The most interesting aspect of today's events (at least to me) was the Stella Octangula Session led by Cindy. Not only did we learn what a Stella Octangula was, but the exercise brought up many other interesting issues that I know I would like to think more about, and maybe someone reading our reports might like to offer some input as well.
I did not sit back passively observing the participants; I joined one of the six groups of two. We all worked on related, complementary projects. Cindy has split up the Stella workbook into these six exercises. As a result, workshop participants or classrooms can be divided into six groups, everyone gets a substantial piece of the action, and finally every group presents its results to the group as a whole. This not only teaches students how to present their results in front of a group, but it also allows the material in the workbook to be presented in a short time period.
Each group worked with a differing number of the following pieces: small tetrahedrons, octahedrons, large tetrahedrons, "cube pieces", "cube corners", and large cubes. We discovered many interesting aspects of these objects, such as the ratio of their volumes, how they fit together to form a cube, and last but not least, what a Stella Octangula looks like. The Stella Octangula is, as one group described it, a tetrahedron with another tetrahedron that has been rotated 180 degrees sliced halfway through it. It is very hard to describe, and maybe I shouldn't attempt it any more here.
Below I will list some interesting/humorous/thought-provoking quotes and comments from the Stella session.
1. "We ignored the formulas, and it became easy." -- Mary Denise
This was a comment that resulted because of the differing approaches Mary Denise and I had to our problem. Mary Denise is very analytical and prefers to have numbers and equations down on paper to solve problems. I, on the other hand, prefer to look at the figure and try to reason and visualize the problem. These different approaches point to interesting ideas about ways of learning and ways of seeing. We also both professed our disappointment at not being able to do this kind of activity in our high school math courses. It was surprising that both of us, from different generations, had had similar stale experiences in our math classes.
2. Mike and Paul introduced a captivating philosophical question. In doing their exercise, they mused on how their Stella Octangula figure was similar to fractal geometry. As you continue to add on small tetrahedrons to the planes that slice through the tetrahedron (I hope I'm reporting this correctly --forgive me for my ignorance of fractals), the limit of the volume of the figure is the volume of the cube. I thought this was a really cool topic. Paul then began to relate fractals and infinity to the immeasurability of a coastline. He claims that you cannot measure the perimeter of a coastline because the perimeter of any coastline is infinite. He explains that you can keep dividing up the things that make up the coastline (into stones, then pebbles, molecules, etc.), and therefore the perimeter is infinite. I am sure he can give you a more detailed account of his explanation if you are interested. Anyone have any opinions out there? I'd be glad to enter into a discussion on this topic. I find the topic of infinity fascinating.
3. Saundra, Ruth, and Sarah mentioned the fact that they find it very difficult to visualize problems. This idea of being 'spatially challenged' was then related to the possibility that this might result from gender differences. Ruth suggested that possibly men, in general, are better at visualizing, and this may be a result of the toys and activities boys are given as children. An example would be building with Legos as opposed to playing with dolls. Does anyone know of any interesting research that has been or is being done on the topic of the problem with visualization and whether or not it is gender-related?
The other three groups presented interesting results, including demonstrations using mirrors, projects with a "cube octahedron" (6 square faces and 8 triangular faces), and the construction of the Stella figure. Ideas were free-flowing, and I even overheard continuing discussions of the Stella session at dinner this evening.
Feel free to comment on any of the above questions. You can send mail to me, and I will forward the questions to the participants.
Good night! --Heather :)
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