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In the role of an advisor, I am often faced with the question

Should I retake this [undergraduate] math class?

My first-order approximation to the answer is:

  • If this is your last math class, a C or better is fine, be done with it.
  • If you are taking more classes after this one, a B or better is fine, but you should retake a C.

However, there is also the possibility of independently checking the student's knowledge to help inform the decision, especially if you are one of the other faculty members teaching the course. For example, you could give the student a chain rule derivative, or a substitution integral, and see how they approach the problem.

This has some pitfalls. For a start, by doing any independent checking, you are implicitly saying that you "know better" than that student's instructor and that you are a better assessor of the student's knowledge than the instructor. This undermines the legitimacy of the student's grades. There are more issues than this, to be sure.

On the other hand, I have been on both ends of these independent assessments.

Is it ever a good idea for an advisor to do impromptu assessments of students? When?

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    $\begingroup$ Does your school have any sort of placement exams that you can offer? I've seen them sometimes used to "test out" of a low level class or to get by with a low grade. $\endgroup$ – Chris C May 14 '15 at 15:43
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    $\begingroup$ I think it is good to keep in mind one of the larger pitfalls: encouraging a student to take the next course, say (2) when they barely pass placing into it from course (1); they fail course (2) and perhaps retake it a few times; they then have to retake the course (1) again, wasting semesters and quite a bit of money. This is why one of my previous universities instituted placement exams for when there is any doubt. $\endgroup$ – Chris C May 14 '15 at 15:49
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    $\begingroup$ I agree with @ChrisC's suggestion to look for pre-existing placement examinations. Another option is to provide the student with a past midterm or final (if possible) to evaluate the student's knowledge. And I would schedule any such assessment (after checking with the department) at a separate time, so that they are able to get themselves mentally ready! During an actual advisor meeting, I probably wouldn't start asking math questions; though I might try to engage around what they remember from a course (maybe remind them of the core topics) and listen to how they talk about the mathematics. $\endgroup$ – Benjamin Dickman May 14 '15 at 18:53
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    $\begingroup$ I don't mean to start a discussion - which is why I'm commenting and not answering - but I'd point out that there's proffesional research in the way of undermining the legitimacy of grades. $\endgroup$ – GPerez May 14 '15 at 18:57
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    $\begingroup$ @GPerez Why not post a bit about that research as an answer? It seems like that would be very productive. $\endgroup$ – Chris Cunningham May 15 '15 at 14:04
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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).

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    $\begingroup$ I very much doubt that intelligence is a linear order. I would think it is some incredibly complicated partial order. $\endgroup$ – Steven Gubkin Jan 7 '18 at 0:04
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The other answers made good points... and suggest continuations and qualifications. For example, some students seemingly benefit from being informed that courses and grades are merely a stylized approximation to the genuine goals... while other students are confused or see opportunities to game the system upon hearing "an advisor" say any such thing. Gauging the tendencies of the given student is a very-necessary preliminary to deciding how much "reality" to admit/explain to them.

Certainly one tipping-point is about the availability of placement exams, or testing-out exams: if they're available and seem to genuinely address the issues (as opposed to being broken), there are marginal advantages to "staying within the (formal) system".

There is the auxiliary issue, suggested by @TomAu's answer, about whether "the advisor" will seek to diagnose context-independent knowledge, or, rather, assess the very-context-dependent issues of a "class". For that matter, the sequels for which a given course is prerequisite may be of wildly varying natures, so that no general prescription is possible if the chief goal is "success in immediately following coursework".

(Indeed, in extensive observation, it is not at all the case that orthodox success in previous courses is optimal preparation for even-the-most-orthodox success in the immediate sequel, apart from general behavioral traits and capacity.)

And, even in not-so-inflated grading systems, a "B" in a "prereq", if the student (as is usual) has no outside information, is like starting a long car trip with a slow leak in a tire...

On net, try to avoid improvising assessment, ... but some students are not "typical", in the sense that they do know more than the grade indicates, and may be ingenuous enough so that their competence would not be accurately appraised by the staid conventional apparatus (which does often appraise compliance or conformity as much as genuine mathematical competence, in my opinion.)

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It helps to place such assessments in light of the goal that inspired the question. I think this goal is most often "How do I manage my time to learn topics from area X?", although it could also be from a desire to master a subject independent of time constraints.

Assuming time management is the goal, it helps to run through some scenarios:

a) the student has mastered (at a B level) enough to move to the next subjects, so he just has to review old material occasionally alongside the new material. Think of this as taking 1 1/8 courses combined.

b) the student has not mastered enough to move to the next subjects. Then he has to relearn the old material as well as learn new material as well as learn how to apply old to new. Think of this as taking 3 courses combined instead of one. The student will need a tutor to handle each of these three areas, as opposed to handling just the material in the new course.

c) the student doesn't know. In which case, the student needs to self-assess, and find whether not retaking means a multiplicative factor of 9/8, 3, or even higher. You can give them guidelines for where you think the number will be, but a good and quick metric is how much of the introductory material in the new course they can handle. If they are struggling with it, a likely cause is insufficient mastery, and they need to apply some (not small) multiplicative factor or postpone the new course until the old course has been mastered.

I think you should give them an additional assessment, not an independent one. If you present them the model above, they may get a more realistic view of whether to retake the course.

Gerhard "Even Bad Models Are Useful" Paseman, 2015.05.14

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    $\begingroup$ "Even bad models are useful" :) $\endgroup$ – paul garrett May 14 '15 at 21:56
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The problem with course grades is that they measure the student's knowledge of the "book," or lecture notes, not the material or subject.

If the student knows things about the subject over and above the "book," I would have a higher assessment of his (or her) ability than the course grade.

On the other hand, if a student knew things only by "rote," and was totally outside his element outside of the class, I would lower my assessment of his ability below his course grade.

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Chris, interesting questions and some good responses and your approximation is thoughtful. Just a few other soft thoughts to consider.

  1. I wouldn't do a test on the kid. I just think that gets too much into the weeds and kind of breaks a little the feeling of support role of an advisor. If you really want to stealthily diagnose it, I would just talk to the fellow ("where did you have problems?" "what did you feel good about--what did you feel bad about") and try to get a feel for the person's own confidence about their level of knowledge. EVen just the form of how they describe their knowledge level will give you some feel for their ability. But I would not quiz them with specific problems. Keep it in more of a friend and supportive coach role. Maybe you could look at their course tests if they want to show them, but even that is maybe pushing it.

  2. You should advise the student to talk to the instructor of the next up class (if you are not very familiar with the course yourself or if the content varies significantly by instructor.) That person is best able to judge what is needed for their class and probably more suitable for that person to be putting the student on the hot spot with a little verbal or even written diagnosis of skills. (And it also starts to get some dialogue between the teacher and the student for next semester, sets them up for success.) [The one case, I am leaving out is if you taught them last semester. In which case, you should have know the student's level of knowledge well and have a gut feel.]

  3. May vary based on grading scale (I guess some places are harder than others). I have progressed with Cs and even a D (high school algebra!) Maybe in general, I think your progression scale is a little harsh (basically saying a B is passing).

  4. Some other soft factors to consider: skill of the student (smart but lazy or distracted kid can probably succeed more in next class than a slower student who needs a good foundation or he is lost.) Personal needs of the student: are they OK with hanging out or do they feel the need to move on. What their major or career goal is. (There are some that need stronger math skills than others.)

  5. (minor) You can maybe be a little practical when considering areas of weakness. If the kid says (and you believe him), I got all the calculus except the series convergence stuff, I would progress him as that is something he won't need (comparatively!) unless he is a math major.

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