Launchings from the CUPM Curriculum Guide:
Only connect!

David M. Bressoud May, 2005

Recommendation 3: Communicate the breadth and interconnections of the mathematical sciences.

Every course should strive to

“Only connect! … Live in fragments no longer” [Howard’s End, E.M. Forster] is a plea to unite the dual aspects of our nature: the monk and the beast. I find it appropriate advice to those of us who teach mathematics, a plea to unite the multiple aspects of the mathematical concepts that we present to our students.

    As mathematicians, we recognize that the power of mathematics comes from the web of connections in which each idea rests. The derivative is a powerful idea precisely because it unites symbolic manipulation with the geometric concept of a tangent line and the dynamic concept of a rate of change. It tells us about the microscopic behavior of a function and so provides a means of approximation. It can be interpreted as a one-dimensional Jacobian, expressing local magnification. It is the inverse of an operation that accumulates, i.e. integration. Most importantly, it enables the language of differential equations with which we can model almost any phenomenon that changes in response to external stimuli. In other words, it is a concept at the nexus of many mathematical ideas. The connections it establishes are what make it important.

    How many of our students finish first-year calculus with an appreciation for the power of this concept? I remember my own sense of amazement mixed with horror the first time I sat down with a student to probe his understanding of the derivative. He had successfully completed calculus through several variables. He understood differentiation as a process that “turns functions into simpler functions, like x cubed into three x squared.” As much as I questioned, hinted, and suggested, I could evoke no other connections. Is there any sense in which such a student has learned calculus?

    This was the question that led to the calculus reform movement. The first bullet of this recommendation echoes its rallying cry: to teach the ideas not just symbolically but also graphically, numerically, and verbally. This is not just a call for us to draw pictures, analyze data, and describe situations. Students do not learn from what we do. They learn from what they do. They must build their own repertoires of experiences with pictures, data, and concrete situations.

    This recommendation is more than a request to embrace the rule of four. This recommendation says that we need to provide opportunities for our students to discover the rich texture of mathematics, especially contemporary concepts. The CUPM-Illustrative Resources lists many sources with suggestions for enriching undergraduate courses. A brief sampling of what can be found in the CUPM-IR includes:

    There is yet another aspect to this recommendation. That is the challenge to think about traditional courses from radically different perspectives. I believe that there are students who benefit from a calculus course that is taught from a logical, rigorously deductive point of view. I know there are students who would best learn what they need in calculus from a geometrically grounded, conceptual course. Linear algebra can and often should be taught from a geometric point of view. Traditionally, statistical analysis has been built from probability theory. George Cobb at Mount Holyoke and Danny Kaplan at Macalester have developed courses that build these same statistical tools from Monte Carlo simulations. Computational science, an orphan that often finds itself abandoned in the gap between mathematics and computer science, is in fact an important and exciting field that can motivate and lend its perspectives to much of the undergraduate curriculum. References to many of these courses can be found in the CUPM-IR.

    It is not easy to teach this way. We need to find ways to help our students discover the power of these connections for themselves, and we need to find means of determining whether or not they really are getting the message. There is a large and growing community of mathematicians who strive for this level of teaching. We need to continue to learn from each other.

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