Teaching undergraduate students is an activity that to some extent divides into two different components, that require slightly different approaches. On the one hand there is the teaching of mathematics students, or students in very mathematics-heavy subject areas, who can be expected to show independent interest in the subject matter, and who can be expected to continue on to several further classes that will build on the current contents and require a large scale understanding of mathematics to be developed. For these students, I believe that the role of the teacher is to stimulate the interest, exhibit passion for, and demonstrate techniques to learn the material at hand. It is not possible for a teacher to reach in and imbue a student with knowledge, but it is clearly within the reach of a teacher to show enthusiasm and interest in the material, to demonstrate the wonder and beauty inherent, and to show how to approach the literature and tools at hand for the students to learn the field. Naturally, one component of doing this is to give the students a guided tour of the field, to show them what it contains, and lead them through the approach to definitions, facts, proofs and proof techniques, but ultimately the act of learning is in the hands of the students themselves. Ultimately, as a teacher of mathematics majors I cannot teach the students mathematics, but instead I try to teach them how to learn mathematics.
This holds doubly true for graduate teaching; where assumption of interest can be even more strongly asserted than with undergraduate mathematics majors. Here, the task for the teacher is that of native guide, and support when the student gets stuck, of showing interesting and promising patches of mathematics to go learn, or to go discover, and to be around with help and ideas whenever the wild undergrowth or arid emptiness of beginning research overwhelms the student.
The care and nurturing of interested students is something that lies close to my heart. I can trace my own passion for mathematics to my participation in the Junior Mathematical Congress series, and I have continued to involve myself into mathematics camps, the JMC congresses and other efforts to reach out to promising high school students, including mentoring and teaching algebraic topology to a small group of high school students during my doctoral studies. My interest in sharing the passion I have leads to a strong interest in teaching as well: while my positions at Stanford and St Andrews have been pure research positions, I have still taken initiative to teach two courses here. The first one, MATH 198, a category theory course I planned out on my own, publishing the lecture notes (http://haskell.org/haskellwiki/User:Michiexile/MATH198) online. The second one, MATH 20, is a yearly occuring service course in calculus — the second one in a cycle of three courses covering differential and integral calculus, with some ordinary differential equations and series.
On the other hand, there is the teaching of mathematics as a service science, as a toolbox for the application in completely unrelated fields. In these courses, it is no longer a reasonable assumption that the students take the course out of sheer interest in the mathematics at hand, or in the future. For these, the approach needs to be more practical, more of teaching a craft than teaching a theory. Care and attention needs to be given on conveying an understanding of the meaning of the basic definitions, and how these relate to anything the student may actually meet in their future studies, as well as to how the student can be prepared to rediscover the parts of the theory they find themselves actually needing when they need it. It isn't, to take an example, crucial for a biology major to understand that an integral is the limit of a Riemann sum in the same way that they need to understand that it represents accumulation of a changing quantity, and the properties and limitations tools they may use: mathematical software that may fail on certain problems, and that approximations come with error estimates that can be looked up in the relevant tables to find the information they need in the end.
Nevertheless, regardless of the students' motivations and origins, I believe in a continuous engagement with the material. Small, but constant homework keeps the students from shoving their studies until the week before the exam, and leads to a much more even rate of studies, with better results in the end. In this, the advent of web-based and automated systems for homework questions provides crucial and immensely powerful teaching tools. Through my academic history, I have involved myself in both development — through the PrimaLatina web based teaching system for a first course in Latin at Stockholm University (http://primalatina.klassiska.su.se), through teaching a web based course in mathematics at Stockholm University, as well as through using WebWorks for the calculus service course I taught at Stanford University.
Another axis along which teaching technique differentiates is with the size of the class. For very small classes — up to maybe 15 students — the atmosphere in the class can be made very collegial, turning the course into a conversation, a dialogue even, on the course material, advancing the whole class together through a planned course of study, but doing it with very low barriers to participation and explicit encouragement of interruptions. As class sizes grow, though, the feasibility of the conversation falls, and when class grow up to more than 30-40 students, enough will be too shy to ever voice questions or interruptions during class. For large enough classes — definitely those above about 50 students — I find it crucial to organize recitations with smaller groups to ensure everyone gets a chance to voice their questions and misunderstandings in a secure enough setting to get teacher help understanding everything.
I put this into practice while in Jena: in my recitation classes I worked hard at defusing the hierarchal thinking that is more dominant in the German culture than in the Swedish, and I pushed for a conversation in the class room with many different approaches. One of these was to explicitly ask two students every lesson to show their solutions to the homework on the board, removing as far as it was sensible the division between the lecturing teacher and the note-taking student.
As the class grows though, any hope at providing this tone is lost, and what remains is to coordinate the recitation efforts, to highlight the important developments and the narrative of the material at hand in a lecture, and to use every tool available to make the task of teaching more efficient, so that the lecturer can still be helpful to all the 300 students without becoming completely drained with the experience. Even if recitations are not available, the barriers to participation that shy students may exhibit can still be broken down, by forming small groups in the lecture hall. Again, I believe that the various web-based teaching tools in existence have a lot of value to bring, a lot of potential for these problems, and will pay close attention to how I can keep my own use of technology in the class room up to date and appropriate.
To summarize, I find that the most important part of teaching is to remain adaptable to and aware of your audience as a teacher. I try to hone this flexibility as I teach more and more, and have already solicited both midterm evaluations and pedagogical advice from Stanford's Center for Teaching and Learning in order to grow as a teacher and continue the ongoing process of learning to teach.