The Common Sense of Teaching Mathematics

"I could have chosen almost any book by Caleb Gattegno and in fact one of the exciting things is that most of his books are now available for free on a website, www.calebgattegno.org . Gattegno was a maths educator who talked about, in his own practice, teaching the entire five-year secondary maths curriculum in 18 months — and teaching it to mastery. What he meant by mastery was that students were as good at performing these things as he was. Gattegno did a lot of work in teaching languages, and he basically felt that learning mathematics was like learning a language. He looked at how young children learn languages, and he developed a curriculum which, certainly in his own hands, demonstrated it was possible to access all those powers we have as learners, for learning our mother tongue, in the learning of mathematics. Hence the speed and skill which he was able to impart to his students. He advocated a route through this whole debate about “should I teach for understanding or should I teach rote learning?” What he did was devise activities where children could very quickly gain a symbolic mastery and then through a more creative exploration within those symbols, would develop their own understanding. In a sense, to quote a man called Dick Tahta who is actually the author of one of my other books, the teacher takes care of the symbols, the teacher is there to design activities which will mean the children get a sense of how these symbols work, and the sense takes care of itself. We leave it up to the children to actually make their own connections about how these things work. So it’s not denying the importance of understanding, but it’s suggesting that if that’s what you focus on right from the beginning, that’s quite an inefficient way to learn mathematics. By devising game-like activities which involve these mathematical symbols, you can get children into using symbols in a very sophisticated way, and through that use, develop their understanding. So to try and ground that suggestion, again going back to working with primary children, Gattegno from a very early age would get children working with very large numbers. He had a chart where it would be possible to get children working with hundreds and thousands and tens of thousands and writing those symbols accurately from a very early age. I don’t think it matters what the children understand by being able to say, age 4, 10,000 plus 10,000 is 20,000. It doesn’t really matter what that means – they can get excited by using these symbols and names. I’ve certainly seen that. They do get very excited. In a sense the understanding or the meaning of those, and the relationships between those words develops over time. It’s not like I need to understand what 10,000 means before I can start using it and playing with it. At that time he was a well-known figure, he was written about in Time magazine and I don’t know if you know the Cuisenaire rods ? Cuisenaire was a Belgian teacher. They’re these different coloured rods starting with a white cube, going up to an orange one that is length ten. Gattegno was involved in the promotion of those materials, and I think at one stage every primary school in Canada had them and they were very well used in the UK. He certainly has had influence, but I don’t think many people today would recognize his name. It’s not clear to me why that has happened. Partly, it’s his writing. The book I’ve chosen here is one of his more accessible ones, it’s quite curricular focused. When he writes in a more general way about mathematics, it’s almost impenetrable. He has a whole system of thought about what it is to be human and learning, and really I’ve only been able to access his work through working with it in a group, with groups of people who are interested in the same thing. It’s not an easy approach. What he’s advocating depends a lot on the awareness of the teacher. He developed these materials – like the rods – and I think they can be spectacularly successful, but they’re not easy to use. They’re not the kind of thing that you can just pick up and go into a classroom and from day one expect to be successful… One answer I have to that links with a research project I’ve been involved with at a primary level, where teachers have been getting into this idea of positioning the students as “becoming mathematicians”. This has allowed some of these teachers to let go of their worries about not necessarily knowing what mathematics will come out of a particular activity. The teachers can place themselves alongside the students and say, “Well, actually, as a mathematician I’m not sure what the outcome will be, but maybe as a mathematician this is the question I’d ask, or I’d try to pursue that pattern.” On one level I think you’re right, but there are alternatives. The learning of mathematics can turn into something where the teacher is not needing to offer a very clear explanation, but a process of exploring and discovering things together with the children. I think you’re right. In this primary project, the kids have been loving even just trying to say the word mathematician. It’s powerful, I think."