## From the wrong to the ridiculous

A cautionary tale that I’ve shared with many students over the years: when I was doing my Standard Grades and Highers, I had a bad habit — which, like most teenage bad habits, I regarded as a matter of pride — of aiming to finish the paper and leave the exam hall early. On the occasion of my Higher Maths exam, this came back to bite me in the buttocks when I somehow turned over a page without having done the question on it, didn’t go back to check my answers, and consequently swanned out of the hall blithely unaware — until a few hours later — that the question existed. My Higher Maths result, unsurprisingly, was not impressive: a few marks lower still and my career would have taken a different direction, and this would have been entirely my fault and deserved no sympathy whatsoever.

I wanted to start with that tale to place myself firmly in the camp of the idiots before adding my nanocent’s-worth to the ongoing discussion of student “howlers” and whether it’s right to share them. The voice of dissent here is provided by Dr Katie Alcock and Dr Alice Bell’s THE article and Dr Bell’s accompanying blog post, which I referred to in passing in an earlier post. I’d originally described the tone of that article as “faintly sanctimonious”: returning to it at a friend’s prompting I removed that description, feeling that it might be defensible but was also unfair. As Screwtape points out with his usual diabolical shrewdness, the effect of treating something as a joke is too often to remove it entirely from discussion, and it is well-nigh impossible to criticise the use of humour without coming across as puritanical. So, having made it clear that I am almost certainly a hypocrite in these matters, what is there left to be said on the topic?

I think Dr Alcock’s “for” part of the THE article conflates two points, although this may be the fault of the word limit and/or the subeditors. One is the role of gallows humour as a survival tool for teachers, and this is where most discussion I’ve seen has focussed so I’ll leave that alone. The other is the nature of the mistakes that tend to get shared:

The howlers we most enjoy are the ones that indicate some kind of twisted understanding or that a student is trying to get away without knowing anything. The spelling mistakes may be funnier to the general public, but it is the misconceptions that amuse colleagues.

That’s the point that I feel needs to be expanded, because until we distinguish between types of mistake (beyond asking inadequate questions like “is the student dyslexic?”) I don’t think we can reach a sensible conclusion about whether or when it’s defensible to hold students’ errors up for ridicule.

There are plenty of ways to make mistakes in an exam. We’re all capable of slips of the pen, momentary brain farts and the like: the professional mathematician who can get through a long algebraic argument first time without misplacing a minus sign, a 2 or a $\pi$ is a very rare beast indeed. By and large these mistakes aren’t particularly funny, and wouldn’t be particularly funny in any context. (I’m reminded of a Myles na gCopaleen column where he presents the misprinted classified ad “Wanted: wife, capable of being bent”, before explaining that in fact he has just made this up, but that if we feel it would be funnier for appearing on the classified page there is nothing to stop him paying for space and inserting it there.)

A more serious category are the errors that indicate genuine lack of understanding of a point. These are the ones we see so often that they become wearily exasperating: the product rule applied in place of the chain rule; the inequality that reads $-1 > x > 1$; the function $e^x$ integrated mechanically to $e^{x+1}/x$… We can and do warn students about these, and hope that some of them will heed the warning; these errors, though, provide most of the soul-annihilating drizzle of stupidity that makes exam marking such an unappealing task. They are also, by and large, not amusing.

The errors that are worth sharing, and that form the currency of conversation in staff rooms and corridors during the exam season, are the ones that go beyond error and into gobsmacking and culpable stupidity. These are the statements that not only are wrong, but that anybody sitting the exam should be able to see are ridiculous: the essence of culpable stupidity is not that someone isn’t intellectually up to the task in front of them, but that they aren’t applying such sense as they do have to that task. To my mind, the inadvertent quantisation of velocity falls into this category. It isn’t simply that it relies on an error (omitting the variable of integration) that every maths student has been warned against repeatedly; it’s that the result claimed is so universal that it should immediately have set off the alarm bells. Similarly, the student who cited the linearity of theorems surely ought to have been aware that if there were such a principle it would have had wider consequences than enabling him or her to answer a six mark question in three lines.

I suspect (or hope?) that errors of this ridiculous kind are rarer in essay-based subjects than in maths, which is why the THE‘s annual celebration of howlers is generally underwhelming. The “heinous/anus crimes” example is faintly amusing in a Two Ronnies single-entendre way, and one could argue that any student who produced it (without the excuse of a verifiable orthographical disability) probably hadn’t done as much reading as one should expect from a university student. To qualify as ridiculous, though, the student would have had to write the entire essay on the understanding that “heinous crimes” were crimes relating to the bottom. (I find it hard to believe that this was the case.) A real inadvertent gem — “the early Britons built their houses out of wattle and daub and there was rough mating on the floors” — might still be worth sharing to raise a chuckle, but the chuckle should be sympathetic rather than mocking. Culpable stupidity, on the other hand, deserves a degree of ridicule, because the purpose of ridicule is to expose stupidity for what it is.

If somebody presents an error to me as ridiculous, the hope is that if I find myself falling into the same error, then rather than trying to find excuses for myself or to brush it off I will (a) feel silly and (b) try not do it again. On the basis that I’m not the only person who reacts like this, I will feel quite happy presenting the UAM “howlers”, suitably anonymised or even misattributed to myself, to future classes; and I feel quite happy sharing these howlers with other people and encouraging them to do the same. This sharing certainly has the social benefit (if benefit it be) of letting off steam and allowing markers to feel that we’re all up to the eyeballs in the same mire, but the ultimate hope is that like other forms of satire it will have a purgative effect.

(In a sense, incidentally, the “Professor Bumlick” example cited by Dr Bell is exactly the opposite of what I’ve called a ridiculous error: it suggests that the student, far from failing to engage their brain, has engaged it to the extent of seeing clean through the literature in question. This student, it seems, has arrived at the same sort of conclusion I’m pushing here: absurdity is absurdity, and it should be pointed at as such.)

The principles, if there are any, of acceptable satire are therefore the ones that apply to ridiculous errors. Satire is never likely to appear sensitive or in good taste. It must have at least a touch of anger driving it — another reason why the THE howlers tend to be unimpressive — or it degenerates into a kind of cosy banter. And its great weakness is that it may readily become unjust:

It is such a spur to one’s genius, such an opening for wit to have a dislike of that kind. One may be continually abusive without saying any thing just; but one cannot be always laughing at a man without now and then stumbling on something witty.

(Jane Austen, Pride and Prejudice, chapter 40)

I can, I think, justify to myself holding the culpable stupidity of students up to ridicule as long as I can argue that this ridicule serves a purpose; that it has a cause behind it that is independent of whether the stupidity is funny or not. As a teacher, I can harm my students by bruising their self-esteem or by leaving their intellects unchallenged. The ideal is to avoid either crime, but if I have to be convicted of one I’d far rather it was the former than the latter.

This, then, brings me to a point about which both Dr Alcock’s and Dr Bell’s articles are dead right: to some extent the joke is also on us. What is a student doing in a second-, third- or fourth-year maths exam who is apparently incapable of conducting the simplest check on his or her own elementary reasoning? Who let this student get to that stage; who was prepared to let their culpable stupidity go unchallenged; and (to answer the first two questions by raising a third) why did we do this? The danger is that we have become like the audiences of physics and engineering students singing along to Tom Lehrer’s “Wernher von Braun” song  and never realising that it was actually about them. Students and their errors are not our creation, but if we’ve built and inhabit an absurd structure of ersatz education and meaningless assessment, we shouldn’t be surprised to find other absurdities moving in to share the space with us.

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