When I studied General Relativity using Misner, Thorne and Wheeler's "Gravitation", it was eye-opening to me to learn the geometric meanings of vectors, tensors, etc. The way such objects were taught in introductory physics classes were heavily dependent on coordinates: "A vector is a collection of 3 (or 4) numbers that transform in such and such way under rotations..." or whatever. I much prefer the geometric definition of a vector as a "tangent" to a smooth parametrized path, and a covector as a linear operator on vectors, etc. But once you introduce spinors, it seems that the geometric way of understanding things fails. People are reduced to saying "a spinor is something that transforms in such-and-such a way under rotations" again.The mathematically sophisticated way of describing them in terms of representations of the group of rotations (or Lorentz boosts) don't really provide much more understanding of what they really "are". Is there some geometric way of understanding spinors that is more in line with the "tangent to a parametrized path" understanding of a vector?