Man vs. nature. Red Sox vs. Yankees. Mothra vs. Godzilla. Our world is rife with conflict. The one I want to focus on today is pedagogy vs. curriculum.

Pedagogy refers to the art and science of teaching. Curriculum refers to the material taught. Put simply, curriculum is the "what" and pedagogy is the "how." While the line between pedagogy and curriculum can be blurry at times, it's a crucial distinction nonetheless.

Within the world of college/university mathematics, curriculum has pummeled pedagogy into a bloody pulp. I've observed mathematicians engaging in discourse on teaching all over the country, and 99 times out of 100, these conversations are about curriculum. With good intentions, we obsess over "covering enough material." We worry about the sequencing of our departmental classes. We strive to find new and interesting mathematical examples, homework problems, and projects to incorporate into our classes. These are laudable efforts, but they only get at half the picture.

If you are an instructor, I want to challenge you with a question: Just because you cover something, does that mean that your students now know it? Educational research has for decades told us that the answer is no. (Here's a classic example from "A Private Universe," showing graduating Harvard students unable to explain what causes the changing of the seasons.) And yet this notion seems to underlie much of what still goes on in today's college mathematics classrooms.

The neglect of pedagogy relative to curriculum at the college level is hardly surprising. The deans, department chairs, and instructors are trained experts in mathematical disciplines, and not in teaching. But just like Sherlock Holmes needed his foil of Moriarty to thrive, curriculum needs pedagogy to have its full potential unleashed.