Now that I have graded the first problem set in an applied abstract algebra course, let me tell you a few things: there are people whose errors would suggest that they don’t know the material (or that they made technical mistakes) and there are people whose errors would suggest that they don’t understand the material but know how to use it. For this reason, I gave out multidimensional grades.

Suppose that, here, N is the number of dimensions in a grade; for all intents and purposes, N=2 for what follows. Of course, multidimensional grading is not without its complications; it could seem a little arbitrary to set only two dimensions, but there are reasons why two dimensions are best: if one uses N > 2, in general, one will receive one grade with its modulus but with N-1 phases because the grades would then be expressed in N-spherical coordinates. Here are the highlights of the method, as used in two dimensions:

• Set Gmax as the maximum numerical value of a question or an assignment
• The perfect phase is at 45 degrees
• Set i and j, orthonormal basis vectors (i.e. modulus 1 and their inner product is zero), as representing the two criteria in use
• Each component cannot have more than a certain value
• If there are multiple questions in an assignment, calculate the total grade as a vector before calculating its modulus and its phase
• Components cannot take negative values (this effectively limits phases to be between 0 and 90 degrees)

Perhaps this is a crazy idea but I wasn’t satisfied with standard numerical grades (one-dimensional) so I am trying something new.