During the past ten years, I have taught a broad spectrum of mathematics courses to students at three Universities. I was a graduate teaching assistant at the University of Georgia from June 1991 until June 1997. During that time, I taught precalculus courses, introduction to mathematics courses for non-science majors and also assisted with calculus courses. From July 1997 until May 2000, I held an S. Chowla Research Assistant Professorship at The Pennsylvania State University. At Penn State I taught Calculus, an introduction to higher mathematics, factoring and primality testing, an honors ordinary differential equations course, a course in linear algebra, a number theory course and a one-hour differential equations course all at the undergraduate level. I also taught a graduate course in modular forms and elliptic curves. Since July 2000, I have been at Clemson University. At Clemson, I have taught three Calculus classes, an introduction to higher mathematics for computer scientists and a graduate course in linear algebra. Thus I have had the opportunity of working with students at many levels and of varying backgrounds. These experiences have taught me a great deal and helped shape me as a better teacher.
My first class was precalculus at the University of Georgia. The most challenging aspect of teaching this course was keeping the students' attention. I found that spending time working through many examples seemed to make the abstract concepts that we were discussing much more concrete and approachable for them. I assigned homework regularly and gave short weekly quizes in order to encourage them to keep up between tests. This seemed to help them remain disciplined throughout the quarter.
In the introduction to mathematics courses for non-science majors taught at UGA, the abilities of the students were wide ranging. Many of these students found mathematics to be especially challenging and in some cases were convinced that they simply could not learn it. On the other hand, there were some who were quite bright. Thus, it was quite challenging to pace the class correctly. I found it useful to spend most of our time in discussion rather than simply lecturing. Involving the students in the class as much as possible allowed me to better gage students' understanding of the material. At the same time, this provided motivation for those who understood the material by giving them a chance to demonstrate their knowledge. This also afforded students who were struggling an opportunity to ask questions and to build their understanding. I learned to share my own enthusiasm for mathematics with my students as much as possible. This seems to motivate students and to help hold their interests throughout the class time.
At Penn State and at Clemson, I have taught several first semester Calculus classes which have almost identical syllabi. Since these courses are completely standardized, the syllabi tend to be inflexible and quite extensive. Due to the large amount of material to be covered and the inflexible nature of the syllabus, I am forced to take a bit more control in the classroom in order to ensure that all of the material is covered in a timely fashion. I have continued to try to find ways to hold the students' interest in the course in order to make the material more interesting. For instance, I try to mention many applications of the material, especially those to which I feel all of the students can relate, such as the use of the Mean Value Theorem in issuing speeding tickets on the New Jersey turn pike. This usually gets a few looks of disbelief and some laughs but it helps brighten the mood of the students in a very fast paced and unforgiving course.
One difficult issue that confronts calculus teachers is the problem of getting students into the right class. Many students with a weak mathematical background enroll in calculus during their first semester in college. This is can be devastating. Since first semester calculus classes tend to be very fast paced, such students are almost immediately lost and cannot recover. This gives them a very rude introduction to college life and in many cases their first failing grade ever. Thus it is extremely important for calculus teachers to make sure that ill-prepared students are encouraged to get into an appropriate beginning math class. I confront such students right away and encourage them to drop into a mathematics course at a more appropriate level. I have found that giving these students a brutally honest evaluation of their probability of passing the course based on my past experience with students of similar backgrounds is fairly effective in getting them into a more suitable course. I also relate my own lowly beginnings in precalculus in order to ease their minds about their future. Many of these students later drop by and thank me for this tough but honest advice.
At Penn. State, I taught an introduction to higher mathematics course. This course is designed to help perspective mathematics majors develop abstract thinking skills and to help them learn to write proofs. While teaching this course, I was constantly reminded of when I was learning to write proofs and dealing with more abstract concepts for the first time. This reflection allowed me to better relate to my students by helping me to recognize more clearly their level of understanding. It also helped me see more clearly how to help them bridge the gap from simple computational ability to the more abstract thinking required for writing and understanding rigorous proofs. Most of our time was spent trying to understand proofs from elementary number theory and basic group theory. We usually began by working through examples to build our intuition. We would then state a theorem and attempt to prove it. I usually allowed the students to attempt a proof of the theorems themselves while I simply recorded their observations and deductions on the chalk board. When they encountered obstacles or dead ends, I would offer suggestions and attempt to guide them to a more fruitful line of attack.
I found that students at this stage of learning required much encouragement. For many of them this was the first class in which they had struggled and they had difficulty grasping the fact that not everything is straight forward and easy to understand. One day at the beginning of class, I noticed that everyone seemed to be feeling a bit down. I soon learned the reason. They were all struggling with the homework. After answering a few questions on the assignment, I told them that it was OK if they found the homework to be challenging, because that was the way it was supposed to be. I explained that they were entering a new level of mathematics and that most of the problems at that level were indeed difficult. After I said this a few students actually laughed and the mood of the whole class was somewhat lifted.
The experience of teaching this introduction to higher math was a great introduction to teaching upper level undergraduate classes. Indeed, I had similar experiences while teaching undergraduate number theory and linear algebra at Penn State and found that the above mentioned teaching techniques were beneficial in these courses as well. Currently I am teaching a beginning graduate level linear algebra course. Since the students in this course have just finished their undergraduate work, the course is very similar to the senior-level undergraduate linear algebra course that I taught at Penn State and I proceed in almost the same manner.
Another course I taught while at Penn State was factoring and primality testing. This course was made up of both computer science and mathematics majors at the junior and senior level. This presents a special challenge to the instructor. The mathematics majors in the class tended to grasp the proofs of theorems more quickly and to appreciate the necessity of such proofs. On the other hand, the computer science students had more of an appreciation of the algorithms which were discussed and for the most part were much better at implementing these algorithms. I found it quite challenging to simultaneously hold the interest of both groups of students. Balancing homework assignments between theoretical problems and computational projects seemed to be an effective way of holding the attention of most students.
Since, much of the discussion in this class was centered around various number theoretic algorithms, I found it quite useful to take advantage of Penn State's technology classroom. This is a normal classroom with the addition of a SUN workstation and a projector allowing the computer's display to be viewed on a screen in the front of the room (very similar to Clemson's smart classrooms). This allowed us to not only discuss the algorithms but to see them in action in the classroom. This was quite effective in comparing the running times of various algorithms and demonstrating the effectiveness of certain ideas for speeding up an algorithm. Also, it seems to break up the monotony of the normal routine in the classroom.
Another class in which I found the technology classroom useful was an honors course in ordinary differential equations. We often used Mathematica and Matlab to graph the solutions of various types of equations. In this class I found it essential to graph many of the solutions in order to convey a better understanding of the behavior of solutions especially near infinity and near boundary values. We obtained general solutions to some types of differential equations and graphed them with various choices of parameters. We used Matlab to graph approximate Fourier series solutions to various equations. It is quite interesting to graph the approximate solutions with an increasing number of terms in the Fourier expansion and watch these approximations converge to the true solution. Mathematica and Matlab are capable of producing extremely nice graphs very quickly which allowed us to see the solutions to many differential equations and to many variations of each equation. This greatly enhanced our ability to see the effects of slight changes in the parameters of the equation and the given initial conditions. The students seemed to enjoy this part of the class and to find it beneficial.
Teaching an honors class was a nice experience. The students were very eager to learn. I occasionally posed problems for further study, which I found interesting but would not be collected for grading. On more than one occasion, some of the students actually thought about them and wanted to discuss their work. They often asked questions which were well beyond the scope of the course. I learned quite a bit from working with these students and found teaching this class to be quite refreshing.
While at Penn State, I was also afforded an opportunity to teach a graduate course in my subject area, modular forms and elliptic curves. I enjoyed teaching this course immensely. I was impressed with the speed at which we could move through the material. I found that the preparation time for this class each day was at least twice as long as the preparation time for an upper level undergraduate course. It was fun being able to discuss my own past work and current research interests with serious students and to share enthusiasm for this subject area. I look forward to teaching such a special topics to students at Clemson.
My preferred style of teaching is not a traditional lecture, but consists of a small amount of lecturing and much discussion. I ask the students many questions during the class period, and I am careful to restrain myself from answering the questions for them. I want the students to think through the material for themselves. I view my role as one of simply guiding the students through the material while allowing them to do as much of the work as possible. I try to guide them through as many detailed examples in class as possible, and then I ask the students to work through similar exercises for homework. I have found that working out several examples is often a very useful tool for me when I am trying to understand new concepts in mathematics, and my students seem to benefit from this approach as well. After seeing several examples, they are often able to formulate more general statements which are very close to the truth. I then encourage them to be more precise, and together we work on making their statements correct. Sometimes, I am required to take a little more control of the class in order to keep us moving through the syllabus. However, I wait as long as possible before doing this, because it is more important for students to learn the basics of a course well than it is for them to see all the details covered and have little understanding of them. Once the students understand the basics, we are able to cover the later material in the course more quickly.
This style of teaching has the added benefit of allowing me to become better acquainted with my students. This allows me to quickly spot students who are having difficulty understanding the material. I can then help these students before they get behind. I also encourage students to come to my office to discuss difficult material. When dealing with students individually, I listen carefully to the student and try to understand their questions clearly. After, I understand their question, I then try to lead them to the answer. This often sheds more light on the reason that the student is having difficulty understanding material in class. I have found that discussing problems with students individually or in small groups is often the most fun part of teaching, and I encourage my students to stop by my office as often as they like to talk about the course.
In addition to the courses that I have taught, I have been an active participant in various seminars and at conferences. At the University of Georgia, I helped to organize and spoke several times in a student number theory seminar. I also gave talks in the regular number theory and arithmetic geometry seminars at UGA. At Penn State, I spoke several times in the regular and informal number theory seminars. At Clemson, I have been a regular participant in the Algebra and Discrete Math seminar and am organizing and participating in the Informal Algebra and Number Theory seminar. In addition to local seminars, I have been invited to give lectures at conferences and seminars both nationally and internationally at various institutions. I also regularly attend conferences in my subject area and give contributed lectures. So, I have gained experience giving expositions of mathematics at many levels from precalculus to conference lectures. All of these experiences have aided my ability to clearly explain mathematics and have helped to shape me as a solid teacher.