The momentum - a cotangent vector?

  In mathematical classical mechanics, the momentum is a cotangent
  vector, while the velocity is a tangent vector. I don't get this!

  A cotangent vector maps a tangent vector to a scalar (real number).
  That much I know. But since when is the momentum (in physics)
  a function that maps a velocity to a real number, and what is
  the physical interpretation (meaning) of that real number?

  Using the Lagrange function L, the momentum is p = dL/dq', where

d  is the sign for the partial derivative (this newsgroup does
   not do Unicode) and
q' is a "q" with a dot above.

  How can I see that (given that q' is a tangential vector)
  p is a cotangential vector?