We need to talk about mathematics

Colin Billett
22nd December 2015 at 15:01


Colin Billett, Subject Genius, We need to talk about mathematics

In my early years of teaching a colleague had a nice easy mantra, that you needed to have a class quiet before you could teach them anything.  It made some sense then, and to some extent it still does now.  But it has never sat comfortably in my mind, and the more I see of learners who have failed in some way, or showing signs of struggling, the more I question this approach.  I meet learners who have had little opportunity or encouragement to think for themselves, learners who lack the vocabulary, and particularly the mathematical vocabulary, to describe and expand on the problems with which they are struggling, and more importantly, to talk aloud about what they perceive as the question, what they might try, what connections they see, and how such things might be resolved.

I fear that such a requirement for silence still pervades our classrooms.  The current Ofsted handbook rather suggests it is almost obligatory for part of a lesson: ‘Pupils may rightly be expected to sit and listen to teachers, which of itself is an ‘active’ method through which knowledge and understanding can be acquired effectively. Inspectors should not criticise ‘passivity’ as a matter of course and certainly not unless it is evidently stopping pupils from learning new knowledge or gaining skills and understanding.’

So what is there to challenge – a happy, industrious classroom in which learners are quietly engaged, and all on task, seems to be all that any teacher might ask?  But Ofsted are clearly missing the point, or being terribly disingenuous.  Learning cannot be passive and active at the same time.  And even if it could, the active learning takes place when learners talk themselves through the process, ask themselves pertinent questions and test their own answers, compare new knowledge with existing knowledge and check if there is any cognitive conflict.  If a learner does this entirely inside her or his own head, then s/he will have only their existing vocabulary to unravel, test and internalise this new knowledge, the new skills.  And from the learners I’ve met over the years, this dialogue is not taking place, not with themselves or anyone else.

What is clear to me, and the theory is there to support it, is that language is essential to thought, and that talking is necessary to that thinking process.  Unless the learner is given the opportunity and encouragement to clarify thought, she or he may be left as unknowing after the period of teacher exposition as before it.  Vygotsky states that  ‘The child begins to perceive the world not only through his eyes but also through his speech’ (1978, p.32).  And furthermore ‘Words and other signs are those means that direct our mental operations, control their course, and channel them toward the solution of the problem confronting us.’ (1986, pp.106-7).  Words are needed to explore problems, relationships and patterns, and to make connections.  And we can’t explain what we mean if we don’t have the words to do so.  As Wittgenstein says ‘The limits of my language are the limits of my mind. All I know is what I have words for.’

So it is clear to me that we can’t learn, share, or show learning if we don’t have the vocabulary to do so.  And I, like the learners, learn those words by using them in context, by talking aloud to clarify my thinking and consolidate my knowledge, and by formulating problems for myself in words I know and understand.  Deprive learners the opportunity or encouragement to talk through what they are doing and we deny them the chance to take ownership of the intended learning.  Until they express it themselves, that knowledge is still deposited in the teacher, and the learner is merely trying to imitate what the teacher has said or done, often unsuccessfully.

A classic case and pertinent to Key Stage 1 is the introduction of ‘proper’ words to describe the attributes of solids – edges, faces and vertices - a statutory requirement for Year 2.  I recently sat with a mature learner who could do no more than point to them and say ‘these things’ for each of those aspects.  I doubt if she had ever handled a cuboid in the classroom and discussed the faces, and what shape each was. Another time, in a classroom situation we were confronted by a question about diagonals on quadrilaterals, and after teaching the same lesson to three groups, I asked the fourth group if they knew the meaning of ‘diagonal’ – not a single reply.   Yet another was a GCSE learner confounded by a shape drawn on a square grid.  Encouraging him to identify it to me, he said in a dismissive way ‘a square…, or a cube – whatever.’  I told him that he ought to know the difference between 2D and 3D objects, it being part of KS1.  I’ve no doubt that he did know the difference, but things went downhill from thereon.  And finally, a learner confronted with a GCSE question that required her to measure an angle.  I asked her what equipment she needed, and she said she’d never measured an angle before.  After much encouragement she decided it might be a ‘protector’, which we duly went to the cupboard to hunt. I took it as yet another indication that without the words she could not function, and it was not her fault that her vocabulary was so deficient.

Talking about mathematics isn’t then just being given opportunity to learn together, to share thinking and insights, to pool knowledge and experience to tackle problems, but also to develop those words as part of a rich mathematical vocabulary, without which we will not have the facility to engage with the tasks at hand.  And remember, next time you ask a learner a question, remember that she or he isn’t using that vocabulary for six or seven hours each day, and may be struggling a little to find the words to respond.  So give them a break, give them time to think and formulate their thinking, and for once ignore Ofted and ‘pace’ before you jump in with the answer.  And once you’ve had sufficient time to say what you need to say to a quiet audience, hand over to them to do the talking. They need the practise, and you don’t.


Ofsted (2015) Handbook for inspecting schools in England under section 5 of the Education Act 2005 (as amended by the Education Act 2011)

Vygotsky, L.S. (1978) Mind in Society. Cambridge, MA: Harvard.

Vygotsky, L.S. (1986) Thought and Language. Cambridge, MA: MIT Press.

Wittgenstein, L (1922) Tractatus Logico-Philosohicus