# Teaching

Teaching mathematics is about far more than instructing students in specific pieces of content. Twenty years after completing one of my courses, it is unlikely that any of my students will remember a specific piece of content she learned. However, I do hope that she will be more capable of solving challenging, complex problems; able to discuss technical material in a precise way; and able to participate in technical discussions by asking good questions to deepen her understanding. I have found active, inquiry-based pedagogies to be the most effective way for me to help students achieve these broad goals. My transition from a careful lecturer to a practitioner of learning-centered active pedagogies began in graduate school, when I took a course on teaching, learning, and course design in higher education. As I assumed a position at Washington & Lee University (W&L), I became a Project NExT fellow and began learning about inquiry-based learning (IBL) in mathematics. After attending an IBL workshop in 2014, I came to realize that some of the activities I had developed in graduate school or as a postdoc were IBL activities and began working to maximize the amount of student-driven inquiry in my courses.

There are some distinguishing characteristics of my courses that cut across the mathematics curriculum. Communication is at the core of every course that I teach. As I have gained experience, I have come to believe that working with students at all levels on clear and precise communication of mathematical ideas is critical for them to develop a deep understanding of those ideas. Mathematicians use language in a very specific way, and because of habits developed through years of practice, we do this across all levels of teaching. However, first-year students are rarely equipped to process these precise statements when presented with them, so communicating their understanding of mathematical ideas can be a significant challenge. If a student cannot convey his understanding, faculty have a hard time assessing (formally or informally) a student's work in order to help him improve. To help overcome this barrier, I make sure that my classes talk early and often (at a level appropriate to the course and students) about how to communicate mathematical ideas. Through a mixture of small-group discussions, whole class discussions, and graded work, my students obtain a significant amount of practice with guidance and support from me. I have found that when students get in the habit of explaining what they are doing, they are better able to self-assess in order to identify the areas where they need to devote effort to improve their understanding. Students now recognize the value in this work, writing comments such as "Having to explain and understand each step of a problem deepened my understanding of how calculus worked." on end-of-term evaluations.

Failure has become a dirty word in the American vocabulary, but another theme that distinguishes my courses is the value of failing productively as part of the learning process. For a variety of reasons, it can be a challenge to get students to embrace productive failure. I have found success at the introductory level by having my students read The 5 Elements of Effective Thinking by Burger and Starbird, which usually leads to us adopting the mantra of "fail nine times". In Discrete Mathematics I, students have often reflected (unprompted by me) upon the lessons learned from 5 Elements when completing their course evaluations a couple of months after we finished reading the book. Over time, I have reached a place where I feel comfortable talking with students in class about failure and encouraging them to share their ideas, even if they may not be right. In one calculus class, I had a student describe to the class how her group had sketched the derivatives of a couple of functions based on their graphs. She stated up front that she didn't feel her explanation would be very good, and in the end it did have some issues. However, her missteps gave us the opportunity to discuss some important points related to communication, and so I concluded my remarks by thanking the student for failing and helping everyone learn a bit more that day. By the time students get to be mathematics majors, they have usually become comfortable with the idea of making mistakes and learning from them, and I encourage that by allowing a large number of revisions on written homework so that students may make improvements. The persistence my students have shown in coming to discuss a flawed proof with me as they revise to resubmit over and over and over in some cases is rewarding, and it is hard to suppress my urge to run into class and give them a high five when they write a final revision that shows how they truly understand all the key elements of a complicated proof.

My first-year courses bring together multiple learning-centered pedagogies in ways that are well-suited to helping my students grow mathematically. At W&L, other sections of fall term Calculus I are set aside for students who have never had calculus before. Thus, almost all of my students have had some exposure to calculus before. For most of the course content, I use a flipped learning approach based on Active Calculus by Boelkins et al. but adaptable to other texts. The students' first exposure to the material in my class comes via reading and videos, and then in class I summarize my key thoughts, stimulate thinking and student discussion through the use of clickers for peer instruction, and support the students in group activities for the second half of the class. For some of the more theoretical aspects of first-semester calculus that are not introduced in Active Calculus (e.g., $$\lim_{x\to 0}(\sin(x)/x)$$, the mean value theorem, and the natural logarithm as an integral function), I have developed guided inquiry activities the students do in class, typically devoting an entire class period to each activity. Students are assessed through a mixture of online homework (WeBWorK or WebAssign), challenging written homework completed in groups, group writing projects, and tests. Completion-based credit for preparation assignments and active engagement in class are also part of their grades. My other calculus courses are structured similarly, although with more mathematically mature students in a course such as multivariable calculus, the in-class inquiry activities typically require a greater level of autonomy. Some of my most rewarding teaching moments have been watching students put together the pieces at the end of an in-class activity in order to discover one of the big ideas of a course, such as that gradient vector fields and path-independent vector fields are one and the same.

At the major level, my goal becomes to speak as little as possible during class so that the students can take maximal responsibility for learning. I have been fortunate to have small classes, which has led me to a modified Moore method approach that combines student presentations and group work in class. This approach is flexible enough to work with a broader range of class sizes as well. As I have become more experienced in teaching this way, I am better able to distinguish topics well-suited for presentations from those best done in groups under my guidance. Students in my upper-division courses are assessed based on participation and presentations, homework (daily preparation and mastery-based weekly typed proofs), and midterm and final exams (typically with in-class and take-home components). I encourage collaboration on homework, but the exams are completed independently to ensure individual accountability and understanding of course material. Students routinely report that my courses help them with their communication skills and developing persistence in the face of adversity, such as the student in my Winter 2016 number theory course who wrote "the course helped me to better explain very complex ideas in the form of a well-ordered cohesive argument as well as persist and learn more from my mistakes."

Technology plays a role in my teaching when it is the right tool for the job. With flipped classes, I typically use the campus learning management system for pre-class assignments. In Calculus I, the free online Desmos graphing calculator has become a favorite tool, and I am working on developing more activities that explicity have students use it. With multivariable calculus, I have typically used Mathematica with my students, but SageMath and Cocalc.com will likely be my tools of choice the next time I teach the course. (I strive to use open educational resources where feasible, and the Python-based syntax of SageMath often means some students in the room will have a basic familiarity.) I expect nearly all assignments and take-home tests for major-level courses to be typeset using $$\LaTeX$$, with submission and grading typically handled electronically through the learning management system.