Mathematics is almost unique among academic subject areas in that at least some knowledge of it is required for a wide variety of other disciplines. Another thing that sets the area of mathematics apart is the particularly important role that quality of teaching plays in instigating enthusiasm and a commitment towards greater understanding. I have heard time and time again from students who have a strong dislike or fear of math that they felt that a particularly bad experience with one teacher had permanently turned them off to the subject. On the other hand, there are many who claim to have first discovered the power and beauty of mathematics through the guidance of an excellent teacher. In light of all this, I have felt a strong sense of responsibility towards the many college students I have instructed at Penn State, and have tried to emulate the characteristics of those instructors who inspire excitement and interest in math.
Through my experience as an instructor and as a tutor, I have discovered that possibly the most widespread misconception about studying math is that it consists of rote memorization of arbitrary rules. To remedy this, I make a conscious effort to teach in a manner that emphasizes conceptual understanding. When introducing a new concept, I often begin by instigating a class discussion on the heuristics involved, drawing on my students' intuition to motivate the definitions of the new terms I will present. I also attempt to draw attention to the need for certain theorems and formulas before presenting them, as well as anticipate what form these theorems and formulas should take.
Another strategy that I have found successful is to prepare worksheets for the students to work on in small groups during class. I design these worksheets so that they assist the students in the discovery of complicated new formulas and properties. (For example, I have used this technique to teach them help them come up with formulas for the multivariate chain rule.) I have been told a number of times by students, either in person or through written evaluations, that this method of teaching was particularly effective for them. Allowing them to discover mathematical truths for themselves, rather than presenting these truths as a list of facts they will have to copy down and memorize, not only helps in the learning of that material but also conveys a truer sense of the process of learning math.
In my experience, oftentimes what holds a student back in math classes is a general fear of the subject. In particular, many students assume that there is always "one correct way" to approach a problem, and that if this approach does not come naturally to them, then they are "bad at math". When helping students one-on-one, I emphasize the fact that there are often many valid angles from which to view a concept, and I make a point of trying to follow the approach which comes most naturally to that individual. Even while working through an example during class, I welcome suggestions of alternate methods of solving it, and sometimes I will redo it using the suggested alternate method.
Of course, sometimes it is necessary to focus primarily on preparing my students for midterms and finals. To this end, I frequently point out common mistakes and share mnemonic techniques that helped me when I was first learning the material. When from time to time I make a careless mistake when working an example on the board, I use it as a learning opportunity. I explain why the incorrect answer I arrived at was not plausible, and how to catch that type of error quickly in a testing situation.
More recently, in teaching my graduate-level course on elliptic curves, I have learned that teaching graduate students requires a very different mindset from what was needed when I taught undergraduate students who are were accustomed to a proof-based approach for learning mathematics. However, I have found that aspects such as pacing of material and clarity of exposition of new concepts are just as crucial as they were for my earlier teaching jobs. I hope to be able to teach a sequel course on elliptic curves in coming months and to increase my engagement with graduate students in general. As my area of mathematical expertise includes topics that can be made quite accessible to early graduate students and senior undergraduate students, I look forward to being able to work with them on projects that relate to our shared research interests (e.g. elliptic curves).
I feel that I have grown a lot as an instructor over the last seven years since I first began teaching, but I will always remain open to suggestions for improvement from my colleagues so that I can continue growing. I frequently discuss teaching strategies with them and, in some cases, use their success as my own inspiration. During my last couple of years of graduate school, I took part in a student seminar run jointly by math and education graduate students, where we exchanged ideas and opinions on teaching strategies and the principles of math education, and I would be very interested in finding (or organizing) a similar discussion group at the next institution where I have teaching duties.