  Magnetic fields are traditionally presented as a (pseudo)vector quantity,
tied closely to the cross product: familiar ideas to experts but full of
challenges and subtleties for students. Building on earlier work in rotational
physics, we present an alternative approach to teaching the subject in which
magnetic fields are instead described using bivectors, which can be visualized
as "tiles" with area and orientation whose components form an antisymmetric
matrix. Bivectors have historically been considered mostly in specialized
contexts like spacetime classification or geometric algebra, but they are no
more complicated to understand than cross products. Teaching magnetism in this
language addresses common student difficulties, opens the door to understanding
the topic in relativity and extra dimensions, and can give fresh insight into
these familiar ideas.
