(The following is something that I prepared for a job application. It turned out not too bad, so I thought I'd share it here.)
Over the course of my teaching experience, I have developed a teaching philosophy that has proven to be practical and effective, yet retains its intellectual integrity. A major component of my teaching philosophy is based on the so-called Socratic method. From experience, I know that for many students, math appears to be very much like a magician’s sideshow – first we do this, then we do that, and voilà! – out pops a solution. If a student does not follow what the teacher did, or fails to understand the steps taken in solving a problem, then the student stands little chance of mastering the concept, and will at best demonstrate mere competency that is most likely arrived at with difficulty. To minimize this seemingly magical aspect of math, I prefer instead to approach problems from the students’ point of view: Here is the problem, and we have a fairly good idea as to where we are supposed to end up; now, how do we get there, using what we know at this time? Thus, I think out loud for the students, soliciting their input and trying their suggestions. If an approach is suggested which I know will not work, I still play around with it briefly before informing that it will not work, and explaining why it fails. In my opinion, this is an excellent way to teach mathematics: lecture, which is the equivalent of equipping the students with the tools necessary for the task, and problem-solving, whereby the students learn vicariously how to put the tools to use and why we do it in the manner that we do, as opposed to some other way.
Other aspects of my teaching philosophy complement this Socratic approach. I believe that all students are capable of achieving what is expected of them – all students can master the material, given enough encouragement and practice. Thus, I believe that homework is an integral part of a math class, for it provides the student with the practice that is necessary for mastery. But although practice is necessary, it is not sufficient – encouragement is also a key component. All students from time to time will encounter situations and concepts that prove to be challenging to them, but with some encouragement and perhaps a little prodding from the teacher, they can overcome the stumper and press on to victory.
Because all students face a challenge at some point, I believe that it is important for the teacher to take into account the various learning styles that are present in the average classroom, particularly in explaining unfamiliar concepts. Some students learn best visually, some aurally, etc., so the wise teacher utilizes different approaches when it is apparent that students are failing to grasp the thrust of the lesson. The teacher should also foster an open atmosphere where the students understand that no question is too silly, and chances are that if one student has a question, then it is likely that others have a similar question as well.
I believe also that the teacher needs to be able to explain concepts without relying on obscure argot; after all, the goal of teaching is to enlighten, not obfuscate. Cryptic terms and concepts need to be explained clearly before they can be used and mastered with confidence.