Launchings from the CUPM Curriculum Guide:

Keeping the Gates Open

David M. Bressoud February, 2006

B.3: Critically examine course prerequisites

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Mathematical topics and courses should be offered with as few prerequisites as feasible so that they are accessible to students majoring in other disciplines or who have not yet chosen majors. This may require modifying existing courses or creating new ones. In particular,
• Some courses in statistics and discrete mathematics should be offered without a calculus prerequisite;
• Three-dimensional topics should be included in first-year courses;
• Prerequisites other than calculus should be considered for intermediate and advanced non-calculus-based mathematics courses.

As we consider what we as mathematics departments can do to encourage and entice students to take more mathematics, we need to think about the structure of our curriculum. The visualization of the common curriculum, shown below, is taken from the Notices article “Do we need prerequisites?”[1] by Donal O’Shea and Harriet Pollatsek. It illustrates the narrow opening through which most departments allow students to approach our courses. Do students really need Calculus II before they can profitably study linear algebra or modeling or number theory?

For students who know that they are going to major in mathematics, we should encourage them to take Calculus II as early as possible in their career. And we do need prerequisites. One of the strengths of the mathematics curriculum is that students are building on what they have learned before. But what about the students who have completed Calculus I and are reluctant to continue with calculus? Should they be forever barred from learning about number theory or combinatorics or modern geometry? If they do take a mathematics course other than calculus that interests them, is there any way to build on it without having to back up and re-enter the calculus track?

The Mt. Holyoke program described by O’Shea and Pollatsek has many routes through the mathematics curriculum. It is an exciting program worth serious consideration, but I want to speak of our more modest approach at Macalester that still succeeds in widening that gate. Our “proofs” course is Discrete Mathematics, a course whose specific content is at the discretion of the instructor, but it includes some number theory, some combinatorics, some graph theory as it helps students learn to analyze and construct proofs. It is required of all Mathematics and all Computer Science majors, and we encourage those who know they want to major in one of these fields to take it in their first semester at Macalester.

It is a great leveler. The material is challenging to almost all of our students. Our students find it a refreshing change from what many of them have come to see, even before they arrive in college, as the slog through calculus. And it is particularly effective at attracting students into our major.

We run as many sections and teach as many students in Discrete Mathematics each year as we do in Calculus I and as many as Calculus II. Only about one-in-three of the students will major in Mathematics or Computer Science. They take it just because it has a reputation for being an interesting course. Very often, they will then take one more course. They have options. From Discrete Mathematics, they can take Linear Algebra, Number Theory, or Combinatorics. With Linear Algebra, they have opened the door to Modern Geometry, Algebraic Structures, or Discrete Applied Mathematics. Students can and do explore the mathematics that interests them.

Discrete Mathematics is the second widest gate into our curriculum, but students who are hooked by Elementary Statistics or Introduction to Statistical Modeling can move on to Applied Multivariate Statistics, Probability, and Mathematical Statistics with no other prerequisites.

We do need prerequisites, but we also need to examine our prerequisite structure and recognize the unnecessary obstacles that it may present to students with an interest in mathematics.

[1] Donal O’Shea and Harriet Pollatsek, Do we need prerequisites?, Notices of the AMS, vol. 44, 1997, 564–570. www.ams.org/notices/199705/comm-holyoke.pdf

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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 bressoud@macalester.edu.