One of the reasons I feel very fortunate to have been in Penn State's doctoral program for six years is that this program came with a substantial teaching load and we were given the opportunity to teach a variety of undergraduate math classes. I additionally taught two renditions of a special topics course for graduate students ("Elliptic curves and complex abelian varieties") at the University of Milan where my position was reserach-focused, but my next position at Emory University, which I held for three years from 2019 to 2022, was entirely focused around teaching large undergraduate math courses. In fact, because this was my first position where teaching was the primary focus, and because of the high standards expected of both students and faculty at Emory, I feel that I traveled farther in my journey as an instructor just in those six semesters than at any other time in my career. I am pleased that my new position at Wesleyan University still involves some teaching and is allowing me to continue on my path of development as an instructor.
One thing that sets the area of mathematics apart from other disciplines is the particularly important role that quality of teaching plays in instigating enthusiasm and a commitment towards greater understanding. I have heard time and time again from students who have a strong dislike or fear of math that they felt that a particularly bad experience with one teacher had permanently turned them off to the subject. On the other hand, there are many who claim to have first discovered the power and beauty of mathematics through the guidance of an excellent instructor. I have therefore felt a strong sense of responsibility towards the hundreds of college students I have instructed, and I have tried to emulate the characteristics of those instructors who inspire excitement and interest in math.
Through my experience as an instructor (and as a tutor), one issue for math students has stuck out to me over and over is the common misconception of math as consisting of rote memorization of arbitrary rules. In the case of many students it contributes to a resentment towards the subject as a whole or to traumatic experiences of already feeling like a hopeless failure at math that many students bring with them when they enter my math classes. A conscious effort to mitigate (and in many cases, undo) this damage therefore forms a core component of my teaching philosophy. My strategies for combatting the common perception of math as dreary at best and terrifying at worst because of revolving around rigid rules can be briefly summarized as follows.
1) The material in any particular course should be presented as the narrative of a story of discovery, with motivation provided to anticipate new results, concepts, and definitions, which should be enhanced afterwards by ``sanity checks" and explanations of what they now allow us to do.
2) It should be stressed that answers on assessments are a \textit{form of communication} between the student and the grader (myself), and therefore there are no strict rules for formatting or notation as long as the student is able to effectively communicate to me that they understand what is going on.
3) There should be emphasis on distinct but equally valid approaches or viewpoints when solving certain kinds of problems rather than ``the one correct way". When giving one-on-one help on a concept, this includes feeling out the approach comes most naturally to an individual student.
It was visible to me from the moment I entered the Emory mathematics department that this environment is one where innovation and experimentation in teaching techniques is strongly encouraged, and I seized on the opportunity to contribute more ideas in my coordinated courses and to strike out more independently in structuring in the courses that have fewer sections per semester. Some of the things I have tried for the first time at Emory include the following:
• group quizzes taken during class (pre-COVID), often directly preceded by variants of the same problems for students to think about individually;
• mastery-based quizzing, where grades are counted for individual topics and where students get multiple attempts to demonstrate understanding on each topic over several quizzes (nearly every comment of student feedback on this has been a positive one!);
• active learning during class time, with worksheets designed by me so that they guide students through discovery of corollaries and relevant techniques regarding big topics learned in lectures (this became my default program for synchronous class meetings online while I have become proficient at creating topic-by-topic lecture videos for my students to watch asynchronously).
Like most of us university instructors, I was forced by the event of the recent pandemic to gain substantial experience teaching remotely. Overall a silver lining of the very unfortunate situation that has resulted in the switch to online teaching is the way that it has provided me with insights into the nature of mathematical learning and opportunities to try out even more new things as an instructor, as well as compelling me towards greater conscientiousness with regard to vast diversity of student backgrounds and a stronger drive to structure my courses and lesson styles to mitigate the resulting inequities as much as possible. I have always taken advantage of the wider community of instructors with whom I am able to discuss specific teaching strategies and technological tools as well as the principles of math education (currently I am doing this through a regular cross-departmental teaching seminar at Wesleyan). I hope that my future academic positions will allow me the opportunity to continue to be engaged in higher math education and to continue to grow as an instructor.
A further exposition of my teaching philosophy may be seen through an "extra page" that I include in every syllabus, which contains what I call "meta" guidelines for having a smooth journey in my math courses. These guidelines can be found here.