My opinions about class format, grading policy, use of technology, and other issues derive mainly from my primary goal as a teacher: I want each student to learn and understand the concepts of the course. The computations that form much of the curriculum of low-level math classes are forgotten after the final exam if the mathematical ideas were not understood. Although applications can provide a focus, students are not necessarily interested in the same applications. A student who understands the concepts will be able to apply mathematics in other contexts, overcoming differences in terminology or situation. The underlying mathematical ideas are the natural focus for a student who does not need math for a particular application, but studies it in order to learn how to think in a mathematical manner.

An engaging classroom atmosphere, a fair grading policy, the feeling of accomplishment - all of these keep students happy and can help them succeed. However, the center of their satisfaction should be the mathematics itself, and the vast majority of students are capable of appreciating its beauty (even if some would not admit it). Without it, we teachers don't really have anything substantial to offer. Of course, happy students are easier to teach and more pleasant to be with. The key is that student happiness is a secondary goal, desirable when it does not conflict with the primary goal of student learning. I focus on these two global goals as a teacher, and my experience teaching many different classes in different formats gives me the ability to reach these goals. (Disclaimer: I recognize that my ideas will not apply so well to students taking courses beyond multivariable calculus, and I will have to adjust my class policies accordingly when that opportunity arises.)

Students want good grades, regardless of how they are achieved. In addition to communicating the ideas, part of my job is to ensure that good grades are meaningful as well as attainable. My tests are for evaluation only and focus on core material, with no surprises. I emphasize learning and discovery to classwork and homework, which do not affect grades as strongly as tests. I generally don't assign extra credit; a student needn't jump through hoops in order to convince me that he or she is willing to work hard for a grade. I also don't assign additional material for extra credit, since students need to understand core material before being able to branch out, and a student who needs extra credit has not demonstrated competence with core material. I don't mind giving students a second chance to learn and understand the ideas, so I do allow late work and occasional make-up exams, with small penalties. These small penalties encourage keeping up with the material. This is important because almost all students need good study habits to succeed, but many think that they can get by without them, obtaining the grades they want without necessarily learning anything.

Communication between the students and the teacher is helpful to both. Different students have different needs, and in order to make adjustments I seek timely feedback from them. My best source is the classroom, where I pay close attention to students' reactions and regularly prompt them with leading questions. I also encourage them to ask questions in class; this tells me what needs further clarification and allows me to correct misunderstandings as they arise. Both office hours and alternative classroom formats give me invaluable one-on-one contact with students. I always make time for meeting students outside of class, because this interaction is such an effective tool both for student learning and for me to assess their level of comprehension. Discussing graded exams and homework is a useful catalyst; for it to occur, comments on graded work must indicate the mistakes.

Many techniques help keep lectures from being boring and help students concentrate during class. I find that daily handouts with outlines of the lecture are very effective in doing this. The organization and progression of the lecture is made clear and easier to follow. Note-taking is reduced and students are less likely to copy the chalkboard verbatim. As students have remarked repeatedly in teaching evaluations, I know how to use examples to motivate and provide the right amount of explanation for my audience. Some teachers use gimmicks to break routine, which is fine as long it doesn't distract from the real focus of the class. Humor related to the substance of a lecture helps keep students mentally fresh, but I don't tell unrelated jokes or try to be overly witty. Lectures should not be rushed; it is better to assign some material for reading if the text is well written. However, I do make it a point to go over all exam topics in class at least once.

Working in small groups and using computers are other ways to help focus attention on the material. When departing from the lecture format, there various other issues that must be considered. I have found that small groups are great for keeping students engaged as long as the group dynamics don't become too much of a distraction. Small groups also match well with discovery-themed classes. However, the individualized attention from the instructor is somewhat offset by the likelihood that sometimes a group will get stranded and confused while the instructor is with other groups.

I have taught classes based on interactive computer lessons, and I think that the main benefits of technology are the graphics capabilities. Hypertext computer books have advantages in organizing and reviewing material, but there are still many difficulties with present technology. Machines are often used as a means to sidestep a lot of calculation and use realistic numbers, which I would love to do. Unfortunately I've come to the conclusion that only confident students will believe a result without seeing the details, and these students tend to be either the very good or the very bad.

Different teachers have different styles and so must make conscious adjustments based on their natural strengths and weaknesses. After several years of experience, including completing the requirements for the Graduate Teacher Certificate at the University of Illinois, I know what works for me as a teacher. As a part of this, I have had to adjust to many different teaching situations. This ability and my strong sense of responsibility towards my students give me confidence that I will be a good teacher in every situation that presents itself in the future.