Summer Research Internship Program
In Summer 2012, I participated in Cooper Union's Summer Research Internship Program, an intensive six-week session in STEM subjects for very gifted high school students. I designed and led the program in mathematics, Symmetry: Geometry and Elementary Group Theory. The central objective of our projects was for students to discover and explore a profound, beautiful, and central theme of modern mathematics - the fundamental interconnections between geometry and algebra.
After discussing some basic number theory and modular arithmetic, we began our research into the main topic - how geometric symmetries in two and three dimensions can be described algebraically in terms of groups. The first project investigated planar symmetries. Students saw that the rotational and reflective symmetries of the unit circle and of regular polygons could be completely understood in terms of multiplication and conjugation of complex numbers. The second project investigated the geometry and rotational symmetries of three-dimensional regular polyhedra - the Platonic Solids. Understanding the rotational symmetries of these objects was seen to be the same as understanding certain permutation groups.
As I wanted students to discover results for themselves, lecturing was minimized. Instead, I created questions which built naturally on each other, so that students were led gradually and organically to their final theorems. For instance, difficulty in directly calculating the composition of rotational symmetries of a regular tetrahedron motivated the labeling of its vertices, naturally leading to an investigation of permutations and studying rotational symmetries in terms of these. At the end of the program, students wrote a final research report and gave a power-point presentation to the rest of the internship program.