# Mathematical Work

In this section, Prof. Haynes Miller and Susan Ruff describe the mathematical work that students do in the course. Under the close mentorship of the course instructors, students get a taste of what it is like to do research as they explore carefully crafted, open-ended project topics.

## Giving Students a Taste of Research

Andrew Wiles, the British mathematician most notable for his proof of Fermat’s Last Theorem, described the process of mathematical research as a journey through a dark, unexplored mansion: Stumbling about in the darkness slowly leads to an impression of where various pieces of furniture are. Eventually, perhaps by accident, the light switch is found, and it all becomes clear and logically connected. Then the process is repeated in each subsequent room.

The course attempts to provide some sense of this experience to undergraduates. Students explore new mathematical situations, perhaps by working out examples, running computer simulations, reading the literature, or pursuing ideas with a pencil and paper. They identify a version of the problem that they find approachable, seek out regularities, and attempt to explain them mathematically. They experience some of the struggles of research, and in the end, develop an understanding of some aspect of the original problem, often far removed from the concept motivating the project topic. This is fine; research in mathematics is largely about pursuing one's own angle.

## Criteria for Good Project Topics

Students begin by choosing from a list of over 40 project descriptions that have been developed by the mathematics faculty over the past decade. Each project description presents an open-ended mathematical situation, suggests some relevant questions, and allows students to define and pursue a range of research directions. The success of the course depends upon a list of good project topics.

The following are characteristics that we have found to be important features of a good project topic:

- Open-ended, with a variety of research directions. In mathematical research, we rarely aim at one specific target. Instead, we look at a general area and try to find the parts of it that are both interesting and accessible. We want students to experience the challenge of defining and pursuing their own specific research directions.
- Accessible to students across a spectrum of backgrounds, interests, and research abilities. It is important that the projects appeal to students with different levels of preparation or areas of expertise. Our projects have spanned pure mathematics, applied mathematics, and combinatorics. It has been quite remarkable to see what different teams have done with the same projects and to appreciate the diversity of results that students have produced.
- Manageable in a one-month period. We want the students to have a taste of what it is like to do mathematical research, but we only have about one month for each project, and students often enter the course with little to no research experience. In contrast, professional mathematicians often spend months or years on a project and submit a paper only after substantial progress has been made. Our research topics are open-ended but focused enough that we believe most teams can make reasonable progress within the short timeframe.
- Difficult to resolve via external resources. We have found that absorbing existing research can be distracting or overly time-consuming for students, and we recommend that they focus on exploring the mathematics themselves instead of diving into a full-blown literature search. Most of our topics are not associated with extensive literature and are not easily searchable on the web. When there is a giveaway term, we often rephrase and disguise the problem in a way that makes it much harder to find. It is fine if students find existing research on a topic, but they are then expected to build upon what they find or pursue a different direction.

## Developing the Project List

Developing a project topic list appropriate for the course has been challenging and has benefited from the input of many individuals.

### The Original Project Topic List

In 2003, we started to develop the project topics for the course. We began with projects used in similar courses at places like the University of Chicago and the University of Cambridge, but we typically found that they were too prescriptive for our purposes.

We modified the project topics to make them more open-ended and flexible, and we also proposed many new problems. We hired students to try them out as undergraduate research projects. Prof. Michael Artin was one of the main people gathering these projects and working with testers. He would pull colleagues into his office and demand undergraduate research topics. Many people in the department contributed their own favorite problem of this level to the list. This process created an initial list of two dozen or so projects.

### Refining the Project List over Time

The project topic list constantly evolves as we learn what works and what doesn’t; what warrants clarification, additional explanation, or less description; how challenging or accessible various problems are; and so on. Some project topics turn out better than others, and projects are retired from use if deemed ineffective. Project descriptions are often re-written. Each faculty lead is encouraged to contribute new projects, and many do.

Many of the project topics are accompanied by comments for the mentor. It is important that students are allowed to find their own way, but knowing something about the mathematics behind the project can help the mentor direct students to more fruitful avenues of investigation. Each semester, the instructors are encouraged to add more comments for use by future instructors.

## Sample Project Topics

The following project descriptions are examples from the project topic list given to our students.

- Decimal Expansion (PDF), as described in the "A Taste of Research" video in the Introduction section
- Coloring Knots (PDF), as featured in the writing workshop video in the Writing section
- Number Squares (PDF), as featured on the Sample Student Papers page (student paper draft, instructor comments, final paper, and debriefing video) and on the Sample Student Presentation page (practice presentation and final presentation videos)
- Tossing a Coin (PDF), as featured on the Sample Student Papers page (student paper draft, instructor comments, and final paper)

## Students' Research Findings

With open-ended projects, students are not the only ones who might worry about what they will be able to produce. When the course first started, we as instructors wondered: Would students be able to accomplish anything in a month-long, research-type project? Would the course be too challenging for the most inexperienced students, many of whom enter non-mathematical fields after they graduate? Would it be too basic for the most advanced students who enter the course having already published papers in professional research journals? It turns out that the opposite is true: students almost always find something new and interesting to contribute while working on the projects, and learn something over the course of the semester. It is rewarding to see the students’ creativity and imagination at work in this open setting.

We expect every student to make progress that is commensurate with his or her background and experience. Some students barely make any progress on their first project but eventually become more confident and capable at approaching research problems and at describing their findings. Because the course is so open-ended, students can find a project that is accessible to them and define a sub-question or attempt an approach within their grasp. Our most accomplished students can learn to be more patient and improve their teamwork skills. They can gain experience tackling new research projects, and sometimes they find brilliant solutions. Many projects lead right to the frontier of current research, and students gain an understanding of the effect of deeper mathematics on these fairly naïve questions. These are all important experiences for budding mathematicians.

## Research Mentorship

For each project, each course instructor mentors three of the student groups. The mentor typically meets with each group once a week. These meetings are similar to meetings that a faculty member might have with a graduate student. Often, we as mentors do not know any more about the project than the students do, and we enjoy thinking through the issues together. We try to be honest and sincere. We do not feign ignorance. We do what we can to help the students, while at the same time giving them enough leeway to find their own direction, since the whole point of the course is for students to experience research themselves. If an instructor dominates the conversation or sets the research direction, the instructor can rob the students of the research experience.

At the same time, we recognize that most students are truly novice researchers. Sometimes we need to help students understand the research process, identify ways to tackle a problem, and feel comfortable defining a research direction. We help students cope with the challenge and uncertainty of confronting a question that doesn’t have a known answer. Sometimes weeks go by with no visible progress, and we might offer a gentle push in some direction. We try to be sensitive to students’ worries, encourage the students, and help them overcome obstacles.

After a team submits a first draft of a paper, the mentor and perhaps the faculty lead instructor and the communication instructor read and mark up the draft. Then a “debriefing” meeting is held. This meeting is attended by the students on the team, their mentor, the lead instructor, and sometimes the communication instructor. At this meeting, the students give an informal description of their research, and then everyone discusses both the mathematics and the writing. This meeting serves several purposes. It provides students with some practice at presenting mathematical ideas. It serves as an opportunity for the instructors and students to discuss the students’ research, ask each other questions, and explore possible ways to push the students' work further or in new directions. And it gives the instructors a chance to talk through improvements for the paper, rather than depending entirely upon written commentary.

More broadly, the students as well as the instructors can appreciate this close, working relationship. Many students have never worked with a faculty member, postdoc, or graduate student this closely before, and the course can give them a personal connection to all three. For the instructors, one of the most enjoyable things about the course can be getting to know a particular group of students very well.

## A Recommendation Against Using Existing Literature

In the first class of the semester, we explain to the students that using literature is part of doing research, and they can look at the literature if they want to. We specifically designed the problems so that no matter how much students search, there will still be open directions for them to pursue. The only thing they are not allowed to look at is projects done by students in earlier iterations of the course.

In the first class session of the Spring 2013 offering of the course, we told the students that the course would be more fun if they did not consult the literature. To our surprise, almost none of the students turned to existing literature. We thought it was very successful this way because students seemed to gain more from developing their own ideas and trying things for themselves. This focus on individual discovery seemed to have a positive effect on the course.