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What Do Grades Mean

Of course I can only comment on what they mean to me; the thorniness of the question comes from its subjectivity.  It has been decades since I was on the receiving end so I shall only comment on giving them.

For me, having taught full-time for over 34 years, the grades are categories into which I have historically been comfortable sorting students into based on their performance in my classes.

I offer a good deal of assessment (weekly and monthly) of different kinds (computation problems sets, essay exams) and yet I view my assessment as a form of statistical sampling of student performance, mindful of all the noise (a roommate needing to go the ER the night before an exam, etc.) that affects measurement of student performance. In particular I do not feel as if my estimate has the precision to justify +/- grades but follow the system that is part of our culture. 

We did not have +/- grades when I began and I was pretty comfortable with the five categories (Excellent, Good, Average, Unsatisfactory, Failure) of letter grades despite the non-canonical meaning of Average in a culture where the majority of students are above Average. Nowadays when a student challenges a grade as being borderline I must only state that the time to discuss the grading scheme is before the final grade is given.

As someone who occasionally reviews transcripts (e.g. for a job search) I find it nearly impossible to evaluate grades given by other professors. Some institutions are known for filtering admissions very competitively and then giving grades that reinforce the perception that the filter was merit-based. To understand a grade's meaning I would probably need to know the corresponding percentile rank in the class. An A that puts you at the 50th percentile means something different than an A that puts you at the 90th percentile.

Different disciplines and groups of disciplines handle grading differently, which does make our system of Cum Laude, Magna Cum Laude, and Summa Cum Laude honors a bit silly for relying upon GPA's.

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