I recently wrote about a pupil who uses a subtraction technique he invented for himself; this is a rather similar situation.
There are two reasons I particularly remember the inspection I did at Hamphill Middle School. The first happened when I sat down at the back of the classroom. Now, like most of us, I’m pretty experienced at sitting down, so I wasn’t concentrating on the process very hard, but I became aware that something strange was happening. I couldn’t work out what it was, and it continued to happen very much in slow motion until I found myself on the floor – one of the back legs of the chair had given way. I was hugely impressed by the reaction of the children, who were most solicitous and greatly concerned for my welfare even though no inspector has ever made a more undignified and hilarious spectacle.
After this dramatic start the lesson continued with the children doing some division examples. Partway through, I noticed a girl with her back to me, behaving in a manner so furtive it would have resulted in instant arrest in the outside world. She looked around to check the teacher wasn’t watching, hunched herself up to conceal what she was doing, and scribbled something on a piece of paper. She wrote something in her exercise book and scrunched up the paper and stuffed it into her pocket.
The calculation she was doing was 95÷5 and her method is totally transparent. She’s seeing it as a sharing, with 95 to be distributed into five packages. She makes an initial distribution of 10 into each package and follows this with a further 5. She’s keeping track of the amount distributed, and decides she can make another distribution of 5 into each package. You can almost hear her exclamation of frustration as she realises that will be too many and the 5 needs to be downgraded to 4.
At that point she knows she’s distributed all 95; she adds the 10, 5, and 4 in any package and has the answer of 19. She transfers this to her exercise book in the approved format, and the piece of paper has now served its purpose and goes into her pocket.
I talked in detail with her, and what she said went along these lines:
(1) My teacher’s taught us how to do these, but I don’t understand her method, so I’m not a very good pupil.
(2) However, I can do them using my own method, which of course is really cheating, because it can’t be as good as the correct method.
(3) I love my teacher, and if she thinks I can’t understand her method she’ll be disappointed because she’ll think she’s let me down.
(4) So I’ll work them out my way and write them up as if I’ve used her method.
Poor Justine! Her emotions involved her perception she was a failure who could only use a method she saw as inferior, coupled with a very real concern that she didn’t want her teacher to be disappointed either in herself or in Justine.
My reaction was of course exactly the opposite. As far as I was concerned, Justine was actually doing better maths than many others who were simply following a rule without insight or understanding. The example shown is pretty low-level, but it’s extensible. As understanding grows, you can divide – should you need to do so – both larger and smaller numbers using the same starting point.