Grading

This week has basically been consumed by teaching. I teach a section of Applied Calculus at UVA--it's my first year ever teaching a class. It's quite an experiment, really. Let's take a bunch of grad students, who have been admitted to the PhD program entirely because of academic merit in mathematics and not whatsoever on teaching merit, and throw them in front of some college freshmen (oh, I'm sorry, "first-years," as we say at UVA). Surely some learning will take place.

Tuesday was my students' first of three mid-term exams. I felt like it went well. There's nothing quite like grading an exam. Even in mathematics, it's highly subjective. The only time it isn't subjective is when the answer is entirely correct. I guess that does make it different from, say, grading a paper in philosophy. Is there ever an entirely correct answer in philosophy? This in itself is a philosophical question, of course... but the practical answer is, "no."

Subjectivity naturally comes into play, because for any given problem, a significant percentage of my students will have written something not 100% correct, and then I'm faced with the tricky problem of trying to measure how much understanding the answer communicates. How do you quantify understanding? Sometimes the whole process seems absurd.

I'm conditioned to expect a certain grade distribution, and I can kind of use this expectation to judge the validity of the grades I have come up with for my students. The truth is, though, I desperately try to find things that I can give credit for. Any signs of intelligent thought are rewarded with points.

Mathematics could be the most easily self-regulated discipline in the world. Every mathematician is highly critical by nature. One simply cannot succeed in mathematics without having this irrepressible urge to spot every important detail in a problem and verify it. Thus, if you tell any mathematician to grade a collection of calculus exams, his first inclination is simply to rip every answer to pieces.

Yet that same critical impulse generally causes me to turn inward and ask myself, what purpose would it serve to be hyper-critical of responses on a calculus exam? That combined with a genuine caring for my students as people causes me to try to find the best in every solution.

I never have to worry that I'm being overly generous. It is simply not physically possible for a mathematician to go above a certain threshold in giving credit for only partially correct solutions. The mistakes that appear in answers students give can range from humorous to painful, but in any case I just can't ignore them. No matter how much credit I want to give a student, I am forever bound by a mysterious force beyond my power to be more honest in my grading than perhaps my students would like me to be.

It's pretty brilliant, isn't it? The department never has to look over my shoulder to see if I'm actually being an ethical grader. They know that I am already constrained by a power far greater than any threats that bureaucracy might employ to keep me on track.

All of this is not to say that all of us grad students grade our students the same way. Sometimes we don't agree on what exactly the students are expected to understand about a problem. Math has this attractive quality to it because it is so objective, but setting learning goals for students (and then measuring success) is a more or less entirely subjective matter.

I could make it entirely objective, of course--each answer would be either right or wrong, no in between. But this probably would be the most unethical way to grade, because it would completely rob students of any way to communicate their ability to develop some sort of problem solving strategy, which is actually what we're trying to teach. Math, as it turns out, is not really about the answer.

When it comes right down to it, every academic discipline, math and the hard sciences included, are mostly about creating a dialog about something we wish to understand better. The reason we grade our students is to try to get them to make their ideas clearer, not just "right." There's something deeply personal about the whole thing, actually.

What's sad is how so many students go on asking what's going to be good enough--good enough for you, the teacher, to find them acceptable enough to label with an 'A' so they can go on and accomplish their own goals quite apart from anything you've told them. If only they realized that they'd really earn their 'A' if only they opened themselves up to actually communicating ideas.

I suppose it's really just the difference between being a consumer and thinking of your teacher as a supplier of a good, and being a person and thinking of your teacher as another person. If you're just a consumer, you can certainly get your 'A,' but only because our culture encourages grade inflation (education is subject to market forces, after all). If you really want to understand the material, you have to move from being a consumer to being a real person desiring real communication.

I understand why many students can't move this direction. I'm actually quite okay with most students in my class not caring about math. I'll do my best to be a supplier of a good--maybe if they work hard, I can even supply 'A's for them. There is, after all, a certain level of dignity in that kind of transaction.

But being a teacher, while also being a student, has caused me to reflect that what's really behind all those commands to "show your work" is a desire for something more personal. I wonder if my students will ever get that.