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:

Diophantine Equations

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:

Coloured Equation

As well as looking like:

Terms Separate

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.


About Kris Boulton

Teach First 2011 maths teacher, focussed on curriculum design.
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8 Responses to What do the kids actually see?

  1. Pingback: What does simplicity look like? | …to the real.

  2. Pingback: How might social media help teachers improve education? | Pragmatic Education

  3. Jill Berry says:

    Thanks for this, Kris – I enjoyed it, and it made me remember two things.

    Firstly, in my first few years of teaching I decided I wanted to have singing lessons and riding lessons (you won’t remember Roy Rogers, but…). I found it salutary to be in the formal position of a learner building my expertise in an unfamiliar area at the same time as I was learning my craft as a teacher. It was a humbling experience and I’m sure it helped me to empathise with the pupils’ perspective in my lessons.

    Secondly, as a deputy head starting to apply for headships I realised that I had no whole-school timetabling experience and felt I needed to understand more about basic timetabling principles if I were going to lead a school. So I went on a timetabling course. I learnt more about myself than about timetabling I think! I was surrounded by men and women who were probably a bit like you… and they were just much quicker and more confident with number than I was. I’m an English specialist and generally more comfortable with words. I can DO number but it takes me longer, and I found it incredibly difficult that so many around me were so much quicker, more intuitive and confident than I was. After a while a kind of ‘red mist’ descended and when we were asked to do a quick calculation I started to freeze.

    I took all the material home and worked through everything at my own pace to prove to myself that I COULD do everything we’d been asked to do, but that was without the pressure of so many around me doing it faster (with, it seemed, little appreciation that it wasn’t easy and obvious to all).

    The next day in my lessons I looked at some of the less confident members of my classes and had a far deeper understanding of how they might feel when I asked a question and a small number of hands shot up almost immediately as the keenest and the quickest eagerly showed me they’d got there already. It made me rethink how I handled Q and A, for one thing.

    So thanks for reminding me of these experiences and encouraging me to reflect. I’m not a mathematician/scientist and didn’t ‘get’ your equations but I did get the message!

    • Kris Boulton says:

      Thanks for another insightful post Jill – sorry it took me so long to approve it!

      What’re you’re describing in the red mist sounds like a combination of one or two things, of which you’re probably already aware. On the one hand it could cognitive overload, where the working memory is filled to capacity, and we’re left with an indefinable feeling of nothing… not knowing where to go next, or what to do, just frustration. Alternatively it could be more linked to ‘maths anxiety,’ where timed conditions are placed on an activity with which a person isn’t yet 100% au fait, resulting in anxiety, which then has the unfortunate compound effect of releasing hormones which block certain parts of the brain, actually inhibiting thinking even further! (huge problem for exams, of course)

      Like you, I definitely always try to keep in mind how much work it is to understand something first time round, and how much needs to be in place before the things finally click together, like PapaAlpha describes.

  4. PapaAlpha says:

    That is what teaching is all about, breaking knowledge down into its component parts and building it back up again. Sometimes pupils can build it back up for themselves by making connections to the components you’ve taught before, mostly they need you to show them how to link to the next component. It’s like giving a young child their first Lego set and showing them how to fit two pieces together: the possibilities for construction then become endless.

  5. Pingback: MathsConf14: The Stories of Maths – MathsMuse

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