### by Samuel Nicholas Ramsey, Group in Logic

#### Teaching Effectiveness Award Essay, 2017

In my first year of graduate school, a math professor confessed to me that it was only late in their graduate school career that they learned that most mathematicians spend their time feeling completely confused. This should be obvious, but in subsequent years, I was able to observe just how well this knowledge is kept secret from undergraduates within math departments. Course curriculum tends to be presented in a neat and tidy package with little discussion of how the ideas were hammered out over years of difficult intellectual struggle. Because of their long hours spent acquiring mathematical fluency, graduate student instructors can often answer questions about calculus or linear algebra without much thought, going straight to the correct method of proof without demonstrating how one might know to reach for one tool instead of another. Students enter the classroom with different math backgrounds and, in an interactive discussion section, it can come to seem that some students just *get it* and other students just don’t. This has a significantly detrimental effect on students’ confidence and performance, leading too many students to feel like math just isn’t their thing.

My project was to place confusion at the center of my pedagogical approach. The starting point was my own memories of the sinking anxious feeling I felt in math courses in college. I had never felt so stuck and had no idea how to transcend that feeling. I want to communicate to my students that actually pretty much no one ever does. At least not for long, since learning new math always involves feeling lost. First, I explicitly mentioned confusion as a part of the process in each section, often via personal anecdotes or episodes from mathematical history. Secondly, and most importantly, I planned lessons around the *deliberative* process through which one *chooses* an approach to solving problems first, before describing the correct method. For example, I would often pose a problem and solicit suggestions from the students on various strategies for solving it, opening up the floor for a discussion of the pros and cons of each approach *before* actually giving the solution. Finally, I wrote quizzes that emphasized method choice and had questions that revisited old material over and over to privilege the slow acquisition of mathematical maturity over rapid calculative skill.

I pursued this teaching strategy while I was a GSI for Discrete Mathematics, a course that covers basic topics in finite combinatorics, logic, and probability and often serves as a transition for students who are beginning to learn how to write proofs and reason abstractly about mathematics. I evaluated the success of all this talk about confusion by closely monitoring students’ performance on quiz questions, in which they were asked to make their reasoning explicit, and by taking regular notes on level of participation and atmosphere on discussion section. By making clear to my students that they had permission to feel confused and that feeling stuck was not tantamount to failure, I saw the classroom environment become more open and participatory—students became more willing to share their provisional thoughts and speculate about solutions. And the proofs they wrote on quizzes got better and better, as they had practiced talking about mathematics and making explicit their reasoning, allowing them to organize their thoughts and ultimately discover new paths of logic leading to a solution. Math teachers try to eradicate confusion—but this aim seems to be, well, confused: confusion is a part of learning. Instead of trying to eliminate it, I sought to bring it to the fore, so that students could learn to think critically about their own methods of reasoning.