Launchings from the CUPM Curriculum Guide:
Learning to Think as a Mathematician

David M. Bressoud September, 2006

One of the issues with which CUPM struggles when writing its curriculum guides is the level of specificity for junior- and senior-level mathematics courses. Should all mathematics majors be required to study modern algebra, analysis, probability, complex analysis, modeling? While certain of these courses are needed by students who will continue into graduate work in mathematics (see recommendation D.3), CUPM has long embraced the concept of a flexible major that can meet the needs of a wide range of students, including those who neither need nor want preparation for a doctoral program in mathematics. What the committee does want to see are programs that require students to study at least one topic in sufficient depth that they are analyzing sophisticated mathematical arguments and formulating their own. For this reason, CUPM recommends that all mathematics majors should have a year-long sequence in a single topic so as to build depth of understanding. This sequence could be in algebra or analysis or probability/statistics or applied mathematics. The topic is less important than the experience of grappling with and learning how to master difficult material.

While the first of these recommendations is relatively easy to implement, the second, work on a senior-level project, is much more demanding. Yet it can be the crucial difference between the student who has been credentialed and the student who has begun to operate as a mathematician. One-on-one interaction with a faculty mentor, learning new mathematics to the point of mastery on one’s own, being forced to communicate one’s insights both orally and in writing, these are the experiences that truly prepare students. These require a large investment of time and effort from the faculty, and it should not be surprising that the examples given in the Illustrative Resources predominantly come from small institutions.

But not all the examples are from small colleges. UCLA has a very successful program for its honors students that produces high quality undergraduate research (see The University of Chicago has a successful Directed Reading Program that concentrates on learning how to read original research so as to be able to explain it to others (see

Macalester College instituted a required senior capstone paper for majors in Mathematics or Computer Science in 1996 (see One of the high points is Capstone Day when the seniors give 20-minute presentations on their work. All mathematics and computer science classes are cancelled that day. Students in those classes are required to attend and write brief summaries of at least two capstone presentations. This means that each presentation has a large audience of undergraduate peers who know little or nothing about the topic. A lot of work goes into helping our seniors put together an effective presentation for such an audience. The experience is exciting for our seniors. It is eye-opening for the students in the audience, giving them some idea of the range of topics one can study in mathematics and inspiring them to think about a major in mathematics. And, our seniors find that this is great preparation for job and graduate school interviews. Interviewers are always impressed when the candidate can talk passionately and intelligibly about a mathematical topic explored in depth.

Original research, especially publishable research, is wonderful when it can be done. We have found that that is almost always too high a bar. Most of the benefit can be obtained from exploration, analysis, and synthesis in a topic that is new to the student. This still requires a lot of one-on-one work with faculty, but the rewards are well worth the effort.

Do you know of programs, projects, or ideas that should be included in the CUPM Illustrative Resources?

Submit resources at

We would appreciate more examples that document experiences with the use of technology as well as examples of interdisciplinary cooperation.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at