At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.
Running a teachers’ professional development session recently, I was sharing some children’s mathematics. The children had investigated constructing arrays that allowed them to calculate products in a variety of ways. For example, one child explored how nine times twenty-two could be modelled from an array for ten times twenty-two by slicing off a ‘strip’ to reduce one side to nine.
The array presents a powerful model of why what might become a rule of thumb (to multiply by nine, multiply by ten and subtract the original number) is grounded in mathematical structure.
I got the teachers exploring 75 x 89 in similar ways, for example starting with an array for 100 x 89 and then working with the array and using halving to get to 75 x 89.
Towards the end of the session I got the ‘what’s the point?’ question.
This question often hangs in the air but there is not always someone brave enough to voice it. I asked for elaboration and the teacher declared that such activity was meaningless and had nothing to do with ‘real-life’. Taking a risk I asked the group (of around 120) whether anyone had actually found the activity meaningful. Thankfully around 2/3 of the room put their hands up. The questioner persisted that the activity was meaningless, despite my pointing out that since that clearly was not the case for the majority of people in the room, the activity itself could not be regarded as meaningless, only some people’s response to it.
We talk about mathematical tasks as having ‘meaning’ or being ‘rich’ but course tasks do not have ‘meaning’ or ‘richness’; these are qualities that people engaged in the tasks bring to them.
The discussion with the teachers moved in the direction that schools are advised to always relate mathematics to ‘real-life’, coming back to the utilitarian argument for mathematics. I’m not convinced that ‘utility’ actually does succeed in making mathematics more meaningful for more children. For some, yes, but for many children playing around with ideas can be ‘meaningful’ in much the same way that people find meaning in playing the piano, painting an abstract, or reading poetry; activities each of which could equally be argued as having little immediate impact on people’s lives.
The idea that someone might engage with playing with mathematical ideas simply for the intrinsic pleasure of the play itself is alien to many. But that’s not the fault of with mathematics itself, and all to do with the way we continue to teach it.