On Assessment, III

Today, I’ll continue with the discussion of assessment I began a few weeks ago.  Last week, I ended with the observation that “The performance of our students determines our expectations, rather than the other way around.”  Before continuing, I just want to make clear the context for that remark.  When I wrote this essay, IMSA was fully funded by the state of Illinois.

What would it look like to legislators — who approved our funding — if you brought in the top students (relative to their local regions), and then they regularly received low grades, or worse yet, flunked out?  Not good.  And because the admissions process was fairly involved, would this indicate a major flaw in this process?

Yes, some students couldn’t handle the high-pressure academic environment.  But I know that in several of my classes, a student needed to work hard not to earn at least a C.  In other words, showing up, handing in homework on time, and doing a reasonable amount of studying for quizzes and exams would usually guarantee a grade of at least a C.

Now is not the time to dive more deeply into this artificial adjustment of expectations, but I did want to mention that this issue is a significant one, even at presumably elite schools for mathematics and science students.

So to continue with the essay….

“Now there is a natural give-and-take between evaluating student performance and setting expectations.  And, of course, the above remarks are nothing but generalizations.  But they illustrate some of the important issues at hand, and may bear fruitful discussion.

Moving to more concrete issues, I believe that the assignment of letter grades on exams in BC Fast-Track [recall, this was the colloquial name for the Honors Calculus sequence] was, on the whole, successful.  Without going into unnecessary detail, the classroom environment was such that the assigned grades were meaningful to the students.  To give a few examples, I assigned a grade of A+ for truly outstanding work, perhaps only a half-dozen times throughout the entire semester.  The students knew this, and so that accolade truly meant something.

Moreover, an A meant something as well.  It was truly rewarding to see the real pride of a student who, used to earning grades in the B range, began to earn the occasional, or perhaps more frequent, A.  Admittedly, students who made it to the second semester were essentially guaranteed a grade of no lower than a B-.  But this seemed to make an A that much more meaningful.

So student exams had two letter grades on them — one for the skills portion of the exam, and one for the conceptual portion of the exam.  No points were assigned, and few comments were made.  Students were expected to rework problems on which they made errors.

I bring up this point because I think this system of assigning grades really did motivate students to learn calculus rather than accumulate points.  This is the critical issue: I suggest that the way we assign grades does little to disabuse many students that taking a mathematics course is about accumulating sufficiently many — or losing sufficiently few — points.

Let’s take a particular example.  The past few semesters, I stopped assigning half-points on assessments [the usual practice at IMSA].  I might forgive a sign error now and then, but too many on a single assessment would warrant a point or two off.

In the past, I simply considered a sign error as a half-point off.  And so it was.  But consider that without being able to occasionally perform fairly involved calculations, it is not possible to become a successful mathematician.  Attention to detail is as important in mathematics as it is in any number of other disciplines, and we try to develop that skill punitively — you don’t attend to detail, and we will take off points.

Of course one might argue that points are given for work well done — but any of us could, I think, agree that when discussing the grading of an exam, it’s how many points off for a particular type of error that is discussed as much as, or even more than, how many points are given for work correctly done.

And so the idea of “partial credit” is born.  Perhaps now is not the place to begin this discussion, but consider that a student might meet expectations (that is, earn a B) without ever having done an entire complex problem on an assessment completely correctly.  (Some teachers have even gone so far as to give no partial credit.  See On Partial Credit, Letter to the Editor, MAA Focus, February 2002, p. 17.)

Why this system of points and partial credit?  One may speculate as to its origins, and there is controversy even now about its use on standardized exams.  But I cannot help feeling that one function of partial credit is that it allows a teacher to defend the assignment of a particular grade.  “Every sign error is a half-point off.  That’s why you got a B+ instead of an A-.   I have to use the exact same scale for everyone in order to be fair.”

But doesn’t this simply shift the responsibility for the grade onto a rubric?  I suggest that many of us would feel competent to take a set of calculus exams — with names removed — and within five or ten minutes, separate out all the A papers.  Of course this is subjective — but no less subjective than saying that this problem is worth six points while another is worth ten, or that sign errors are a half-point off, unless, of course, the derivative of the cosine is taken incorrectly, in which case it’s a whole point.

Thus the assigning of points is no more “objective” than giving a letter grade.  As I’m sure that anyone who has graded a complex word problem based on an assignment of points can attest to.  Consider the student who has the entire procedure correct, but because of a few algebra errors, has no intermediate calculation correct.  The problem is worth ten points.  How many points should the student receive?

Well, of course, you say you’d have to see the problem first.  But I say, no.  The student receives a C.  Having no intermediate calculations correct demonstrates — regardless of what else — that the student has not met expectations.

So is it possible to avoid points altogether?  Perhaps.  Consider the following grading system:

The essay continues in the next installment of On Assessment with a discussion of how to implement this system in practice.