## Don’t disturb my circles

Every working mathematician knows that if one does not control oneself (best of all by examples), then after some ten pages half of all the signs in formulae will be wrong and twos will find their way from denominators into numerators.

The technology of combatting such errors is the same external control by experiments or observations as in any experimental science and it should be taught from the very beginning to all juniors in schools.

— V. I. Arnol’d, On teaching mathematics.

The latest addition to my collection of howlers and horrors arrived just too late for the Pi Day celebrations, which is a shame. The context was a “show that” homework question in which the target answer was something like “$k = 20\pi$”. The solution evolved by one second-year student — actually it appeared in four different scripts, but I’m assuming they were not entirely independent — made a simple calculus error to arrive at $k = 40$, and concluded

$k = 40 = 20\pi$, $\quad \rm{when} \quad$ $\pi = 2$.

(Technically speaking, I suppose this is correct. However…)

Once I’d recovered my powers of speech and thought, I started to wonder what lay behind this statement. It came in the first question in the assignment, and the handwriting showed no signs of hurry. The working involved a trig function, making it hard to believe that the students had identified $\pi$ as having anything other than its conventional meaning. Could it be an extreme example of the phenomenon I’ve observed before, in which students make wistful comments such as “I can solve quadratic equations; I just can’t solve them in Mechanics“?

I suspect, on balance, that it’s symptomatic of a different underlying problem, which is the lack of a control mechanism. Everyone who tackles maths problems for a living knows that it’s rare to get them right first time; no matter what theory may say, in practice there is generally a certain amount of iteration and accommodation until a calculation settles into place with all its components neatly related to each other and its sign errors decorously tucked out of sight. One of the big differences between us and our students seems to be how little they realise this, or the aversion they have to checking and correcting their working. Having reached the end of a calculation and spotted a discrepancy, they behave like a cowboy builder faced with a joist that’s a few inches too long: a couple of smart whacks with a sledge, a couple of nails bashed in to hold it, and the offending structural element bows itself into place — throwing out the stresses everywhere else, but hopefully good enough to pass casual inspection until the client’s used tenners are nestling in a grubby inner pocket.

Is that too cynical? Can my students really care so little about the accuracy of their work that they’re prepared to whack $\pi$ with a hammer until it takes the value 2, rather than to look elsewhere for an error? Could they suspect me of such underhandness as setting a question in which $\pi$ denoted 2 without saying so? Are, they, perhaps more plausibly, simply incapable of assessing the validity of any mathematical argument, including their own? (Other evidence suggests that this may sometimes be the case.) And what, after all these years in the job, can I find to do about it?

Poor Archimedes! He thought he was getting
Somewhere, when that soldier came by and a sword
Asked another question and cut short his.

— Norman MacCaig, Discouraging.