Teaching Research: Encouraging Discoveries

What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle-school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me. The title of this article may seem a little puzzling. After all, the words teaching and research usually only appear in the same sentence when separated by the word and, on a list of a faculty member’s obligations. Do these words belong together at all? What does teaching have to do with research or discovery? One of my favorite quotes about teaching comes from one we normally associate with research: The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world. —Henri Poincaré [5, p. 128] But Poincaré also said: It is by logic that we prove, but by intuition that we discover. —Henri Poincaré [5, p. 129] So we teach to build intuition, and intuition enables our students to make discoveries. Notice that Poincaré does not say that the principal aim of teaching is to convey facts or theorems. The principal aim of mathematical teaching is to build qualities of mind that enable students to make discoveries. But how does one do that? How does one turn a learner into a discoverer? When I was first starting out as a new professor, I might have given these answers: • Teach the needed background. • Cultivate maturity. • Inspire them! • Ask good questions. • Select the smartest students. • Give open problems. • Advertise the thrill of research. • Be an expert in what you advise. • Encourage independence. doi:10.4169/000298910X521634 This article is based on the James R. C. Leitzel Lecture, delivered by the author at the MAA MathFest, Knoxville TN, on Friday, August 11, 2006. November 2010] TEACHING RESEARCH: ENCOURAGING DISCOVERIES 759 This content downloaded from 134.173.131.100 on Tue, 2 Jul 2013 17:39:15 PM All use subject to JSTOR Terms and Conditions Now, I shall explain why I believe every one of these pieces of advice is either plainly wrong or, at best, inadequate. Along the way, I will mention some concrete teaching ideas that you may find useful as well. 1. THE PRINCIPAL AIM OF MATHEMATICAL TEACHING. There is one word that is an excellent metaphor for everything I want to say: