As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:
Get students asking questions.
As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.
GeoGebra is an excellent tool and resource.
I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.
There are far more ways to present and explore mathematical ideas than I am currently aware of.
That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like Mathematics Teacher, resources like the mathtwitterblogosphere are another great place to find new ideas.
Geometry and mathematics in general are far more broad than most students get a chance to see.
Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing this with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.