Launchings from the CUPM Curriculum Guide:
Avoiding Dead-End Courses

David M. Bressoud November, 2005

Recommendation A3 : Ensure the effectiveness of introductory courses

General education and introductory courses in the mathematical sciences should be designed to provide appropriate preparation for students taking subsequent courses, such as calculus, statistics, discrete mathematics, or mathematics for elementary school teachers. In particular, departments should

There is something perverse in the way many mathematics departments structure their general and introductory courses. Either a course is outside the prerequisite structure, which means, in effect, that the department sees it as a dead-end course that prepares the student for nothing else in the mathematics curriculum, or it fails to accomplish its purpose of adequately preparing those students who enroll in it. Far too often, few of those who take college algebra or pre-calculus choose to continue, and few of those who do go on manage to succeed in the next course.

There are many students who come to college after a bad experience with mathematics in high school. A good introductory course should be able to turn this attitude around. There should be some students in every introductory course who find their interest in mathematics re-ignited, who finish the course wanting to learn more. Departments need to anticipate these students and find ways for them to build on their successful experience. Too often, the mathematics for Liberal Arts course succeeds in energizing some of its students, but the department then knocks these students down with the message that this course was actually irrelevant to any further mathematics they might want to study. I know of no college that has grappled with this issue more effectively than Mt. Holyoke College. Their “exploration courses” are worthy of consideration and emulation. More information is available at

I am quite proud of the fact that at Macalester College, students can, and sometimes do, work their way up through sophisticated mathematics without ever taking calculus. In addition to the statistics sequence, students can start with Discrete Mathematics and continue through Linear Algebra, Geometry, Number Theory, Combinatorics, Abstract Algebra, and advanced Discrete Modeling, and without ever being told that they need to know how to differentiate or integrate. Calculus is required for a major or minor, but for the student who just wants to explore interesting mathematics, there is an opportunity to do this in depth without studying calculus.

The thrust of recommendation A.3, however, is toward those courses that claim to be part of the pre-requisite structure, but fail to fulfill their role. Most departments that I have known have an anecdotal impression that their college algebra and pre-calculus courses are not working; that very few of the students who take them with the expectation of using these courses as preparation for calculus actually succeed in making it through even a single semester of calculus. The first part of this recommendation asks us to move beyond anecdotal evidence, to find out what really happens to these students. The second part is an appeal to use real information to modify existing courses and to advise students so that they will take the course that will most help them.

The Illustrative Resources can connect you to information from others who have grappled with these issues. Among the college algebra/pre-calculus assessment projects that are described there, several have been conducted with the support of the MAA project SAUM (Supporting Assessment in Undergraduate Mathematics). These include a college algebra study at Cloud County Community College ( and a pre-calculus study at San Jose State University (

As described in October’s column, CRAFTY has launched an extensive, 11-institution study with control groups to explore the effectiveness of modeling-based college algebra. Reliable assessment is a big piece of this project.

Our introductory courses are far too important for us to be able to ignore vague feelings of unease about their effectiveness. We need to know what is really happening to the students who take them. We need to think about how we can support, challenge, and entice these students to continue the task of broadening and deepening their mathematical knowledge, even those students who would prefer a non-traditional route into mathematics.

[1] Tables E.2 and TYR.4 in Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2000 CBMS Survey, David J. Lutzer, James W. Maxwell, Stephen B. Rodi, editors, American Mathematical Society, Providence, RI, 2002

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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