Robert Boltje–Teaching Statement 2001-02
Assistant Professor,Mathematics

First of all let me say that I feel very honored to be nominated for this award by students from our department.

The courses I taught were mostly of the same type: During class time, that is 3.5 hours a week, the material of the course is presented. Homework porblems are assigned on a weekly basis and are to be turned in. In most undergraduate classes Teaching Assistants oversee the grading of the homework and hold weekly problem sessions during which there is time to talk about the solutions of the homework problems and about questions that arose from the material taught in class. For specialized undergraduate courses and all graduate courses we do not have teaching assistants assigned and usually there are no problem sessions offered. In these cases I grade the homework and hold the problem sessions myself. In the following paragraph I try to explain why I feel this is important.

It is my deep conviction that Mathematics cannot be learned properly by just going to classes. There are several different levels of understanding that a student should be helped to attain in the learning process. This can be achieved by discussing problems, solving problems, writing down the solutions, and presenting the solutions. The first level would be to understand the language, that is the definitions that are made during class, the statements of the theorems, and the logical structure of the proofs of the theorems. This is already an active process for a student (very rarely in a mathematics class a student can widen his/her knowledge by just passive listening), but not yet a creative one. The next level of understanding is given by the ability to talk about the new material covered in class. Here it is of utmost importance to work in a group with other students and discuss the material covered in class. Homework problems whose solutions require the newly taught material and the encouragement to work in groups on tese problems provide an ideal stimulant for this. Now the creative part of the learning process starts by solving the problems. After all, the creativity in finding solutions to a problem is one of the most important abilities of a mathematician and I feel that usually there is not enough help offered to develop this ability. the next important step is to write up solutions. Even if they are worked out in a group I insist that everybody writes up their solution individually. There is a big difference between thinking that one has found and understood the solution of a problem and writing it down in a comprehensible way.

The last and highest level of understanding is achieved when a student is so secure about his/her solution that he/she can present it in problem sessions on the board in front of the other students and is able to answer questions of fellow students. It is important to provide an atmosphere of trust and respect during the problem sessions to make students feel comfortable when presenting their solutions. Not that the performance of the presenting student should be the main focus, but the common desire to understand the presented solutuion and to discuss alternative solutions. If a problem was so hard that none of the students could solve it I encourage them to present their ideas. After that we develop a solution together.

Another issue in teaching I would like to address is the use of textbooks. Students like textbooks because it gives them a sense of security. If there is something they did not understand in class they have at least the texbook to turn to. This is a good thing in some cases but it also might take away the incentive for communication among the students and between the student and the lecturer. For the lecturer the use of a textbook opens up the seductive possibility to get around a pedagogically demanding topic by just referring to the book, this way saving time in preparation. I usually do not use a textbook but make my own notes which are complete in the sense that they are self contained and sufficiently detailed that they can take over the function of a textbook. Why do I think it is worth making this effort although there are very good textbooks available? For one thing I was never happy when I tried to follow a textbook. This is because every author has his/her own taste regarding the way and the order material is presented, the notation and terminology used, the emphasis that is put on certain parts, and the imprtance of certain examples. Just imagining that say half of the courses on a certain topic in the country are taught from the same book adopting a singular point of view makes me shiver. I feel that I can pass on my excitement about and the beauty of the subject I teach much better if I use my own point of view instead of somebody else's that I do not share in some details.

Whenever I had periods (on scholarships) during which I did not teach I was wondering why (after some time) I was already looking forward to teaching a course again. Probably it just is that if one has seen things that one thinks are very beautiful, one wants to show them to others in the hope that they also find them beautiful.