*By Steve Balady, graduate student, University of Maryland – College Park
*

**What’s the Directed Reading Program?**

“The Directed Reading Program (DRP) pairs undergraduates with mathematics graduate student mentors for semester-long independent study projects.”

This mission statement isn’t mine — it was the consensus of a group of graduate students at the University of Chicago in 2003. Since then, programs with this mission have been started at Rutgers, UConn, Maryland, MIT, UT-Austin, and UC-Berkeley. I was an undergraduate participant in the program at Chicago, and I founded the Maryland DRP in 2011. Since then our committee has overseen more than a hundred projects — freshmen through seniors, projects on areas as diverse as logic and finance, with student talks ranging from how to multiply complex numbers to a showcase of original research on nonlinear dimension reduction.

Here’s a bit more detail about how we run the program at Maryland. In the first week of the semester, graduate students in mathematics (including applied mathematics and statistics) and undergraduates (mostly math majors, but not all) submit a form telling us what they’re interested in doing. This doesn’t have to be a specific project. Sample undergraduate interests: “I enjoyed real analysis,” “Machine learning with real-world data,” “I talked about differential geometry with a professor in the department,” “My friend told me about topology and I think it sounds really cool.” Graduate interests tend to be more focused, often suggesting specific projects and the background a student needs for it. The DRP committee (a handful of graduate students) does our best to pair the students with mentors. After the first meeting, the mentor drafts a prospectus outlining the specific goals of the project. The undergraduate is expected to do about four hours of independent reading per week, to meet for a *conversation* with their mentor for an hour a week, and to give a 12-minute talk to other participants in the program at the end of the semester.

**That sounds neat, but why should I, a graduate student, start a DRP at my university? What are the benefits?**

This is a serious question — it’s critical that a school’s DRP be started and run by its graduate students. While our department gives us some money for pizza, mentors don’t get paid and students don’t get course credit.

One perspective is that this program can be a big deal for the undergraduates who participate, so it’s something that should be done. “The DRP program has given me a feel for what research is all about. Since I’ve gotten to explore my interests more, I’ve started to do research with a professor and am definitely planning on going to graduate school in mathematics.” Said differently, the DRP allows structured access to the cultural norms of the graduate community. Graduate students share anecdotes and legends of proofs gone right and wrong, we talk about professors and their weird teaching habits, and (shockingly!) we’re real people. Meeting one-on-one with an undergraduate allows us to bring some of our reality to them, and that allows them to see graduate school as a potential reality.

The other side of this is not why mentors should participate (graduate students should be doing lots of things!) but why people actually do sign up to mentor. “Because it’s fun!” “I taught a college algebra course this semester; it was great to spend some time working with a math major.” “He was excited about the stuff that I find interesting.”

Here’s a longer story, from a project I mentored — my student applied to the program to find out “how mathematicians think.” We started talking about intro analysis, and after a bit I suggested a pretty classic epsilon-delta problem — show that \(x^3 + ax^2 + bx + c\) tends to infinity as \(x\) gets large. The usual proof divides out the \(x^3\), then bounds the rest above \(0\). This method is so standard that I wasn’t expecting anything else. In particular, I definitely wasn’t expecting induction on the degree of the polynomial. If you take the derivative enough times, you get a linear function which goes to infinity, which means that eventually it’s bigger than \(1\). Now apply the Mean Value Theorem. (Awesome, right?) This proof won’t get either of us published, and I doubt that we’re the first ones to do it this way. But it’s ours to me, and the ownership of knowledge — knowing what I know, and knowing that there’s so much more — is a large part of my identity as a mathematician.

**Okay, I want to start a DRP. Do you have any logistical advice?**

- Having a diverse committee of graduate students is important. Because the success of a project is about mutual subject interest and trust between the student and mentor, pairings are much more likely to work when there’s someone on the committee who knows the mathematical content (and the personalities!) of each mentor who applies. Some of my friends who work in abstract algebra are happy to run a project on introductory number theory with a rising sophomore; others would prefer only to participate with a student who was ready to work through a chunk of a graduate-level text. (And that’s fine! We want both of these.) Maryland’s mathematics department in particular is massive, so it’s necessary for us to have people we can talk to about those same personalities of the graduate students in signal processing, or geometry, or applied statistics, or…
- Relatedly, decentralization works well. We used to have a big meeting at the start of the semester, but the one-on-one nature led to lots of timing conflicts. Things have gone much smoother since we’ve assigned committee liaisons for each pairing and let them schedule things individually.
- We try to be as inclusive as possible, but it’s tricky to reach people who aren’t math majors! We put up flyers in the math building and send out emails to the math department listserv each semester, but a lot of interest in the program comes from word-of-mouth.
- Projects can vary widely in scope, and that’s great. The important thing is that the student and mentor agree on a specific goal soon after their first meeting. Having a specific theorem/application/result in mind gives pairings a lot more direction and keeps students from feeling overwhelmed. This also helps with choosing an appropriate resource: for many students this will be a chapter or two from a textbook, but we’ve also had very successful projects that used an online open course.
- If you make the end-of-semester talks a priority, the talks will be
*really, really good*. Our policy now is to make sure that students have two full weeks to prepare their talks. Something I’ve noticed is that my students have often wanted to use some of this time as an opportunity to learn how to TeX. Sharing ideas is part of math; give them time to practice that!

**Is there anything else that would make it easier to start a DRP at my school?**

The Maryland DRP committee has made it a priority to share our resources — we want getting started to be as easy as possible. There are links on our website to all the forms that we use each semester. You’ll also find a list of all the talks that have been given since we started the program here (some of them even have slides!). If you have specific questions, you can also send me an email (sbalady at gmail dot com).

In my experience, the biggest obstacle to starting a DRP happened before I did anything — I spent a long time trying to convince my friends and professors that a program like this could work at our university. It was a lot easier just to do it! “The Directed Reading Program (DRP) pairs undergraduates with mathematics graduate student mentors for semester-long independent study projects.” Send out an email to the grad students in your department with that sentence and ask for volunteers; that’s really all it takes.