We outline the structure of a term-long graduate course on teaching undergraduate mathematics. We analyze reflections written by the graduate students participating in the course to understand how they experienced the course and how the course impacted their beliefs about teaching. We find that students enjoy the course and find it to be useful. They report learning a number of basic teaching techniques and develop their personal teaching philosophy. While some students start to think about novel teaching techniques, they are still in need of in-service professional development to truly hone their craft.