Mathematics/Computers/Computational Sciences

Leader: Thomas F. Banchoff, Professor of Mathematics, Brown University

Recorder: Kathryn J. Kent, PhD, Conference Consultant

Main Points

Ways to engage undergraduates in mathematics research are far less obvious than in the laboratory sciences. Increased use of computers in mathematical research has in some cases established coding as an analog to washing beakers in a lab setting, but there are still widespread perceptions by many faculty that it is impossible for undergraduates, particularly first and second year students, to contribute to productive research. There are, however, examples to the contrary, in which students develop their own research projects or faculty research is carefully and thoughtfully broken down into projects that students can tackle.

Mathematics research was formerly a solitary activity and one that did not typically include undergraduates. In recent years, however, there has been more interaction between mathematicians and those with laboratory models such as physicists and bio-statisticians as well as computer scientists. Students with more and more computer skills present an obvious resource, since the skills are needed in new research endeavors. The National Science Foundation has also spurred undergraduate research in mathematics through its new requirement that grant proposals include undergraduates and through curriculum development projects.

Experimentation is important for undergraduate research, first with pencil and paper and now with computers. Together, experimentation and theory offer a good way for students to obtain a broader background and study things beyond traditional classes.

Challenges

The main challenge is demonstrating to and convincing faculty that it is possible to have undergraduate students conduct successful, productive research. New opportunities for students to present work at national conferences are one means to introduce this.

A second challenge is to afford faculty the time and experience to break down a large research question into something undergraduates can work on. This can be alleviated in part by budgeting for these activities in grant proposals, which now encourage applicants to include undergraduates.

It is also a challenge to get students involved in pure math research other than coding (the "moral equivalent" of washing test tubes). To a certain extent, the level of involvement depends on the scale of the project. By partitioning a problem sufficiently, faculty should be able to involve students in the problem, but that takes time. The kinds of activities where students get involved in applied projects from industry are more prevalent.

A final challenge is to devise ways to engage graduate students. Graduate students are not encouraged to and do not have time to mentor undergraduates because their time lines are so tight. They consider mentoring a teaching duty, but the quality of undergraduates varies enough so that it is hard to get them to see the value in their supervising them in research. The graduate student needs to see a benefit in serving as a supervisor, both to the undergraduates and to themselves. If they could see the benefit they personally would derive, they might be more responsive. Gaduate student needs help/training in how to make their own research more accessible to undergraduate students.

Opportunities

Cross-disciplinary work is beginning to be exploited as an opportunity for involving undergraduates. Such work is encouraged in grant proposals. In addition, what is meant by "applied mathematics" today is different from what was meant 10-20 years ago. There are areas of mathematics that are accessible without four years of prior study that can be applied to life sciences and other disciplines.

The professional associations are now recognizing the importance of undergraduate research. MAA conferences are starting to encourage undergraduate students to present research. Several years ago the AMS held a symposium on undergraduate research in mathematics and described several different programs at specific schools.

Examples of Effective Programs

The NSF-sponsored VIGRE (Vertical Integration of Research and Education) programs at New York University's Courant Institute (http://www.cims.nyu.edu/vigrenew/index.htm) and Brown (http://leibniz.math.brown.edu/vigre/) and the Courant math club provide opportunities to involve both graduate and undergraduate students in ongoing research and encourage exploration outside of traditional classroom settings, working on open-ended problems.

The Courant Institute had had students doing research as part of its numerous small working groups, but it was the students who approached the faculty. The VIGRE program formalized their role and helped create a culture that encouraged undergraduate research. There were some options for applied mathematics, but they tried to find more opportunities in pure math. Some of this was accomplished by borrowing an idea from Princeton for using computers to work on pure mathematics topics such as the Riemann hypothesis.

The Courant Institute also offers a course in which faculty, post docs and graduate students each teach separate sections, but work together to choose topics and construct and grade homework and exams.

Recommendations

We need to get students involved in more creative approaches to mathematics. This includes offering fewer traditional courses and more courses in which students are given open-ended problems where the answers are not apparent and there is not a clearly delineated approach.

We should establish more interdisciplinary discussion of undergraduate research to enable faculty learn what their colleagues in other areas are doing and develop ideas. It often helps to have more creative faculty discuss how they began research and how they found their area of research. When did they ask a question different from peers, or wonder about something that wasn't presented?

It is important to develop a culture of research, which can be done through the creation of math clubs, seminar-style courses, thesis options or requirements and by encouraging students to participate in conferences. This culture can also be achieved during classes and seminars by pointing out the types of research being done in different areas and by asking interesting open questions. Success stories should be advertised. In some cases an REU program can get the word out about math research and the ability of undergraduates to do it.

To attract students to undergraduate research:

• Give hard problems to students and see who comes up with new approaches and ideas, and encourage them further.
• Offer seminars in which faculty and graduate students present problems, point out some of the challenging questions, and then break down the problem to make it accessible. Some students who do well have gone on to publish work as undergraduates; others often do not go anywhere.
• Having a math club and undergraduate club helps encourage students to get involved.
• Poach good students from the Computer Science department.
• Introduce them to something for their end of year assignment, then have them do research over the summer and TA the course the next year. Word of mouth from upperclassmen helps, and word that this will help you get a job at the top companies also helps.
• Develop a culture of research. Mentor students, develop a club, have some success stories.

Calculus and other introductory math courses need to be more interesting and give students a different idea of what mathematics is about. It's a frightening thought that for many people out there, their only experience of mathematics is a calculus course. We need to do more to expose students to the idea of problems that have no solution or many solutions. Not all students will want to do research or will be inspired to do so from introductory courses, as demonstrated from this teaching evaluation:
Q: Is the professor enthusiastic?
A: Yes, he loves calculus. He wishes we would love calculus. I wish he wouldn't do that.