General Philosophy behind the course
Teaching Philosophy We want to get students back to a form of experimental mathematics. Often texts start with the theory and delve in to examples last. We want to turn that around. We want to start with examples, and build up the student’s intuition. After looking at examples we want to build our knowledge from the ground up. It is only after working through several special cases that we look for patterns. Once we see the patterns we develop the theory.
Lecturing We give short lectures to introduce a topic, but hour-long lectures are rare. It is however useful to give a “fact finding” lecture at the end of a section. It is useful to summarize all our findings. This ensures that students haven’t overlooked any observations, and understand the material in sufficient detail. Students like these types of lectures, and it is easy to turn them into a class discussion.
Homework Homework is an important component of any mathematics class. Homework can include independent reading, finishing up a worksheet, or distinct assignments. We discourage students from asking questions the day homework is due. Students should get into the habit of starting their assignments early. There are times however when the homework assignments deserve more attention. The homework assignments from the Non-euclidean sections, and the assignments concerning isometries and group theory (the last is an optional topic) benefit from in class discussions. One technique that has worked very well in the past is to have the students write their answers on the board. Having student work available helps us launch into a good class discussion. It is also a great way to talk about what is required in a good answer. Some students need help in figuring out how much to write down. We do point out to students that since no one will really remember what is on the board, there is a certain degree of anonymity with this process. Students are often afraid to volunteer ideas that may prove to be wrong. Point out that this is a good way to start a discussion, and that answers that are incomplete or incorrect provide a great opportunity for everyone to learn more. We also encourage students to write down or at least mention during the discussion any alternative solutions to a given problem. It is also useful to use this technique when discussing the first few worksheets. In the beginning of the class, students tend to not delve into the material in sufficient dept, and this is a good way to set the standards for the course.