>You don't have to remember the defintion. You just have to remember a >bunch of examples. For vector spaces, I just think of R^n, and then >generalize. It relates to algebra to it has to have something to do >with operations. So, vectors have to be additive, and scalars have to >be nice. The definition is pretty obvious. Memorization is only really >needed for geometry.
I think this works only for those students that already have a fair amount of sophistication. Most of the students at my institution have very weak logic skills, let alone generalization skills. The definition for them is very far from 'obvious', even after several different examples have been given. I strongly suspect the OP is in this category. For these students, the first step is memorization of a precise definition.