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I spent much of this summer working at a summer camp for high school students. I worked with math classes: knot theory and non-Euclidean geometry. I was surrounded by interesting people and it was a lot of fun.
I didn’t know much knot theory and hadn’t ever seen a formal development of non-Euclidean geometry, so I learned some things this summer. I stayed barely ahead of the classes (and probably wasn’t as helpful with homework as I would have liked), but having seen each of these subjects before, I was able to say broad generalities about big ideas. I appreciated knot theory more than non-Euclidean geometry. The knots course was a course in waving your hands and playing with rope, and I found that much more enjoyable than the formalism of the geometry course.
What should be the ultimate goal of a short math course at a summer camp? Any material that is taught will probably dissipate fairly rapidly; assuming their memory is like mine, I would be surprised if any of the students would be able to produce a coherent explanation of the subject even after a few months. Even so, mathematical maturity and an interest in mathematics would likely remain.
With this in mind, such a short math course should contain an introduction to rigor and proof, demonstrating how mathematics is done in the real world. Beyond building mathematical maturity though, it should have content in the form of big ideas. Maybe it would not be possible to communicate those ideas perfectly precisely, but as long as the spirit and the coolness of those ideas is communicated, I consider the course to be a success.
The summer has been entertaining for nonmathematical reasons as well, such as water guns, a war hammer, and a sword named Dragonsbane.
For the remainder of the summer, the plan is hibernation. Observe that penguins live in Antarctica, where it is currently winter, so it is obviously the correct season for hibernation!