Teaching Philosophy

Teaching Objectives

I am an avid proponent of the role of mathematics as a fundamentally enabling discipline that underpins research and development in all spheres of science and technology. In a society where science and computation play an ever more prominent role, there is an increasing need for students to understand the language of mathematics, not only its grammar and vocabulary, but also its descriptive potential, so as to articulate mathematical ideas clearly and fluently to a wide and diverse audience. Despite its inherent interdisciplinary nature, however, mathematics is often perceived by students as abstract, obscure and inaccessible. To me, the challenge as a teacher lies in making mathematics user-friendly to students, training them to critically read mathematics and communicate their ideas effectively. I try to achieve this by conveying my enthusiasm for the subject, demonstrating its usefulness, explaining concepts clearly, and helping them develop their technical skills.

Teaching Style

In my six years of teaching students from diverse cultural and social backgrounds with varying academic interests and levels of mathematical proficiency, I have learned that the success of any teaching strategy depends largely on engaging students to participate actively in the learning process, while striking a delicate balance between challenging them and building their confidence. Underscoring the relevance of the course material to their research- or career goals or at least demonstrating the potential value of the mathematical theory they learn, can be an effective motivator. The pervasive presence of linear algebra in applications ranging from search algorithms and machine learning to numerical solutions of flow problems is a case in point. I remember distinctly changing my opinion on the usefulness of linear algebra upon realizing that polynomials can be thought of as vectors! On the first day of class, I often ask students to write down their fields of interest as well as what they expect to learn in the course.

In the classroom, I believe in maintaining an atmosphere based on mutual respect and co-operation, in which students can feel free to ask questions without fear of recrimination. I also value constructive feedback from both students and colleagues, which has contributed considerably to my development as a teacher in the past. Since a substantial part of learning mathematics is through practice, I try to empower students to take a personal responsibility for their mastery of the subject, by promoting co-operative learning through in-class group assignments, compiling worksheets designed to review technical skills necessary for the course, providing suggestions for online resources and encouraging them to regularly attend office hours.

Mathematics can be quite technical and frustrating for students to understand. It is therefore crucial to explain mathematical concepts simply and clearly. I like to include a brief outline of each lesson, relating the current material to concepts discussed earlier and foreshadowing its relevance to subsequent lessons. This gives the course a narrative structure according to which students may orient themselves, regardless of the possibly messy, technical details encountered during the lesson. Geometric explanations of mathematical ideas often avoid reference to technical formalisms, while practical examples are a great vehicle for exploring their usefulness, as well as their limitations. The ubiquity of user-friendly programming languages, such as Matlab, Mathematica and Python also presents exciting opportunities for exploring mathematical ideas visually as well as algorithmically, while allowing for the inclusion of more realistic, albeit more complex illustrative examples in the syllabus. I have successfully incorporated this type of exploration into my sophomore differential equations course.

Homework, Exams, and Grading

Exams usually carry the most substantial weight towards final grades and are therefore often a source of anxiety for many otherwise competent students. In order to mitigate the stress brought on by exams, I like to be clear, timely and transparent not only in conveying my academic expectations to students, but also in grading their exams. It is also important to ensure that expectations are consistent with the level of difficulty encountered in class. The flexibility of homework assignments and projects in terms of time and format, allows for questions of a more pedagogical nature and I often prefer to use homework assignments and projects to explore trickier, thought provoking questions. Sometimes I ask students to write an essay on a topic we have been discussing in class, challenging them to translate abstract mathematical arguments into everyday language. Homework thus provides a more continuous form of evaluation that represents a valuable barometer of students' overall level of understanding.

Technology

The role of technology in the classroom is a complex, multi-faceted, and ever-evolving issue and the topic of much current debate. As a rule, the supplementary use of technology should always be informed by underlying pedagogical objectives. The use of software alluded to above may be a helpful exploratory tool, although often a chalk-on-slate exposition more effectively lays bare the mathematical structure and mechanisms that give rise to observed phenomena. While online resources may be current and easily accessible, information is often scattered and shallow in comparison to the coherence of a classic text. Online testing provides a convenient way for the assessment of certain technical skills, but may be lacking in providing an accurate gauge of the students' overall understanding of the subject. On the other hand, developments in technology often change the role of the modern workforce and should be considered in the development of any curriculum.

Teaching Plan

Courses I would like to teach include calculus, pre-calculus, linear algebra, real- and functional analysis, numerical analysis, scientific computing, stochastic processes, statistics and estimation theory. In the future, I also plan to develop a curriculum for a special topics course on statistical inverse problems. I am currently involved in developing and teaching a scientific computing course, with John Burkardt (jburkardt@fsu.edu), that introduces students to the multitude of Finite Element software packages, such as FreeFem++ (C++) and Fenics (Python), that are freely available to aid simulation and design for systems governed by partial differential equations.