Teaching
Earlier in my career, I viewed teaching as a performance. This approach has some strengths but can lead to a passive student experience. Today I instead emphasize active and group learning methods in class, using a mixture of slides, worksheets, discussions, and computer exercises. I think that homework should feature deep and interesting questions while exams are better suited for predictable checks of core definitions and techniques.
I've recently started to experiment with collaborative exams.
I like Terry Tao's essay: Does one have to be a genius to do maths?
Nonstandard content in standard courses
Here are some course topics listings with some nontraditional choices:
Calculus III includes the multidimensional Newton method, for zero-finding and for optimization (slides)
Differential Equations
Computational Linear Algebra includes discussion of the barycentric formulas, Chebyshev polynomials, interpolative matrix decomposition, nonnegative matrix factorization
Student-centered syllabi
Here are some readable syllabi from recent courses:
Calculus II