I think it's probably a little difficult to get people started on bases and vector spaces without introducing lines first, and the concept of relative rates of change (as evidenced in the slope of a line) is fairly important, and good preparation for calculus.
Fine, so introduce lines, circles, points, rectangles, triangles. All this stuff can be introduced without delving in to maths. All you need is graph paper, a pencil, and a ruler. With these, introduce co-ordinate systems. The triangle is relative to the center of the circle, which is relative to the center of the world. Oh look, we can create shapes between any points, combine shapes to create new ones, etc. Oh look, the circle has a center point, and oh, by the way, so does the rectangle.
All these concepts, rate of change, etc, should be taught. Just about every mathematic taught has a practical use, but the students (and faculty) are oblivious to it.
I almost also think math should be taught with an easy programming language. Seeing graphics on screen do something is a heck of a lot more rewarding than a checkmark.
