On Grading

A few weeks ago, I posted a short satire on how I view one of the most common ways of assigning grades — taking points off for mistakes, and determining students’ grades by how many points are left.  Today, I’d like to take a more practical look, as well as describe a system I currently use in my teaching.

I can recall two events which started  me thinking more critically about grading practices. The first was a letter to the editor written in a mathematics journal.  The university professor described a final exam in a differential equations course — ten questions, worth ten points each.  To his dismay, no student earned ten points on any problem.  They got by with partial credit.

So he thought he’d try the following grading scheme:  no partial credit.  Each question was either correct (10 points) or incorrect (0 points).  To his surprise, the course average didn’t change significantly — students actually were much more careful because they knew the stakes were high.

Second, when teaching multiple-section precalculus courses at the high school level, I would need to give the same exams as my colleagues.  We’d sit around and ask questions like “How many points off for a sign error?”  “What if they just make a minor arithmetic mistake?”  We’d try to make sure everyone graded pretty much the same.  We’d even talk at the (insane) level of half-points….

So when developing an honors-level calculus course (at the high school level), I thought I’d try something different.  First, I separated exams into two sections:  Skills and Concepts.  And second, I’d grade problems as Completely Correct (CC), Essentially Correct (EC) — meaning a student knew how to approach a problem, but had significant issues in following through, or not correct (X) — indicating lack of a viable solution strategy.

I’d then assign a letter grade based on the CC/EC distribution.  This is along the lines of the university professor’s thought — you can’t earn an A if you don’t have a certain number of problems Completely Correct.  In other words, a student must demonstrate significant mastery in solving problems, not just get by on partial credit.

I do know of professors who use a “2–1–0” scheme — and this is certainly similar.  Assigning a CC 2 points, an EC 1 point, and an X zero points can help in giving an approximate idea of where a student stands.  But I also use grades of CC- (perhaps more than a few arithmetic errors), EC+ (almost CC), and EC- (got the right idea, but just barely).  If a lot of the EC’s are in fact EC+’s, I might bump the grade up a notch.

I also differentiate between Skills and Concepts questions.  Skills questions are more-or-less textbook problems — routine, checking that a student knows the mechanics.  They are relative short, although I might sometimes assign two CC/EC/X grades if the problem is a little more involved.

I purposely avoid problems which scaffold in the Skills section.  For example, if you want to assess integration by parts through a volume problem, and the student sets up the volume integral incorrectly, then you may not be able to assess their ability to perform integration by parts.  I address this issue by writing two separate Skills questions:  first, a “set up but do not evaluate” volume problem, and an integration by parts problem.  I don’t feel anything is lost here.

The Concepts questions are intended to assess whether students really have some conceptual understanding.  They are typically open-ended, and require some argument (though a formal proof is not necessary in calculus).  As an example, here is a Concepts problem from a Calculus II exam given last semester:

For each of the following statements, either justify why it is true, or give a counterexample to show that it is false.

(1) If \displaystyle\sum_{n=0}^\infty |a_n| and \displaystyle\sum_{n=0}^\infty |b_n| converge, then \displaystyle\sum_{n=0}^\infty |a_n+b_n| converges.

(2) If \displaystyle\sum_{n=0}^\infty |a_n+b_n| converges, then both \displaystyle\sum_{n=0}^\infty |a_n| and \displaystyle\sum_{n=0}^\infty |b_n| converge.

This is certainly a non-routine question.  As such, I grade Concepts problems more leniently, assigning an EC if it seems that a student has shown some insight into the problem.

On this 65-minute exam, there were seven Skills problems and four Concepts problems.  To earn an A, a student needed 5 CC and 3 EC.  The 5 CC meant that there had to be significant Skills mastery.  But note that if an A student earned at least an EC on all the Skills problems (which they should certainly do), they only needed to make progress on one of the four Concepts problems to earn an A.

I try to design the Skills part so that it takes about 45–50 minutes to complete, leaving 15–20 minutes to think about one (or more) Concepts problem(s).  And as the grading scheme implies, some progress needs to be made on the Concepts problems to earn an A, which is as it should be — an A student should be able to demonstrate some level of conceptual understanding.

Further, I might bump up the grade if a student can make progress on more than one Concepts problem, or perhaps gets one CC (which does not occur all that often in my classes).

I’ve been using this scheme for about seven years now, and I like it.  Grading is more pleasant (no agonizing over points), students rarely argue about EC/CC (it’s not that fine a distinction), and I can count on one hand (with fingers left over) the number of students who have argued about the assignment of a letter grade.  What’s also nice about this scheme is that it’s easy to adjust for exams which are (inadvertently) too difficult — just relax the requirements needed for an A.

Now to some extent, this may seem highly subjective.  But I maintain that it is no more subjective than assigning points.  Precisely how are those assignments made?  If a quotient rule problem is worth 10 points and the student switches the order of the terms on the numerator, is it 1 point off?  2 points?  Maybe 3?  How many “tenths” of the 10-point problem is the order of the terms on the numerator worth, anyway?  It’s truly an arbitrary decision.

A point-based system does suggest that problem-solving can be broken down into separate chunks which can be assembled to make a whole — something mathematicians know to be ridiculously hard to quantify, if it is possible at all.

Now I do admit to giving point-based exams in courses like Business Statistics — a 13-section course of which I taught two sections last semester.  I had almost 50 students, and the course is essentially a skills-oriented course.  Like any other teaching strategy, a grading style may seem more appropriate in one context than another.

So I am not advocating a “one size fits all” grading practice here.  My intent is to suggest that there are successful alternatives to a strictly point-based grading system.  And while every system has its drawbacks, I believe that one of the main strengths of the system described in this post lies in the ability to meaningfully assess conceptual understanding — something I have found virtually impossible in a point-based system.