I agree about the importance of alternative representations, but, people should be somewhat careful about which ones they're espousing. Sometimes people get quite enthusiastic about wedge products and then think what they're excited about is geometric algebra. Personally I would like to see wedge products taught alongside vector algebra and calculus. But I don't see a useful place to include the geometric product, except as more better way of stating things about actual Clifford algebras (quaternions and gamma matrices). I do suspect that there is a 'better' version of GA that is important than that, but I haven't seen it described.
Maybe I'm just not on the math departments enough.