On the other hand, maybe GA can help with differential forms. Differential forms involves exterior algebra, and I feel like some aspects of exterior algebra are elaborated upon in an insightful way by GA. For instance, the grade-2 elements of an Exterior Algebra can be understood as angular velocities in many circumstances. In GA, this is captured by the exponential map that sends grade-2 elements to rotors. I don't know if this can be helpful for teaching purposes.