“It is good to express a thing twice right at the outset and so to give it a right foot and also a left one. Truth can surely stand on one leg, but with two it will be able to walk and get around.” (Friedrich Nietzsche)
To me, mathematics is surely a two-legged beast. The left leg is built from definitions, axioms, theorems, and rules, and provides the language required to discuss, prove, or disprove ideas. The right leg is formed by motivation, meaning, and intuition. It is the right leg that provokes questions, seeks to make connections, and provides context for the results described by the left. A student with strong mathematical legs can “get around” the world of mathematics, travelling and exploring independently.
My role as an instructor is to help my students strengthen both of their mathematical legs, and to model how the two can work most effectively together. In addition to ensuring that students have a solid understanding of the course material, my goal is to encourage students to communicate ideas effectively, relate new material to prior knowledge, apply mathematical techniques to various situations, and test the limitations of such techniques.