Here is a list of surprising things that students don't know: 
- basis vectors that go with curvilinear coordinates
- matrices as transformations
- power series as approximations
- eigenvectors as the things that are “unchanged” by a transformation
- a geometric conception of fields
- that zero is a number
- how to add two functions pointwise
- how to read equations as words
- what is planar about plane waves?
- geometric interpretations of dot and cross products
- the meaning of the vertical axis on a graph
- how to shift a graph left or right, up or down
- how to expand a sum (see video from E&E 5/29/09
)
- do not understand what “features” are when asked to describe the features of a physical situation