JoaLDG: no coin flipping in the exam hall!

Leave your calculators and quarters at home: yesterday was the linear algebra final exam. So today it was graded. It took a little under five hours, from nine-thirty to a quarter past two, for maybe a hundred and fifty papers. I had one problem to grade. For five hours. A little numbing, but not so fatiguing as you might think from it being doing math for so long at a stretch: it consumes much more energy to shift to thinking about a new problem than to continue looking at the same kinds of solutions to the same problem. Grading in parallel is easier as well as more consistent.

Actually this one problem was in two parts, a and b, and the two parts didn't have anything to do with one another, so really it is two problems. They were couched as “true or false” questions, but when they add “Explain.” to the end that actually means: “prove or disprove.” Since this isn't the math majors' linear algebra class, it will perhaps be not too surprising to you to read that generally speaking the option taken was “not prove.” I was not unduly burdened by doing “really two problems.” It turned out that the composer of part b badly mistook the difficulty of his problem: the average over the entire class for that problem was about one-and-a-half percent. There were earned a total of twelve points in all those hundred-fifty papers, eleven points from seven hundred and fifty. (After a vast, vast number of zeroes were awarded, after I began to feel that I was back in kindergarten learning fine motor skills by forming the numeral 0 over and over again, well, I started counting.) And of those four people sharing twelve points only one earned full credit, so congratulations to MJA.

This is, I confess, a fair disgrace. (I should mention that part a was rather better done – I wasn't counting but I'd guess the average was somewhere around forty percent, plus or minus ten. So I wasn't just a zero-scribbling bit of broccoli for those hours. Maybe you find that a fair disgrace too, but at least it's a spread.) The other one is really just a waste of everyone's time. I can't blame the problem-poser for this disaster; the problem was merely a great blind-spot for almost everyone.

Something is shameful if it's counter to some law – what law do I mean here, calling this a disgraceful situation? Really just what I said: no spread of grades.

Grading problem sets I am acutely conscious of their existence as a pedagogical tool. Ultimately everything I do is predicated on the need to indicate to the students the deficiencies and successes of their technique. So for example if the problem is to compute some certain numbers or vectors and the bottom line is incorrect and the reason why is exactly that there was an error in carrying out one step of an algorithm a little while earlier then substantial partial credit is earned. This is only natural: I am not grading the answer but rather the mastery of the material and the techniques of the class. This mandates more nuance than up-or-down.

There is, of course, a different point of view, that overly discretised up-or-down view, which we'll call the “either the building is going to stay up or it's going to fall down” vantage. I think we can all agree this is the standard to which we'd like to hold our civil engineers, but it seems a little draconian for first-year math students, and moreover, as I said, unpedagogical. I must even object: what do I care for the right answer? If I wanted the right answer then I'd do the problem myself. And if some miss the nuance in my nuanced feedback, all the worse for them, but I don't mean to pander nor panic.

And yet there is a well-known and popular case of this discrete approach. The Putnam contest for undergraduate math students applies a similar rubric: “not a solution” gets in the range of zero to two points (from ten); “solution” gets in the range of eight to ten. The problems are, to be sure, not easy, but the easiest problem on the Putnam paper isn't so hard that you couldn't manage it in three hours (if you were an energetic young undergraduate math student), and yet the median score, the score which half the contestants don't exceed, is perennially zero or one point, out of one hundred and twenty. (One year the median score was three, and some wag remarked that this was a reflection of how “ridiculously easy” the contest was that year.)

Well! If I graded linear algebra problem sets in this style, I can only think I would quickly discover how expendable I am – just after all but a handful of students from forty drop out in fear of failing the class. And if this seems draconian even in a prize exam, well, in the first place, it is the rule of the game, and everyone comes to play the game. But moreover it has after a fashion a certain logic: every problem in our linear algebra textbook may be all-too-easy for me to solve three different ways – there's a reason I'm qualified to grade this stuff – but if you give an incorrect solution to a problem no one knows how to solve, how can we really say how close you are? Every false theorem is one mistake away from being proven. It's not a hopeless problem to say how near or far a proof is – there's a reason they pay those research mathematicians so well – but the iron prison of the idol called Rigour lets no one free who hasn't really a complete and correct solution.

So when I grade prove-or-disprove problems on a final exam, what am I doing? Really my goal cannot be pedagogical. The students are permitted to look at their papers in the sequel but my understanding is that this is not typically done, and in truth I don't expect it to be otherwise. I sympathise with those who, coming to the end of a difficult course which they have not yet totally arranged into comfortable, familiar parlor room furniture in their head, would rather take a long, quiet, contemplative silence by the koi pond. So I have no belief that I am still teaching anyone anything.

Rather I am evaluating: who has it and who hasn't it? We need our grades to give a spread, to differentiate the students, and form them into some groups, the men and the boys and those in between, linear algebraically speaking. This does inform one's granting of partial credit. For example, it is very far from my mind that a simple response of “true” or “false” should beckon my pen hand to form anything other than a zero. (No doubt you have already guessed this, since it really beggars belief that only four people should have their coins come up tails, even if your name is Rosencrantz.) If there is no evidence that a proof could be in the offing, there is no reason to say that anything good has happened.

This problem of credit for the right single word has been a serious challenge to me: how could it be that any credit at all is earned by an utterly confused answer merely because one key word happened to be correct? Yet at the same time I want to encourage students to make guesses, develop their intuition, and so forth, even if they can't finish it off. After all, a good conjecture is the first step to a theorem. So on problem sets a one-word answer should worth a little bit of partial credit – typically one or two points from five, depending on the depth of the problem in question and the credibility of the notion that the student just didn't want to bother justifying their answer. (That unmathematical practice I strenuously discourage, even to economics students.) But to avoid giving the slightest encouraging word (or numeral) to errant nonsense I have developed what I have been privately (now publicly) calling the egregious weirdness doctrine. (I like to give names to these sorts of things, as you may have gathered from my discussion of the ultimate linear algebraic sin, but typically I'd rather terminologically paint myself in a more temporally-based judicial role.) This doctrine states that if I am considering a range of possible grades for a submission then if there is egregious weirdness present I will err to the low side. To clarify, some silly examples of egregious weirdness; pretend I've added parenthetically “(good grief!)” after every last one of them: claiming a matrix with a row of zeroes is invertible; writing down two vectors that are parallel after applying Gram-Schmidt; citing a theorem's converse despite having disproven it in, well, every single problem for the last two weeks; and so forth. They're the kinds of mistakes that are manifest and impossible for me to believe derive from anything but the coupling of linear algebraic ennui and the sort of deep confusion I called “the fog of the Nothing,” which “roam[s] between the sky and the space which is beyond, fantastic, undreambound,” in the first JoaLDG entry.

If the egregious weirdness doctrine explains how I can give credit for one-word answers on problem sets without seeming to punish people for just “writing a little bit more” (shouldn't it be a principle that writing a few more lines oughtn't worth an answer less credit?), perhaps its dual explains how I can't do the same on the final exam: only if there's something to praise can I say it worths something on the high end.

I just wish we hadn't ended up with a Putnamesque lack of hosannas on that problem.

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Though it's the end of linear algebra for the school year, this is not, I should assure the gentle reader, the end of Journal of a Lower-Division Grader, nor the beginning of a hiatus. At my current rate, and at their tendency to bifurcate themselves into two and three parts, I have stories enough percolating on note cards to keep you and me over the summer.