Teaching in General

During my over twenty years of math teaching career, I taught a lot of different courses to students with a broad range of needs, knowledge, and cultural backgrounds. I taught freshmen classes and high level Ph.D. classes. I even taught a course for university professors, on using computer technology in mathematical education at the Moscow State University in the eighties, where everybody in my class was at least twice of my age at that time. Several important general principles that I try to use in my teaching practice can be formulated as follows:

1. No matter how experienced a teacher is, every new class and every new student pose a challenge. For excellence in teaching, there is always room for improvement.
2. A teacher has to appeal to different learning styles, to offer a variety of instructional experiences, and to keep an open mind to new teaching techniques to give every student the opportunity to participate fully and actively in the learning process.
3. Being a successful teacher depends on creating a learning environment with the open exploration of ideas, a relationship in which students feel respected as well as challenged. Students should be encouraged to stretch themselves a bit beyond their level of comfort and be given an opportunity to leave every class feeling that they have overcome a new challenge successfully.
4. Teaching should be an integral component of the creation of new knowledge, an initial step toward continuing education, and first of all, self-education.
5. Though student satisfaction is important for better learning, teaching -- especially in mathematics -- should not become a popularity contest, or a show. A teacher is responsible to the society in general, and should resist the pressure of lowering academic standards in education.

"The easiest of sciences, a fact which is obvious in that no one's brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon." Roger Bacon, 1214-1294.
"The subject in which we never know what we are talking about, nor whether what we are saying is true." Bertrand Russell, 1872-1970.

As a teacher of mathematics, I like to take advantage of the fact that mathematics is a very special subject:

1. Mathematics is an excellent intellectual game where all the players win.
2. Mathematics is also a model that can be used for developing independent and critical thinking.
3. Mathematics can be seen as a language that allows us to communicate ideas precisely between ourselves.
4. Finally, mathematics is a tool used in natural sciences and, thus, a required discipline for many students.

I consider mathematics to be a challenging subject to teach right, which makes it especially attractive for me . I try to use every possibility to expose students to the exciting world of mathematics, often implicitly, so they may not even notice it at first. My main goal is to help students to become independent mathematical thinkers, capable of approaching, framing, and solving problems on their own. I attempt making my classroom to be an engaging place, where there is more discussion than lecture, and where students always feel free to contribute and ask questions. I think that students’ participation is crucial for learning mathematics. In high level graduate courses, I sometimes even ask students to prepare and give lectures instead of me. From time to time, I try to be mathematically provocative, and do not miss an opportunity, if class time allows, to try to prove that 2+2=5 and to let students find a mistake in my arguments, if there is one.

Teaching Style and Methods

I prepare my classes thoroughly, but I like to improvise whenever possible. I want to be interrupted during my lectures; and I give extra points for good questions and comments from students. I open each session with a brief reminder of the previous session's material and an outline of the day's topic, and I typically conclude with a summary of key points. There is a special session for review before every major test. I usually speak clearly, loudly, and slowly, but enthusiastically. Students are encouraged to learn from each other as their grades are not curved, thus, students are not competing with their classmates. In some classes, I give group projects and homework. When possible, I distribute solutions to homework problems, quizzes, and tests. To reduce students' anxiety about tests, I make old exams available on the Web, and often give practice tests. I usually give a test on prerequisites during the first week of the course. I do not consider memorization to be the most important in math courses; all my tests are open book/notes. Assignments and projects are naturally integrated into my courses. I spend time explaining the assignments and solving similar problems in class for further discussion.

Grading Policy

I generally follow the common practice of the department and traditions on specific courses. Grades are mostly based on evaluation of a content-learning. At the same time, I also implemented several ideas in my classes that seem to be not very typical:

1. Grades are never curved, thus, there is no limit of high/low grades.
2. I use multiple-choice tests and avoid giving partial credits in low-level undergraduate courses.

My policy on retaking tests is unique in the department. Every student has a opportunity to retake every test; however, the grade can go down in the process if the student is not well prepared. I keep students fully informed of their progress during the semester, always announce final grades before the end of the semester, and give every student a last chance to improve the final grade during the week of finals.

Teaching and Technology

Technology is an essential and integrated part of my teaching. It is crucial for students to feel comfortable with the technologies that are becoming available; they need to know how to use a new technology effectively, what its limits are, and what to do when the technology fails. I am quite interested and familiar with recent innovations. Computer simulations and numerical experiments are traditional parts of many courses I teach.

All my course materials are available on the Internet since 1994, including syllabi, homework assignments, and tests. I created interactive Web page for several courses, where students can (anonymously) ask questions and make their comments long before such tools as Blackboard were available. In several recent classes, it was required that all students prepare their projects reports electronically and submit them through the Internet. I encourage students to help me in developing Internet pages for my classes; and there are several successful examples of such collaboration on my teaching Web pages.

At the same time, I strongly oppose the abuse of technology. There are courses in mathematics, where students need to learn how to use their brain rather then calculators/computers.

Teaching and Research

I incorporate research activities into my regular teaching whenever possible. In all my high-level graduate classes student research projects are required. In several cases, project results obtained by students in my classes were significant enough to be included in my journal articles. Scientific programs written by students as their class assignments were made publicly available on the Internet.