I TA at Penn during the academic year and teach at Mathcamp during summers. In undergrad, I was a regular course assistant for math and CS courses.
Teaching at Mathcamp is a unique experience. Over the course of a summer, a Mathcamp instructor designs and teaches (approximately) five (approximately) weeklong courses on subjects of their choosing, targeted towards extremely motivated high-school-aged students. I would be super excited to talk with anyone about the pedagogical choices behind any of these classes, as well as to share content and discuss things I would change :).
An informal introduction to the fundamental group, without relying on any underlying point-set topology.
An inquiry-based introduction to category theory. At the start of the class, students made a list of various families of mathematical constructions they thought could be generalized—like the cartesian product of sets and the direct product of groups—then searched for categorical properties encapsulating these families.
An introduction to the theory of infinity-categories in the model of quasicategories. We discussed the need for such a theory, the difficulties with handling coherence, defined simplicial sets, studied the nerve of a category to motivate the definition of a quasicategory, and sketched the proof that infinity-groupoids are Kan complexes.
A survey of various paradoxes from philosophical and mathematical history. Each day started with a ~10 minute lecture discussing the paradox, and then students discussed the paradox and talked about how various mathematial ideas might help us think about it differently. Paradoxes discussed included Zeno’s paradoxes of motion, the Sorites paradox and other paradoxes of ambiguity, paradoxes of the material conditional, and paradoxes of self-reference like the liar’s and Russell’s paradoxes.
An introduction to the combinatorics of saturated transfer systems and submonoids.
Course-assisting at Reed consists of grading, holding office hours, and being an intermediary between the students and professor.