She had to find two sets of factors of 12, choosing from the six numbers 2, 4, 5, 7, 9, 10. She chose 2 and 10 immediately. I was relieved and thought that Mrs Jenkins, who had told me that Mary would need a lot of time to choose and try out all the different combinations, counting out lots of cubes each time, was wrong.
I should have known better. It turned out that she had been acting on a principle of her own, which was that the answer was always a big number. Therefore she had chosen 10 and she then chose 2 because she could see there was a 2 in 12. As it happens, this works for 10+2=12, but it is hard to see many cases where it would work. Trying to see how we might get to 7+5 by means other than sheer exhaustive testing, which would not, thought I, prove anything to her except the workings of chance, I tried to get her to tell me the difference between 11 and 12.
"You write 1 with a 2 and one with a 1," she explained as if to a very stupid person. "So why is that?" I encouraged, hoping for some explanation about place value. She gave a shrug, no cheekiness in it at all, just a simple, "Search me, guv, no idea".
We ended up counting out 12 cubes and shifting them in piles from side to side, counting factors and counting wholes. We did it five times and I thought she had got the hang of it, enough to know why 7+5=12. Boy, was I kidding myself. I was also feeling pretty smug because I thought I had got her to realise that the difference between successive whole numbers is always 1. Mrs Jenkins wound up the afternoon with a spot of mental arithmetic and one of the questions was 10+1. I gave Mary a little encouraging smile and she put her hand up. I won't go so far as to say I thought it was in the bag, but I did think we had a chance. "12," she said firmly.
My heart sank. In that moment I saw it all. She was now convinced 1 was a factor of 12 and so was 10: all answers for these numbers were now going to be 12. Or, even worse but just as likely, she simply had no idea, despite her assiduous counting, what any of these numbers meant at all.
I looked back on our 20 minutes with the cubes and realised that she had never taken any numerical value for granted. Just because she had counted a pile once and there had been 7, and even though none had been added or taken away since her counting, that was no reason for them still to be the same. Sadly, just because there had always been 12 to be divided into factors did not mean that the two piles would always add up to 12. In fact, number did not seem to be an unalterable property of things: it was a spell you applied in class to make people pleased with you - or not. What an unnerving world to live in.
Or so I mused. But Mary is a happy girl. Perhaps she just doesn't need maths, or perhaps she doesn't need it yet and, when she does, it will become available to her. In the meantime, the national curriculum and new guidelines on teaching numeracy are all very well, but how many children are like Mary?
Earlier, I had played a maths game with the clever group. "Let's play the more difficult one," said the cleverest mathematician in the class. Try teaching Mary, I thought afterwards. That's really difficult.