The second implication for instruction is related to the
emergence and persistence of sign errors after instruction in
magnetism. Because cross products are a novel mathematical
operation for many students, more emphasis on the nature
and execution of the cross product may be needed, including
explicit emphasis on the noncommutative nature of the cross
product and identifying and practicing proper right-hand
rules. The latter point is particularly important in courses
where the instructor teaches multiple right-hand rules. Some
sign errors were due to a misunderstanding about the direction
of the field between magnetic poles. For these students,
requiring the comparison between isomorphic arrangements
of field lines and field sources or some other instruction designed
to illustrate the relation between field lines and field
sources could be beneficial.