My teaching is careful, considered, and energetic; I think through every aspect of my teaching, continually try to improve, and approach my interactions with students with gusto. The following describes my teaching philosophy and practice.
My guiding pedagogical principle is that students learn most deeply and retain concepts most completely when they participate actively in the learning process and engage with the subject material. I encourage my students to construct as much knowledge as they can on their own, then help them progress when they have reached their limits. I also help them learn how they can construct knowledge that is new to them. This is true independent of the level of the class or students. In having high expectations of my students, I keep pushing them to learn, and they work hard in response. As a result, I am rarely disappointed.
Among my expectations is that my students will do assigned reading. They quickly learn that this is a real expectation: I do not lecture on most material, so in order to keep up with class, they must read the text. To make this a realistic expectation, I am very careful to choose only the most readable of texts. (My favorites include Trudeau's Introduction to Graph Theory and Andrilli/Hecker's Elementary Linear Algebra.) When possible, I assure that the curricular philosophy of the author(s) matches my own. Sometimes I must combine parts of multiple texts (or, in one case, write my own) in order to create a philosophically coherent course, or supplement a shared-across-sections text with other readings.
One of my favorite approaches to teaching mathematics is to ask students to examine examples, make conjectures, and then communally and individually investigate their conjectures. My use of this technique ranges from a single lesson to the entirety of a course. I have taught inquiry-based courses at the general education level, to gifted high-school students, and to upper-level undergraduates. In any case, I do not spend the majority of class time at the board. Introductory classes often begin with student discussion of conceptual review problems I project. My lecturing is limited to short reviews of main ideas in lower-level classes and short lectures-on-demand in response to conceptual confusion in higher-level classes. For much of a given class, my students work in small groups that report frequently to me and to the whole class. In many classes, I ask the students to solve interesting problems posed in the text. While the students work, I circulate among the groups. I check to see that each group is on-task, coax them to explain their current reasoning to me (and thus to each other), and ask them leading questions that are likely to help them make progress.
I also foster collaborative learning outside the classroom. In lower-level classes such as calculus, I require that some homework be done in a structured group setting. Not only can the students help each other (allowing me to assign more conceptually difficult problems), but they are assigned roles to help them reflect on their learning processes. As they take on different duties, the students concentrate on different aspects of the collaboration. In upper-level classes such as real analysis or abstract algebra, all homework must be written up individually. However, I always advocate that students work together on solving homework problems.
A benefit of collaboration that may not be apparent to the students is that their mathematical and technical writing skills improve. When students are required to work together, they must discuss the write-up format so that the person producing the final draft will know how to accurately reflect the group's conclusions. While at first it may be nearly unreadable, group homework invariably becomes well-written because writing quality is part of the homework assessment. When students are working on proofs together, the topic of how to properly explain their reasoning often arises (even if they need not collaborate on the writing).
I think it is very important for students to learn to effectively communicate mathematics, because no matter how well a student has learned a concept, this knowledge is meaningless unless ze can communicate that concept to others. To this end, I not only give careful feedback on student writing, but also encourage students to share their thoughts in class. In some classes, I require formal presentations either of course material or of independent projects; in other classes, students report on in-class work or on simple homework problems done in preparation for class; in still other classes, a discussion-style format allows the students to regularly practice both informal and formal communication of mathematics.
All of my courses are supported by extensive use of course management software (cms) and email. After each class meeting, I send comments, assignments, and announcements to the students via the cms so that they are accessibly archived. Likewise, the cms hosts supplemental reading and links to online resources. I encourage students to email me at any time, and respond promptly at all hours. Many of my courses are supported with computer algebra systems (Mathematica and Sage) or graphing calculators. While the students may use technology heavily, I do not emphasize it in class. I promote the use of technology for completing calculations and generating examples; it is a tool to expedite mathematical processes, not a substitute for knowing how problems could be done by hand.
I gave a more detailed and less formal description of my classroom practices to the 2004 Project NExT cohort (that's the Mathematical Association of America's Project New Experiences in Teaching, intended to improve the teaching and learning of mathematics by giving fresh Ph.Ds. professional development), and the text is available here.
My courses focus on concepts, and so my homework sets and exams concentrate on students applying these concepts and explaining how they are used. In upper-level courses, I use take-home exams; in mid-level courses such as linear algebra, I generally use both timed/closed-book and take-home exams. Take-home exams allow the students to learn more while being assessed, and help to lower the level of anxiety associated with assessment. Because my classes are interactive, participation is part of my student assessment. I assess classroom participation using an excellent rubric adapted from the educational literature. In some classes, I require a paper or project in order to allow the students a more extensive writing experience than a typical homework problem, as well as to allow me to assess their writing in a context where they have had a chance to write and revise (and thus improve).
Just as assessment of students can be both formative and summative (students can use feedback to improve, while faculty can report on achievement), so can assessment of faculty. In this vein, I have sometimes used a Small Group Instructional Diagnosis (SGID), in which a trained facilitator observes a mid-term class meeting, then uses the end of class to solicit consensus feedback from the students. Later, the facilitator debriefs the instructor, who then responds to the class. Course evaluations are useful to me in planning future classes, but SGIDs are uniquely helpful in improving a course in progress. I have learned to conduct SGIDs, and am happy to do them for others.
Finally, I would like to emphasize my strong commitment to the total education of undergraduates. I am a very hands-on advisor in many contexts (classes, projects within classes, the college experience as a whole, future life paths), and proactive in reaching out to my students. It is important to listen to students describe their experiences---especially during the first year of college---because their perspectives often reveal their needs. This leads me to advise students on how to approach learning in my classes and to give advice that reaches beyond my classes to work on other projects and initiatives. In this way, I like to get to know my students and support them in other areas of their lives. Academic experiences, campus life, and personal life affect not only overall student development, but also performance in my courses; because first-year students face academic and personal transitions, I try to attend to this in courses aimed at first-years. When I live close to campus, I invite students to my house at the end of the term for snacks, puzzles, and chatting. I mentor students over the long term; I see their abilities and potential and work to bring their good qualities out, and then I advocate for those students relative to and in the context of the standard system---writing them letters for REUs and graduate school, nominating them for awards, etc.
Interestingly, I find that working on my research improves my teaching as well. The experience of learning, struggling with and creating new mathematics echoes the challenges my students face, and thus I empathize with student stress. I'm always on the lookout for subproblems and subprojects that would be appropriate for students---nontrivial, yet tractable and requiring minimal background. Twice my collaborations with students have resulted in research publications in topological graph theory.
I am always excited to be with my students and work hard at making our interactions valuable for them. This results in a nontraditional and dynamic classroom experience, in which I ask the students to become highly involved in the construction of their knowledge. In turn, such pedagogy leads to alternative methods of assessment that are tailored to specific course structures. Both my pedagogy and assessment choices generate a great deal of contact with and feedback from my students. This improves my teaching, and the circle is complete.