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.
Invariably, if an instructor makes a mistake in class, it is often stated as a teachable moment--usually while the instructor recovers from the error. This phrasing can backfire for the student who is afraid of being the "stupid" student--rather than the professor acknowledging an error occurred, students unwittingly learn through the thinly veiled cover-up that mathematics is all about perfection and never about mistakes.
I was one of these instructors not so long ago. I was afraid of incorrectly starting a proof or trying the wrong integration technique on the fly, or even something as minor as transposing numbers while writing a problem. It became so stressful that I found myself constantly checking the material and afraid to waver from notes I had carefully written beforehand. I wanted to be a beacon for precision and accuracy, but ended up struggling to engage students.
It occurred to me one class that it was perfectly acceptable for students to see me as a human, as someone who drops a negative sign or carries the wrong variable through the problem. If a student saw me try the wrong integration technique, then perhaps they would learn mistakes happen; more importantly, they might discover methods to realize something went wrong, how to rethink the problem, and steps to correct it. Forgiveness of imperfection allows me to address questions as they come up, even without the solution handy. It creates a dynamic classroom and gives students space to fumble their solutions and, more importantly, to see how to correct those errors.
As a young graduate student in front of an auditorium of 250 students, I thought I had to purposely disregard factors that made students different from each other. I was convinced if I ignored issues of race, gender, or first-generational status, I was leveling the playing field. Students would have equal opportunities to carve their own path, follow their own dreams, realize their own potential. They could truly find themselves.
I was wrong.
Identities matter. Background matters. Stability of life outside of the classroom matters. If a student doesn't know how to afford meals and gas in their tank in the same week, or if they are afraid of police at a traffic stop because of the color of their skin, then that matters. Students bring with them these aspects of their life into the classroom, and I cannot think the learning environment is unchanged by this. As a transgender person, I emphasize the importance of names and pronouns and encourage students to privately--or publicly--tell me theirs. With my commute, I talk about how I always post notes online. As a person involved with the campus community, I encourage students to attend events from a variety of political, religious, and academic perspectives with me. These things may not be rooted in mathematics, but by sharing aspects of my life and my experiences, students see themselves reflected back. When students feel their identities are valued, they demonstrate higher levels of resilience under failure conditions and are more likely to ask for help in the moments they need it.
At the end of the day, students may forget how to use the Integral Test when determining the convergence of a series or that the phrase "Let epsilon < 0" is never the best way to start a Real Analysis proof. Students will, however, remember their experience with professors in weekly class periods and the occasional office hour visit. As I talk about my own research or advise students through post-graduation options, I emphasize sometimes there isn't a perfect decision, but rather just a decision. When students transition to their brave, new, post-graduation world, my goal is that they remember the curiosity that I foster in my classroom and ask their own questions. My teaching isn't about rote memorization and convincing students to complete blasé activities; it is about conveying excitement about the unknown and seeking out questions that don't yet--or may never--have answers.