__Teaching Philosophy__My approach to teaching mathematics is based on two beliefs. First, attaining mastery is a highly active process, consisting in large part of struggling with details and managing an ever-changing degree of understanding. To succeed, one must wade through the murky waters of first concepts while experiencing a perpetual state of partial confusion. Second, mathematics is remarkably useful, not only in its immediate applications but also in the rigorous mind-frame that it cultivates. It has the unique ability to elevate one's familiarity with rigorous thinking, thereby making accessible the myriad of important ideas that mathematics has been central in developing. To put these beliefs to practice, my main objective as a teacher is to build a strong sense of confidence by motivating the kind of guided practice that is essential to the learning process.

After a few years of teaching, I met many students who expressed various incarnations of the infamous phrase: I'm not a math person. The fixed mindset associated with this self identification begins with the misconception that some people are inherently good at math while others are not, and that struggling is a clear sign of incompetence. By the time a student reaches college, this perspective can be deeply ingrained and difficult to dismantle. I begin by replying: you're not a math person until you are, and give reminders throughout the course that almost nothing in the subject is trivial. This simple acknowledgement is highly effective at encouraging participation among those who fear their questions are too naïve to warrant discussion. I also offer various analogies that give a more accurate representation of how people become better at mathematics. One that I am particularly fond of is learning how to drive a car. Most undergraduates have some experience driving, but have spent more time as passengers than drivers. I ask: is it possible to learn this skill with observation alone? The point is that attending lectures is like observing a driver from the passenger's seat. Certainly one can get the gist of the main ideas, but genuine understanding is impossible without active and first-hand engagement.

Another strategy I use to overcome fixed mindsets is giving homework with a specific ratio of technical to theoretical problems. In particular, I emphasize the procedural components of the material. The reason for this is two-fold. First, I believe that successful implementation of procedures is often accompanied by a rise in confidence. Second, homework that focuses on theory can have an intimidating effect on students as it requires a certain facility with abstract reasoning that may be absent in a large proportion of the class. However, when theoretical understanding is gradually built in the process of working through complicated (but interesting) procedures, it becomes grounded in a foundation of technical proficiency.

To create educational experiences that are not bound to in-class lectures, I design my courses to help students build effective work habits. When teaching calculus, for example, I administered online quizzes after every lecture. These were designed to give immediate feedback and encourage the kind of practice that eliminates the need for cramming. The quizzes also allowed for student interaction with the material at an individualized pace, since their work could be saved and resumed within a 48-hour time period. Moreover, the natural flow of lecture was not artificially interrupted with a pressing need to administer, proctor, and collect quizzes. Instead, I used this time to explore new concepts and refine old ones.

To foster proactivity in my students, I make myself readily available for those who need extra help. When I first began teaching, this amounted to little more than holding several weekly office hours and running a handful of review sessions throughout the semester. While this worked for some, I soon realized that a higher degree of flexibility was required on my part to include students with outside commitments. Recently, an alternative and particularly effective mode of interaction has consisted of using email to discuss mathematics. By typesetting symbols alongside diagrams and pictures, I am able to give highly personalized explanations and clarifications. As a result, email succeeded remarkably well in reaching large numbers of previously unengaged students. It also had the unexpected effect of providing students an opportunity to read and write mathematics, a crucial and often overlooked skill.

To improve my effectiveness as a math teacher, an immediate goal is to find new ways to motivate the subject's usefulness while also staying within the scope of the class. Maintaining this balance has been difficult, but I consider it crucial for developing positive attitudes among students. This is especially true in courses where the material can be somewhat dry if taken out of context. To improve my abilities in this regard, I strive to include interesting real-world applications that demonstrate the scope of the ideas being discussed. For example, while teaching linear algebra, an application that was particularly effective at generating interest was the use of orthogonal projections in optimization problems. Many students enrolled in this class were engineering majors, so framing the topic in this way led to a stronger desire to understand it and a heightened awareness of its far-reaching applications. Another example came more recently in my precalculus class. After spending several weeks on the theory of polynomials, I came across an article on the use of cubic Bézier curves in computer graphics. Since the course was designed to focus on polynomials in a somewhat abstract manner, I found that dedicating some time to constructing a Bézier curve kept many students engaged and helped them not lose sight of a main point: polynomials are everywhere.

I believe that a successful teacher of mathematics must instill in students a sense of belonging in the world of aesthetic rigor. Central to this transformation is dismantling the notion that some people are forever destined to struggle with mathematics while others are not. To parallel Ernest Rutherford's thoughts on learning physics,

*I believe that all of mathematics is either impossible or trivial. It is impossible until you understand it and then it becomes trivial.*