What do the kids actually see?

 I’ll show you.  I’m going to remind you what it’s like to be a child in a classroom; a maths classroom especially, but to some extent the same will be true for other classrooms.

To start, this is printed out on my classroom wall:

I love it.  It reminds me every day of how difficult it is for kids to decode and interpret what, to me, is as simple as reading ‘See Spot run.  Run, Spot, run!’  When I want to remind myself what it’s like for them, sometimes I harken back to my studies of relativistic quantum mechanics.  Why did I take this module at university?  Obviously because it sounded badass.  Before the exam came round, most in my cohort were smart enough to drop it like a startled squirrel dropping its nuts.  In my case I figured I never really ‘got it’ during lectures in any module, and through the process of having to revise for the exam I would grow to learn all about this esoteric discipline, shrouded in mystique.  I didn’t.  Our lecturer, alas, didn’t believe in providing us model solutions to past exam papers; he felt that we wouldn’t ‘learn’ anything that way.  Funny, I didn’t learn any more without them… but I digress.  Our lecturer’s name was Boris.  He was a Russian maths genius.  He frequently said two things that we found hilarious, and deeply ironic.  Having just plastered the board with incomprehensible scrawl, he would occasionally pause, gaze pensively for a moment at his masterpiece, and then utter his first favourite collection of words – and you have to imagine this said with a thick Russian accent:

“But that is just mathematics, and the mathematics is simple.”

Sigh.  His second favourite phrase was so profoundly ironic that it urged my fellow suffering would-be-learner to immortalise it in cartoon form, which she has graciously allowed me to share with you.  He would say this when, turning from the board to face us, he would notice at last the deadened stares of the perpetually perplexed gazing back:

Yes, those are actual notes taken from her work, and yes, that is actually how it felt to be in those lectures.  We often quipped that RQM certainly was *not* rocket science; rocket science we had studied in year 1, rocket science we understood!  This, was on some other level!  Perhaps we undersold the complexity of actual rocket science somewhat, perhaps, but it makes the point well enough.

If I put up a couple of equations below, then any mathematicians reading the blog will interpret them with perfect clarity.  Any non-mathematicians might not.  Let’s give it a go:

Gotcha.  So these are known as ‘Diophantine equations,’ after the great Greek mathematician Diophantus who dedicated much of his life to studying equations of this kind.  Modern algebra wasn’t invented until a thousand years ago, and contemporary algebraic notation wasn’t invented until only a few hundred years ago.  Diophantus wrote nearly two and a half thousand years ago, and this was how he represented his equations.  In modern notation these equations look like this:

Any maths specialist will now understand those equations with precise clarity.  For anyone who struggled with maths at school, these may look almost equally impenetrable, and herein lies the problem.  I asked at the start ‘What do the kids see?’  Well really, it has little to do with children.  Daniel Willingham makes a far more useful distinction by talking about ‘novices’ and ‘experts.’  The novice mind is still new to a subject; novices have very little knowledge stored in their long-term memory, and the way in which it is arranged and interrelated is very different from that of the expert.  You can be a novice or an expert at any age, and of course this varies between subject domains, depending on your relative degree of knowledge.

A non-maths expert might read the first equation above as ‘x squared plus ten x equals thirty nine.’  A maths expert on the other hand will also see it as ‘x squared, positive ten x, equals, thirty nine;’ from there they will see an implied summation between all terms, an attempt to ‘collect,’ almost as if all terms are being pulled to the centre of the expression by a singularity.  They will actually compartmentalise each ‘term’ – each part separated by plus and minus signs.  They will simultaneously view that plus sign as representing an operation, ‘add,’ and as representing a sign, ‘positive,’ as opposed to negative.  To them it looks a tad more like this:

As well as looking like:

Their expertise allows them to see those equations in a very, very different way to the novice, who can interpret them in all manner of incorrect ways, or simply not at all!  The Diophantine equations above are perfect for bringing us bold maths-types back down to Earth.  For those of us who struggle to appreciate why kids just don’t get it, or why they can’t see the bleeding obvious, well there’s why, right there.  To them, poor dears, they’re staring at something as much the garbled gibberish as Diophantus’ equations appear to us today.

So what now?  Well the first step is just to appreciate this.  Especially in maths, if we do everything in our power to tear apart what we hope to teach down to its component pieces, we start to see just how much it is that we’re asking the kids to learn; just how much we as experts have internalised without realising it.  I once explained to a year 10 group that to find the area of a circle you square the radius and multiply the result by Pi.  One bemused individual tentatively raised his hand and then asked ‘What does ‘square it’ mean?’  …it was a top set, I hadn’t expected that.  Now, whenever teaching something that I haven’t taught the class myself, I make sure I explain/remind them of every constituent part, even if just a passing reminder; that one experience taught me to assume nothing!

This is probably the single most important thing to gain from the Diophantine equations, simply empathising with our students, seeing how they see, and painstakingly toiling to simplify all that we teach.

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