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Teaching Philosophy

I believe that my philosophy of teaching is relatively simple. I base my teaching on the belief that the simple way to learn mathematics is to do mathematics. While the process of reading examples and proofs in textbooks and from lecture notes is valuable, the real learning comes through one's own efforts at solving mathematical problems, either computational, theoretical, or both. This is achieved mostly through class assignments, but also through in-class discussions and exercises. I view my role as a facilitator for this process. I must design the framework in which learning can take place, and then stimulate and nurture the students' development, giving help in terms of knowledge, techniques, and encouragement.

When teaching, I do my best to help the students learn to think logically, learn problem-solving methods and techniques, and improve writing skills (writing clearly and concisely, explaining step-by-step processes, providing valid reasons for logical arguments). In addition, I try to help students see the course material in a holistic context by requiring them to synthesize the various concepts of the course by applying them together.

Teaching Responsibilities

Courses taught: I have taught a wide variety of courses for a wide variety of students. For mathematicians: mathematical analysis, real analysis sequence, functional series, integral and differential calculus, ordinary differential equations, partial differential equations, analytical geometry, linear programming, optimization, integral transforms, numerical analysis; For physicists and For engineers: special functions of the mathematical physics, equations of the mathematical physics, Fourier series, Fourier and Laplace transforms, engineering Calculus sequence, mathematics for engineering; For computer scientists: scientific calculus and programming, linear programming and discrete optimization, mathematics for computer sciences, numerical calculus; For life sciences: calculus for life sciences.

Grading: The purpose of grading is mostly motivational than judgmental. By requiring students to demonstrate knowledge of course material, this motivates them to do the necessary work required to learn mathematics. I always grade on a flexible curve. I usually curve each exam and also the total assignment scores and quiz scores. The curve is based on a mixture of class performance, percentage correct, and comparison with past classes' performances on the same material. Thus, it is possible for everyone in the class to get a good grade, or the opposite. It is rare for a student who attends class regularly and does all of the class work to receive a grade below a C. On the other hand, to receive an A, one must score consistently high marks on exams.

I generally spend quite a lot of time in the process of grading by taking one exercise, marking it for all students before to pass to the next question. This method allows me to be more equitable for all students.

Feedback: In order to increase communication and feedback both to and from my students, I encourage the use of office hours, and I make myself available at other hours as well. I have a Web page which allows students access to all sorts of class information and which also includes a feedback form for students to submit suggestions and comments. I generally use the email of my students to notify them of changes or hints on assignments, temporary changes in office hours, etc.

Availability: I always hold many office hours per week, and I make myself available at other times as well. I tell my students they are welcome to come by my office at other times, and unless I have an immediate deadline or meeting or class, I will be able to help them. My detailed schedule is posted on my Web homepage.

Teaching Methods

Class sessions - lecture and discussion: I always begin each class with a brief summary of the previous class session, and a reminder of where we are in the topic we are currently working on. At this point I usually ask if there are any questions from the reading, homework, or previous class. After this discussion, I usually give a lecture on new material. I try to begin the lecture with a brief outline and a list of objectives, and I try to always include examples during the lecture.

Usually, I try to present course material in analytical, numerical, and graphical contexts. While this approach of course depends on the particular topic, it is particularly valuable in calculus, differential equations, and numerical analysis courses. I am especially conscious of using figures to help illustrate different concepts, as most students can then at least intuitively understand the concepts even if they have trouble understanding the analysis.

Technique of learning: In the past several years, I have incorporated more cooperative learning techniques into the class sessions. These techniques usually involve working in pairs or groups of three on a short problem, with specific instructions on how to share ideas and come up with a common solution. While the groups are working, I can move around the classroom to help various groups, and at the end we compare and discuss the various groups' solutions.

Homework and level of difficulty: Knowing that the only way to learn mathematics is to do mathematics, I usually assign a lot of homework. I try to assign a mixture of routine and challenging problems so that I can stimulate the more advanced students but still enable the poorer students to at least learn the basics of the course material. Usually, I try to grade as much of the homework as possible. For large classes, students can check their work on routine assignments by using solutions manuals, the help room, and posted solutions, and by asking questions in class.

Exams: When I create an exam, I try to follow several guidelines. First, I try to test over a reasonable range of class material, and I try to stress the important concepts. I also include problems of varying difficulty. However, I usually do not include trivial problems. Before each exam, I spend some time in class discussing what topics will be covered and which are most important. I usually give more detail for undergraduate courses, especially the lower level courses; I sometimes give them the so-called exam model. I also try to be careful not to make the exams too long, but I sometimes fail in this regard. I find this is especially hard in higher level theory courses, where it is very hard to judge how long it will take students to do the creative thinking necessary to come up with a correct proof. Consequently, in these courses I do not expect students to solve all of the problems on the exam.

Course Syllabi and Information

At the first class meeting, I hand out a syllabus which gives the basic information for the course. It lists ways to contact me: office number, phone number, and email address, and also the address of my Web homepage and the homepage for the class. It also informs the students about the prerequisites, text, topics to be covered, the number of exams and quizzes, information on homework, and grading policy. If the class requires use of a calculator and/or computer, the syllabus includes a section describing how they will be used in the class and what will be expected from the students in the areas of programming and calculator/computer expertise. Finally, the syllabus also includes a section on use of the class Web page.

During the first week I remember my office hours (also listed on my Web homepage, along with my more detailed schedule). I do not give out a list of all homework assignments for the semester as some instructors do. I prefer the flexibility of changing the problems and due dates during the semester as I get to know the strengths and weaknesses of the students. This allows me to tailor the assignments to the needs of the class.