Why I hate grades, mathematically speaking

orderd clothesI was thinking, the other day, of grades as being an ordering on the set of students in a class. The classic ABC grading system is a total ordering of students. The problem, of course, is the relation “knows at least as much as” is at best a partial order. The reason is there are multiple talents/skills/abilities when it comes to math. One distinction that comes up often in the algebra classes I teach is conceptual knowledge vs. algorithmic proficiency. The student with conceptual knowledge is able to “see”. He or she understands how to read algebraic expressions/equations, finds creative (non-standard) ways of solution, and often anticipates the next lecture topic in their questions. The student with algorithmic proficiency is able to accurately apply lengthy procedures to algebraic problem types. Their work tends to be organized on the page and they know how to check their answers. The best students combine both of these abilities and the worse students have neither. Students strongly weighted in one of these directions or the other are very difficult to compare. This is the problem for our grading system. Our system requires this comparison.

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