I post this upon request, but I immediately must excuse myself as it doesn't give a practical resolution to what the OP asks. Having said this, this does respond to an underlying component of the question, that of using grades to "assess a student's knowledge", as the title states.
My original comment,
[...] there's proffesional research in the way of undermining the legitimacy of grades.
refers principally to the work of Alfie Kohn, an educator and researcher in the field. Of course, by transitivity I'm referring to his sources, but I haven't read many of those directly so I will limit myself to Kohn's take.
In short, Kohn's work provides empirical evidence for the uselessness of almost all prevalent methods in the educational system (allowing perhaps the restriction to American schools), along with theoretical discourse, and very importantly the observation of a lack of evidence in the opposite direction. Of course, "uselessness" has to raise some eyebrows, so I must make explicit just what "use" we wish to have, that is not had via grading. For indeed, students do acquire information, from a bare perspective. However, it is the interest in said information, complexity, and quality of the same that fail to be present. Taken from The Case Against Grades:
- Grades tend to diminish students’ interest in whatever they’re learning. A “grading orientation” and a “learning orientation” have been shown to be inversely related and, as far as I can tell, every study that has ever investigated the impact on intrinsic motivation of receiving grades (or instructions that emphasize the importance of getting good grades) has found a negative effect.
- Grades create a preference for the easiest possible task. Impress upon students that what they’re doing will count toward their grade, and their response will likely be to avoid taking any unnecessary intellectual risks. They’ll choose a shorter book, or a project on a familiar topic, in order to minimize the chance of doing poorly — not because they’re “unmotivated” but because they’re rational. They’re responding to adults who, by telling them the goal is to get a good mark, have sent the message that success matters more than learning.
- Grades tend to reduce the quality of students’ thinking. They may skim books for what they’ll “need to know.” They’re less likely to wonder, say, “How can we be sure that’s true?” than to ask “Is this going to be on the test?” In one experiment, students told they’d be graded on how well they learned a social studies lesson had more trouble understanding the main point of the text than did students who were told that no grades would be involved. Even on a measure of rote recall, the graded group remembered fewer facts a week later (Grolnick and Ryan, 1987).
I won't continue on the subject because I want to avoid a rant, but I do invite all to peruse the blog I link to!
Now I'll try to "answer the question". A priori it's strange to do so though, because if my accusations stand, my response should be "A grade B or a C means nothing, ..." and that's about it because the "Case Against Grades" obviously advocates an entire across-the-board reform of education as we know it. And I can hardly answer with "well, you should probably consider rejecting the entire collection of practices that the institution which employs you utilizes, and encourage all of your peers to do the same with the hope of forming a brand new educational system". I respect that not every teacher wants to be involved in political struggles.
Therefore let's suppose grades are here to stay. We can still take the above as a guide for how to interpret them though. Much like a psychoanalyst "sees through" the given "face value" of a patients explicit words, and constructs a picture of the subconscious desires, feelings, etc., we can go beyond grades' prescribed measure of a student's adeptness, and construct a deeper notion of how his/her mathematical maturity will cope contra the future subjects they'll face.
An easy example: you know the student semi-personally, they show profound interest and ask good questions; they're always seen with some book or some other, etc. but their grade is a flat C minus. Probably the student got lazy and didn't do enough practice exercises before the exam, but overall they have no difficulty with the concepts (most students I tutor with bad grades fall into this category). Granted, the laziness itself is a problem since at one point or other they will have to actually apply their mathematical facilities. For now though, I think the example illustrates what I mean with the psychoanalyst analogy.
Anyhow, this has been an undergraduate-who-sometimes-tutors opinion, expanded from a comment into a longer than expected answer, I hope it can be useful. I'll end on an idea for a semi-joke that I've always wanted to expose in a mathematical community: $$\textit{Lemma: }\text{The space $\mathcal I_H$ of human intelligences is isomorphic to $([0,10],\leq)$.}$$
I plan on somehow incorporating the "operation" $a\bot b:= (a+b)/2$ into it somehow too. It's just a sarcastic remark though! (not a serious dismissal of grading, that's above).
