## The arrow thing

Every now and then when marking students’ work you find a comment in which they look up from the page to address you directly. One such comment stood out recently in the middle of an especially migraine-inducing submission — one of the ones that might have been produced by Jackson Pollock with the aid of a ouija board:

I’m sorry if I haven’t used the arrow thing properly.

I think the “arrow thing” in question is the implies sign, $\implies$. There was one lurking somewhere nearby, at any rate. It might even have been used properly for all I know: it was rather tricky to identify two distinct statements on that page for it to connect. At least on this occasion the student had the grace to apologise.

The arrow thing is, I must admit, popular. Generally it functions as a sort of turbo-charged equals sign: not just “a equals b” but “a is really, really equal to b”. Sometimes it means “at the end of the day the answer is”, or “putting all these together we get”. Now and again, it extends itself like one of Mr Tickle’s arms, in which case it functions as a substitute for any kind of spatial arrangement of symbols, and means something like “walk this way, please”. (The visual effect is that of a sort of noodle soup with algebra floating in it.) And at least some of the time its role is purely phatic. “Here’s a mathematical symbol. I am a mathematician. You are a mathematician too. We’re all happy mathematicians together, aren’t we?” Sadly, the effect intended is rarely the effect achieved.

A student lines up some logical implications before launching them into the void.

Sorting out students’ use of the arrow thing was the main hope that we allowed ourselves when we started our recent project to improve their mathematical writing and reading. “If just a few of them start using it correctly then our effort will have been worthwhile…” By that standard, I think our effort was worthwhile, but there’s an awfully long way to go yet.

So what is it that makes $\implies$ such a problematic symbol for students? One, no doubt, is that in formal terms it’s a slippery object: not, as Tim Gowers discusses with his characteristic clarity, a logical connective in a strict sense, and certainly not closely tied to the everyday meaning of the word “implies”. It’s very vulnerable to misinterpretation, not least because — at least in applied-maths circles — its chalkboard usage is sloppy: we write statements of the form $P \implies Q$ both when we mean “P implies Q” in the logical if-then sense and, less legitimately, when we mean “P is true so Q is also true”. (In applied maths journals, $\implies$ is often entirely absent, which may reflect our lingering sense of shame about our slovenly behaviour.)

I think that a more fundamental problem, though, is that students meet $\implies$ before they have any need for a tool of such precision. It’s like kids who are cutting out paper dolls being handed a surgeon’s scalpel, and the results are in their own way just as predictable and bloody. I can’t locate among the SQA Mathematics documentation the point at which they believe that the symbol should be introduced (and given they don’t approve of penalising what they call “bad form” in working and I call “gibberish”, they may not insist on introducing it anywhere at all), but it seems certain both that the vast majority of my students have at some point been told to use it and that they have not grasped its grammatical function — because they have not grasped that mathematics has a grammar.

As long as your picture of mathematics remains that of manipulating one formula to get another one, you have no use for $\implies$. As long as your picture of mathematics remains that of manipulating one equation to get another one, you still have no convincing use for $\implies$. The plain English words “so”, “if”, “then” and, at a pinch, “therefore” will do the job nicely, and they have the advantage that they already mean something to you. Only when you are really happy with the notion that mathematics consists of logical relationships between statements — which certainly won’t happen before you understand what a statement is, and no longer confuse a sentence with a noun — is there any point reaching behind your back to fetch $\implies$ out of your quiver. Indeed, it’s questionable whether you need the symbol at all until you arrive at formal logic and start romping among its counterintuitive results — until, that is, you have thoughts to express for which the level of precision represented by $\implies$ is necessary.

And yet, the symbol persists. It supplies students with a comforting assurance that they can do maths without using words (indeed, that the better their mathematics is the fewer words it should involve); it poisons thousands of attempts at logical reasoning every week in maths departments up and down the country; and it sticks in the flesh of university mathematics teachers like a chunk of wickedly-edged and unextractable shrapnel. Maybe my student’s toxic term for it conceals a dimly-sensed psychological truth about the arrow thing.