# Teaching Statement

Unlike many mathematicians, I came to my subject by way of a teaching failure. The standard story tends to be that some positive teaching experience(s) (from instructors, relatives or friends) connected with the future mathematician and conveyed how the subject could be rewarding and addictively curiosity provoking. In contrast, my first exposure to rigorous mathematics arose from sheer frustration. I clearly remember feeling at the time that the teacher's totally unconvincing pretty drawings provided only the comforting veneer of a justification. That described my reaction to most of my math classes in high school. It wasn't until I reached The Fundamental Theorem of Calculus that I was faced with a fact I knew must have some decent explanation; but not one I could figure out while ignoring lecture. Deriving little of value from math lecture wasn't unusual for me at the time, and I might have simply forgotten the matter if my instructor hadn't responded to my skepticism by merely pointing back at triangles he'd drawn and asking what part didn't make sense. Well all of it, of course; rotating polygons in your head seemed about as relevant to understanding why the theorem was true as was the continued failure to find a counterexample. But I was far too stubborn to let my teacher get away with a pseudo-explanation so I purchased a book on analysis. Though I soon found a satisfying explanation for the fundamental theorem, and forgot about showing up my teacher's demonstration, I continued to read the book until I was throughly hooked by mathematics.

Once I started to work as a TA and collaborate on research with my colleagues, I found my thoughts returning to my experiences in high school. Trying to explain limits via epsilons and deltas to first year calculus students, I had discovered, was a total disaster. Naturally I wanted to share my understanding and excitement about math with my students, but the explanations which seemed so natural to me only seemed to confuse the students. The temptation was simply to conclude that the non-majors were too dumb to really understand mathematics. I would probably have simply given up on teaching if I hadn't discovered that one of my good friends found math just incomprehensible symbols until she translated them into a pictorial understanding. I couldn't deny that she was just as talented of a mathematician as I; but if I'd been the one who explained The Fundamental Theorem of Calculus to her, no doubt she would have been the one deeply frustrated. It wasn't that she found precise definitions any less important (she probably spent more time in section on the epsilon-delta definition than I did) it was simply that I understood these concepts in a more linguistic fashion while she understood them more diagrammatically.

Having suffered on many occasions from the frustration and spiral of failure that can result from not being given the kind of explanation you can readily understand, I'm passionate about giving my students explanations in several different styles. I actively take time to give diagrammatic explanations even when the linguistic explanation I've already given seems totally sufficient. For example, this last term when I needed to explain why the derivative is zero at relative mins/maxes, I first argued that a non-zero derivative meant a small change in the right direction would make the value larger/smaller; but, then went ahead and showed several graphs demonstrating that relative minimums and maximums look like places the derivative should be zero. The fact that I don't naturally understand calculus in the diagrammatic fashion that seems most common has created something of a learning curve for me at Notre Dame as I've had to learn the best diagrammatic explanation for various concepts. However, being aware of these different learning styles I've made a point of learning how to think about the subjects I teach from multiple perspectives. Indeed, I've found it most helpful at Notre Dame to collaborate on a course with other instructors since their approaches to problems or demonstrations helps expose me to other ways that some students might think about the subject.

Working with other instructors also creates more of an opportunity for another teaching issue I'm passionate about: the proper use of technology in lecture and course management. While a blackboard is necessary, certain explanations function best if they can be presented in a rotatable 3D environment or if parameters can be dynamically varied. So far I've only had the chance to create a few demonstrations (mostly by modifying existing demonstrations), but I hope to do more in this area when time permits.

Ultimately, our responsibilities as teachers and as human beings, are to improve our students' lives and balance the demands we impose on them with the future benefit we expect they will derive. I take this obligation seriously and worry over how their class and homework time would best be spent. When I started to teach I just implicitly assumed understanding math meant giving answers more like a mathematician would. For instance' I took it for granted that dealing with cases like the limit of $$\sin(\frac{1}{x})$$ at $$0$$ was important since it was the kind of distinction we would expect someone taking an analysis course to understand. Of course I knew that non-majors likely wouldn't need to use these subtle distinctions since they would probably be applying calculus to some concrete domain. However, seeing that wildly different sets of associations than those I had developed could be equally effective in deriving proofs, made me aware that non-majors might well develop intuitions useful for their subjects that bore little resemblance to what a mathematician might need. Since coming to this conclusion, I've worked hard not to spend time on technicalities and edge cases just because mathematicians might expect that kind of knowledge. It's my job as the instructor to know what kind of technicalities or unusual cases might reasonably come up in practical use, like dividing by zero when using Lagrange multipliers, and those which can be sacrificed in favor of other material, like understanding that $$\sin(\frac{1}{x})$$ lacks a limit at $$0$$.

Keeping in mind that what matters isn't how the students do on the exam at the end of the term but how much they take away from the course, it's important to realize that students may forget a great deal before they actually have to apply what they have learned. There is solid empirical evidence that students tend to quickly forget material learned for an exam, but that taking exams on that material improves recall. Thus, if resources allow, either frequent exams or regular quizzes should be used to maximize retention. We should also recognize that it's probably more important the students be able to remind themselves of how to solve a problem using the textbook than to perform the technique from memory. For instance, a useful technique is to provide an extra credit problem on exams/quizzes that offers students a few points for solving a simple problem from a section you haven't (yet) covered. Ideally students should be well enough prepared so that they can answer these problems correctly, as only then do they gain the benefits of improved recall and confidence in their ability to use written references on their own.

Since the students I've taught have usually taken prior math courses emphasizing memorizing details with a pedantic focus on manual computation, I make it a point to be casual about the memorization of easily looked up rules, e.g., integrals of trigonometric forms, and try to encourage students to look up material they've forgotten in the book when they come in for office hours. Finally, the techniques taught in non-major courses are likely to be ultimately applied in the real world with computational software, so minor computational missteps are far less important than identification of the correct solution technique and intuitions about what a correct answer should look like, e.g., vector, scalar, non-zero, curve, etc. Ideally, I would like to spend more time educating the students on the use of mathematical software to solve the problems in the course, but I have yet to teach a course with the time or resources to allow such an approach.

# Teaching Experience

### Notre Dame Teaching Experience

2008 Fall Taught "Calculus B" Taught "Calculus III" (Vector Calculus) Taught "Probability and Statistics" Taught "Topics in Logic" (covering higher recursion theory) Taught "Calculus III" (Vector Calculus)

### Berkeley Teaching Experience

2007 Spring TA for "Analytic Geometry and Calculus" TA for "Analytic Geometry and Calculus" TA for "Philosophy of Science" TA for "Introduction to Philosophy of Science" TA for "Introduction to Logic" TA for "Introduction to Logic" TA for "Linear Algebra & Differential Equations" TA for (freshman) "Calculus" TA for (freshman) "Calculus" TA for "Linear Algebra & Differential Equations" TA for (freshman) "Calculus" Attended Berkeley Mathematics Department TA training course.