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Teaching Portfolio

"The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader an impression being shut up in brightly illuminated cell where every detail sticks out with same dazzling clarity but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby detail gradually fades away towards the horizon." ~Hermann Weyl

Welcome to my teaching portfolio. Here you will find my teaching philosophy, a summary of my experience both in the classroom and as a mentor for undergraduate research, and some thoughts on pedagogy. Expanding the sections below, you will also find links to sample course materials for classes on applied topics as well as proof-based courses.

Philosophy

In my view, the task of the teacher is to lead the students on a clear and efficient path through the given subject. The tour should not only be informative, but also entertaining. The route should be easy to navigate, allowing those students who lag behind the group a chance to catch up, yet novel enough to give those wanting to move ahead of the pace the opportunity to explore intriguing side roads.

More often than not, the mathematics instructor is leading a tour parallel to that led by the author of the accompanying textbook. The quote at the top of the page was taken from the preface to Weyl`s book The Classical Groups: Their Invariants and Representations (London, H. Milford, Oxford University Press, 1939) and it is equally valid today. My teaching style is deeply inspired by these remarks and I endeavor to expose my students to both points of view. Without perspective, it will be difficult for them to communicate with other people who use mathematics. Without knowledge of the detailed structure of the subject, it will be difficult for them to use mathematics productively. While the style of the textbook is static, I have the advantage as the instructor of flexibility in communication. I strive to adapt to the needs of the students throughout the course while continually seeking the maintain balance between perspective and detail in my communication of mathematics.

Teachers are necessary for efficient learning. Given enough time, patience, and access to information, I believe that anyone can learn anything. That being said, it should be noted that by watching a carpenter swing a hammer one can easily acquire the idea that a hammer is the tool to be used for driving nails into wood. However, to use that hammer productively one must learn that it is best to hold it as low as possible on the handle and to apply a slight snap of the wrist at the end of the swing, thus maximizing the magnitude of inertia of the hammerhead with the least amount of effort. To learn that efficiently, most people need a teacher. I strive to teach mathematics in a way that optimizes the following minimax problem: As a result of my instruction, I want my students to comprehend as much as possible in the least amount of time. It is my pleasure to continually pursue this goal.

Experience

Both as a visiting assistant professor at the University of Notre Dame and as a graduate student at the University of Arizona, I have served as the instructor of record for a variety of courses ranging from a proof-based honors calculus course to vector calculus with applications, various versions of calculus, and elementary algebra. I have also worked as an assistant for graduate courses in differential geometry and topology and complex analysis, and designed curriculum for an entrance workshop for first year graduate students. In the spring of 2011, I will be giving a short course on Quantization to exceptional undergraduates as part of the Notre Dame Mathematics Institute. Each class level presents its own set of pedagogical challenges and I have enjoyed tackling those challenges time and time again.

The following table summarizes my experience as instructor of record, giving the names of courses taught, the number of times (if more than once), and my overall effectiveness rating from student evaluations, together with comparison data from the University of Arizona and the University of Notre Dame during the same time period. The indices are on a five point scale, with 5 meaning "most effective" and 1 being "least effective." For courses that I have taught more than once, the average of the indices is displayed, along with the average of the comparison indices.

University of Notre DameUniversity of Arizona
CourseOver. Eff.Dept. MeanCourseOver. Eff.Dept. Mean
Honors Calculus II4.93.8Vector Calculus (3)4.64.4
Honors Calculus I4.83.8Calculus II w/ Apps (2)4.54.4
Business Calculus II (2)3.83.5Calculus I w/ Apps 4.44.1
Elements of Calculus4.13.7
College Algebra (4)3.93.5

My efforts in teaching have not been confined to the classroom. I have also given a number of public lectures on mathematics to various audiences. I welcome the challenge of communicating scientific ideas to broad audiences. The titles are listed below together with some brief remarks on their content and the intended audience.

  • Mathematics and Astronomy: Kepler`s Laws of Planetary Motion. Audience: Teachers participating in the Notre Dame Math Circle Institute, along with local middle and high-school students. Summer 2010. Using the immersive environment of the digital planetarium at the University of Notre Dame, this talk motivated and defined the coordinate system on the celestial sphere used for astronomical observation of the planets, and guided the audience through Kepler`s empirical reasoning which led to his celebrated laws of motion. Part of this presentation is also part of the conclusion of my honors calculus II course in which I use calculus to derive Kepler`s Laws from Newton`s Law of Universal Gravitation.
  • Laudibilis Lemniscate Audience: Notre Dame Mathematics Club, REU students and Wabash College. Summer 2010. This talk is just about having fun with an interesting object in mathematics, Bernoulli`s lemniscate. The lemniscate is an algebraic plane curve that Bernoulli discovered and named lemniscus, or ribbon, for its figure. In a sense it is akin to other familiar curves such as the ellipse, hyperbola, and line, and shows up in many places in mathematics. As the title suggests, the point of this talk is to convince the audience that the lemniscate is laudabilis, or worthy of praise.

In the same vein, though at a different level, I have given colloquia lectures at a number of departments around the world, mostly on topics in mathematical physics. The titles are listed below, along with a brief discussion of their content.

  • The View from Poisson Geometry University of Wyoming Mathematics Colloquium. Spring 2010. This talk follows a historical arc from celestial mechanics through the study of integrable systems and the of introduction of symplectic/Poisson geometry into mechanics to motivate Poisson geometry as a subject of independent interest with connections to dynamics, representation theory, algebraic geometry and combinatorics.
  • Perspectives from Poisson Geometry University College Cork, Ireland. Winter 2008. My goal with this lecture was to illustrate results of my thesis showing how Poisson geometry and integrable systems arise naturally in the algebra of matrix factorization (and the corresponding generalizations to Lie groups and symmetric spaces).
  • Mathematical Physics of Music and Musical Instruments California State University, Hayward. Spring 2001. I gave this talk the year after my senior year and the point was to illustrate the natural way in which harmonic analysis enters the design and and analysis of musical instruments.

Mentoring

I have had the privilege of directing the undergraduate research and senior theses of three honors mathematics students over the past five years, two at the University of Notre Dame, and one at the University of Arizona. Two have continued on to graduate study in mathematics and one is pursuing a graduate degree in education. Work with such students is much more than regular meetings to discuss readings, it is an opportunity to expose them to the fascinating world of my profession and help them shape their post-collegiate future.

Undergraduate Thesis Students:
  • Vivian Healey
  • BA 2010 Mathematics and Theology, University of Notre Dame
  • Graduate School: Brown University, Ph.D. Mathematics
  • Thesis: Penrose and Ammann Tilings: Structure, Dynamics, and Noncommutative Geometry
  • Primarily original research. Invited talks at MathFest, and the Young Mathematicians Conference. Award for Outstanding Senior Thesis in the Honors College at Notre Dame. Project evolved out of an REU experience at Canisius College.
  • Sarah Pastorek
  • BS 2010 Mathematics, University of Notre Dame
  • Graduate School: Creighton University, M.Ed. Education
  • Thesis: Weyl`s Asymptotic Law
  • Primarily an exposition. Project evolved out of a reading course with me on functional analysis during the spring of her junior year and connected with an REU experience at the University of Tennessee Knoxville on spectral geometry.
  • Christopher McMurdie
  • BS 2007 Mathematics and Music, University of Arizona; MS Mathematics 2009, University of Paris Sud
  • Graduate School: Washington University, Ph.D. Mathematics
  • Thesis: The Non-Commutative Geometry of Penrose Tilings
  • Primarily an exposition. Project was carried out while I was a graduate student and overseen by my dissertation director. Followed a series of exercises I designed, researching background material, to construct a proof of Gelfand`s theorem on commutative C*-algebras, then worked to understand the set of all Penrose tilings as a noncommutative space.

I also served in a mentoring role for several other undergraduate students at the University of Arizona. In the spring semester of 2006, while teaching vector calculus, I recruited four students from the previous semester to act as preceptors in the spring course. In combination with the Teaching Teams Program at the UA, I mentored their development as tutors and review leaders. I sought to bring out and enhance the individual qualities in each student, whose presence motivated me to ask them to return as preceptors, and taught them to lead by example so as to pass on these qualities to the students of the course. It was a remarkably successful program. In exit evaluations, the preceptors praised their own learning experience through the position, and the improvement in student performance in the course spoke for itself. I would like to explore the generation of an analogous program in the future. In keeping with my ideas on course design, I envision a integrated scheme by which the preceptor experience complements the learning experience of our best undergraduate students, leading to well-rounded graduates of our program. One student enjoyed the experience so much that he volunteered to do it again for each of the courses that I taught after that. Two of the students names below are accompanied by their post-graduate engagements for which I later provided application support at their request.

  • Preceptors: Keenan Coleman (US Naval Nuclear Engineering School), Siobhan Innes-Gawn (Duke University Law School), Christopher Limbach, Sarah Gerdes.

Course Design

This is an arena in which I have experimented thoroughly and which is centrally important to my effectiveness as a teacher. From classroom interaction via ConcepTests and java applets, to problem design, to syllabus scheduling, to lecture and examinations, every component my courses is designed to reveal a part of the story I am trying to tell.

I employ a variety of methods in the classroom, always trying adapt to the needs of the individual group of students. While most of my experience has been with teaching classes in the traditional lecture format, more and more I have attempted to make courses focus on student participation. I am very fond of using visual aids (both digital and physical) during lecture. I've found that this helps to make the subject real for my students. Whenever possible, I try to include a simple, yet interesting, application of the concept we are studying. In order to design work-sheets, ConcepTests, and in-class activities for students, one needs a good imagination and experience teaching that course. Each time I teach a course again, I make it a goal to rework a portion of my lectures, create and implement a new worksheet or computer based activity, or include a new type of application. By following this process of revision I can do my best to accomplish my teaching goals.

To be effective, I have found that I must employ a multifaceted approach. Subordinate to the principal task of conveying the mathematical ideas of the course, one must subtly train students, developing their artistic and communication skills, spatial and analytic reasoning, logic, organization, and knowledge of history. What’s more, one must teach the students to read, write, and speak the language of mathematics which is largely foreign to them. In the short term, I want my students to develop a heuristic understanding of the concepts and basic computational faculty. In the long term, I want students to develop a rigorous conceptual understanding and computational efficiency. One facet of my approach is the design of layered problem sets which force the students to revisit material throughout the course. Student feedback has indicated that this helped them to build a cohesive understanding of the material. For sophomore or freshman level courses, such as vector calculus or college algebra, I also employ a regular system of brief in-class quizzes to keep the students on top of the material. These quizzes ask the students to recall the technical definitions of vocabulary from the course, and explain in their own words a concept from the reading or previous lectures. With the problem sets and the quizzes, I can ensure that the students receive ample feedback on their progress prior to exams. By clicking on the following links you can view some sample materials from my applications focused vector calculus course or my proof-based honors calculus course:

ConcepTests were developed for use in large lecture physics courses by physicist Eric Mazur of Harvard University and were later shown to have drastically improved student comprehension of introductory physics concepts. Since, ConcepTests have been developed for a variety of subjects including mathematics. In brief, students are shown a question, typically multiple choice, and asked to vote an answer using some sort of feedback system. They are then asked to discuss the question with their neighbor and try to persuade the other person that their answer is correct. Following that discussion, the class is asked to vote again and the answer is finally revealed by the lecturer.

I've found this type of peer instruction to be very effective. When written well, ConcepTests can be quite versatile, and may be used for review or even exploration of new material. The examples below are ConcepTests that I have used in teaching Vector Calculus. Notice that the first question is mainly concerned with visualization. The labels on the axes are left off intentionally in order to stimulate discussion amongst the students in the class. The second question is actually reinforcing a concept from trigonometry, but by using the equations from the previous ConcepTest it encourages the students to associate that knowledge with parametric equations. I particularly like to use ConcepTests when teaching Calculus and I think they would be of great benefit in teaching linear algebra. Most of all, I like the fact that ConcepTests give the students an active way to participate in class and provide me with valuable real-time feedback during lecture.

Recently, I have been creating java applets with the open source program GeoGebra to explain various concepts or definitions in my honors calculus courses. Follow the links below for some examples of my efforts in this direction. Many more can be found on the course web-page for my current course.