheng (Olivet Nazarene University), Albert Kim (Middlebury College), Qin Lu (Lafayette College), Neelesh Tiruviluamala (University of Southern California), Talitha Washington (Howard University), and Ping Ye (Quincy University).
Maintain your new study habits: As an undergraduate math major, students typically go to class to listen to the professor lecture, take notes, read sections of the textbook, do homework and study the day before a test. It is common for students to spend only a short time working on each problem and move on. During the REU, things were different. You most likely worked on a single problem for days, maybe weeks. The important lesson is that research mathematics means advancing the boundaries of what is known, which is not done in a few minutes. It takes perseverance and the pursuit of many ideas that might lead to new insights. Most of the time the ideas will not work but they will hopefully provide the building blocks for a final breakthrough. So what is the point? The point is that this way of looking at research problems translates into a more mature way of looking at all problems: you think of ideas of how to go about solving a problem. But don’t give up too quickly. If your idea does not work, try a new one and reflect on what went wrong and what went right. It is even helpful to consider new solutions of solved problems. The whole point is to learn.
Open endedness of problems: One thing you learned at the REU is that research mathematics is about addressing problems that are not always neatly presented. They are usually open ended so that the assumptions or hypotheses have to be formulated by you as part of solving the problem. Research also involves reading journal articles and textbooks to try to find connections between the problem at hand and related problems that have already been solved. This gives you an idea of what graduate-level mathematics is like, especially in Ph.D. programs.
Keep the work going during the academic year: If you enjoyed the research experience, you likely had to wrap up your work because the REU program ended, not because all the problems were solved. Students should consider continuing the work during the next academic year even if it has to be done long distance by email, Skype and Dropbox. Research mentors sometimes offer this possibility but even if they don’t, the student can suggest it. The goal of this follow-up work should be to develop a publishable paper. This makes the time investment worth it.
Maintain a relationship with your mentor: Your research mentor automatically becomes a great source of support, especially regarding other internships, graduate school and fellowship applications. It is important to maintain a lasting professional relationship with the research mentor. Stay in touch, send email updates each semester, send ideas of follow-up problems related to the REU topic, etc. This applies even if, for some reason, the research mentor and the student are not on the best terms. Maintaining a professional relationship is important.
Communicate mathematics effectively: At the end of your REU you prepared a professional presentation on your work and wrote a report with your results. Both of these, as well as the group discussions, are forms of communicating mathematics effectively. Keep on practicing this type of communication by volunteering to give presentations at your math club or undergraduate seminar. Go to math conferences where undergraduates have opportunities to present. Being able to communicate mathematics effectively is hugely important for the future regardless of whether you end up working in academia, industry or government.
As Talitha says, “math on!”