There are four main tenets in my teaching that I carry into each classroom and office experience:
To Err is Human
Keep the Endgame in Sight
Making Eclectic Connections
It's not enough in my courses to simply read the results from a calculator or guess an answer by fumbling at a theorem we just discussed. I expect my students to talk about why an answer is correct or how to work through problems with different methods.
To ask--and answer--these insightful questions, students must not only connect with me as the professor, but more importantly, with their peers. Before thoughtful peer-to-peer discussions can happen, students must feel comfortable voicing opinions. To ease into an engaging dialogue, I typically start material with a seemingly unconnected topic to introduce the concepts of the day. These topics range from the classes' favorite doughnut flavors (one-to-one and onto mappings in a non-math-major course) to Super Mario Cart (Frenet frames in Calculus III) to the current opioid epidemic in New England (compartmental models in Differential Equations). This helps to establish both a conversational tone in the classroom and space for discussion.
The oddball introduction can carry into the day with examples. By encouraging students to provide their own questions and help me refine my examples shows that mathematics is a dynamic discipline, ready to be investigated in a myriad of directions. While constructing examples, students get defensive if I pause the stream of ideas to simplify the problem: Students want realism and descriptions of the world they experience. As we realize that the details of the example don't always fit in the context of the original equation, there are laughs as we collectively fumble an explanation of the unrealistic situation while maintaining a sense of mathematical décor.