General Philosophy behind the course
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.
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 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 s
The homework on non-euclidean geometry is challenging for our students. It may be useful to start out treating the homework as an exploration and allowing students to to start the homework in class. This gives them some time to look at the questions and make sure that thay understand what is expected from them. This approach is usually not necessary for the other topics.
Projects are a great way to have students experience mathematics in a more informal manner. There are different approaches possible when constructing a project. It is important to clearly communicate what we expect from the tessellation they produce. The students may not be artistically inclined, but the point is that the geometry they have learned will allow them to create an interesting tessellations. The final product should be a finished work. It should be at least 8 x 11 inches, created on sturdy paper, either in ink or in color.
- The little artwork and long paper approach: Students are asked to develop one outstanding tessellation (more sophisticated than just translations) and use the artwork to write an indepth paper (8-10 pages) describing everything they know about symmetry, isometries, wallpapergroups, etc. The weight when grading is 30% art - 70% paper (or possibly 40-60).
- Significant experimentation with the artwork and a short paper. Students are asked to use what they learned to experiment and produce sketches of several different tessellations. They choose the one they like best and turn this tessellation into a finished product. In a short paper they describe what techniques they used and what the symmetry group of their tessellation of choice is. When grading, the weight is more heavily based on the tessellations produced, specifically those that were done as sketches. A possible division here would be 70% art - 30% paper.
Fieldtrips really help bring the material alive for the students and helps them see how and where geometry can be found in the world around them. Some possibilities are:
- Fieldtrip to the Cathedral Basilica: The students are asked to go on a scavenger hunt at the local Basilica and find all possible Frieze patterns as well as as many rozette patterns and wallpaper patterns as possible. We ask they take digital pictures and use those to show that they have indeed found these symmetry groups.
- Fieldtrip to the Art Museum: This Fieldtrip can even be used as part of the final in the course. The students are asked to go to the local art museum and find at least two different artworks that can be used as examples of geometry in art. They should be two different concepts. A short paragraph describing the artwork and how it uses geometry would be a good form of assessment.