I wanted to call this blog - Now you see me now you don't. But this is quite misleading. Firstly, the ‘me’ represents the maths, as opposed to me, the maths adviser. Secondly ‘Now you don’t see me – now you do’ doesn’t have quite the same ring to it! It then occurred to me that sometimes the specific information we need to solve a problem is clothed in camouflage and, therefore we need to look at it in quite a different way - just like finding the grasshopper on the tree, we may have to adjust our view.
That's why I want to talk about representing. My boss, David Cook, talks about representing something as simply re – presenting. Presenting something in a different way. Making something look different so that the information we most need pops into the foreground. Having worked with a number of Year 6 pupils in the run up to SATS, I asked which SATs paper they liked most, a large majority of these children chose the arithmetic paper. Why? Well because they are told which calculations to do of course! This makes me rather sad. Mathematics should foster curiosity and creativity in our children and even when it is about pages of calculations to apply procedures to, I believe there should be thought given to what they will learn mathematically on the way. Don’t get me wrong – children need to have mental and written methods for solving calculations and I have no problem with this, but in the big wide world, they will need to decide which operation to use when solving real life problems – they won’t be told.
So for those children who find the reasoning paper trickier, how can we create the magic of now you don’t see me – now you do, seeing through the camouflage? I think it is all about re – presenting. In addition, for those children who ‘see it’ quickly, the re – presenting can pose the challenge and deeper thinking – but that’s a whole different blog!
What does this ‘now you don’t see it – now you do’ actually look like?
Take this question from reasoning paper 3 from 2018:
As I watched a low prior attaining pupil, who at the start of year 6 had suffered from huge maths anxiety, tackle this question, she had a real ‘now I see it’ moment. Her response was: ‘I can see that there are two hexagons in the first picture and only one in the second picture. The numbers are different as well. 147 – 111 is 36 so a hexagon must be worth 36.’ Proud teacher moment right there I can tell you. She had isolated the key information needed to solve this problem by considering what was the same and what was different.
But how could we re-present this question to help children who might not have seen it as confidently?
Could we re – present it as a bar model? Could we re – present it in number sentences?
Here is how you could re – present this problem:
In a number sentence(s);
As a bar model;
Presented in the original way, at first the similarities and differences might not be obvious. The absence of one of the hexagons is perhaps more obvious than the absence of the 36. But re-present it and bam - now you see me!
It might not do the calculations for us – we still need to find the difference between 147 and 111, subtract this from 111 and divide the difference by 3 but, at least we can now see the maths required.
When analysing this years’ SATs papers I had my own ‘now I see it’ moment with the following question:
I will admit that I spent a little longer tackling this question than I had expected – I’m a maths adviser after all! But the satisfaction of that magic ‘I see it!’ moment was thoroughly satisfying. In fact, I announced quite proudly: ‘Got you!’ I had found the maths I needed.
For me, the re-presenting came from ignoring the triangles and thinking about the rectangle and what I know about them. Allowing some information to fall back and bringing some forward is a key strategy - isolating the important information from the camouflage. If the width of one side was 7cm than the opposite side must also be 7cm. I annotated the diagram and then I was cooking! Once I could see the dimensions of each triangle, 7cm + 3.5cm was simple.
Another question which required some re-presenting for me, was question 21:
Here I decided that the way that I would re – present this question was to use a more familiar representation – a number line. This helped me to separate what I knew about the values of the x axis to the values that I knew for the y axis - to isolate each part.
Yes I had to bring other knowledge to this problem: I needed to know which values in the given coordinates related to which axis and I needed to know that any point on a vertical line will have the same value along the y axis. This question certainly required some fluency that not all Year 6 pupils might have secured yet. But if the questions weren’t testing, then it wouldn’t really be a test would it?
Re-presenting this another way could be as a sequence of numbers:
30, 22, … and 25, 40, …
Perhaps you ‘see’ it in a different way?
Which way is best? I don’t think it matters but I would happily listen to children debate over it!
So how do we foster this idea of our children being creative with the problems that they come across? I suppose the answer is easy – we need to model re-presenting and then provide opportunities for them to be creative themselves. We need to let them explore multiple strategies; share what they have learnt with each other; celebrate the successes; pick apart the misconceptions; and then try out each other’s strategies. It will take time of course to implement and then embed this ethos within the classroom and across the school but with the new inspection framework and the question of the intent of our curriculum, perhaps we could indulge in some re-presenting to encourage creativity as well as isolating the specific maths needed.