‘Use what you have’ is a wise maxim that can be applied to so many situations, and I always try to bring to any problem more than my own mind-set I often post little questions on social media because friends enjoy them – anything from KS2 questions to time-honoured puzzles. So when I was helping my daughter to plan a maths lesson, and she suggested the question of how many arrangements are there in a Braille cell, we wondered how her learners might respond. I said I’d post it on Face Book and we could test it, a bit like ‘we asked 100 people…’ on Pointless. Simon came in early with an answer of 50, with no explanation. Others followed with different numbers, some correct and some not, and some with brief explanations. Then Simon came back with ‘What’s the answer?’, to which I replied ‘I’ve already asked that question!’ Cruel? What is the point of a puzzle if it ends when the first person hits the right answer?
I honestly think the first and best rule of teaching is to never answer a question, except by perhaps responding with a different question. My job is to challenge and stretch the learners, and I do this by facilitating a set of learning situations that get the learners engaged with the subject, and most important of all, thinking mathematically. And I do this by posing questions, setting problems, devising puzzles, and using all the resources available to me, including an almost endless supply of past questions, now so easily available for any level from KS2 to A’level.
But why do I never answer a question? Even to a rhetorical question such as that I find it almost impossible to reply. Why would I want to? Almost every teacher knows that by explaining something to another person we deepen our own understanding. And my understanding of maths is pretty good, so I don’t need the benefit; I give that opportunity to someone else. And that’s as far as I will go. I can think of many reasons, and I’m sure the reader can. What so many teachers need is the confidence, skills, mental resources, often physical resources, and a stock of suitable responses, so that we can always answer a question with a question, or by finding something that might help.
Writing in ‘Mathematics Teaching’, issue 243, November 2014, Barbara Ball describes the struggles of changing the traditional classroom approach in the 1980s.
‘Some students thrived in this atmosphere. (…) But others felt threatened:
I wish you would stand at the board, Miss, and give us sums to do, instead of walking about all the time.
We had complaints from parents too:
My son asked his teacher a question and he refused to answer it.’
I had the great privilege of working with Barbara, and she supported me as I tried to put the ownership and responsibility back to the learners. But it can lead to difficulties and conflict. My worst case recently was Peggy, an FE learner who had already failed GCSE a few times. In response to a past paper question, she asked me if six was a prime number. Over the years I have got used to such inane questions, and responded almost by reflex – ‘What do you understand by a prime number?’ At which point, very quickly I thought, she got most indignant and almost shouted ‘Just tell me if it is a prime number!’ What does one say to that? ‘Think for bloody yourself for a change!’ Not an allowed response. So I said what I could to diffuse the situation, and realised that one of us would have to go. She did, to a colleague who readily gives an answer, and an explanation too, I’m sure. And I also left, in the same year. The annual offer from the Teacher’s Pension Agency suddenly seemed quite attractive.
Learners have to think for themselves, and also to begin to ask themselves those important questions. So ‘Is six a prime number?’ should have been for the learner ‘What do I know about prime numbers?’, and if the answer is damn-all, then ‘does anyone at my table know, are there any posters on the walls, can I look it up anywhere?’ But we, as a profession, have sadly brought so many to this stage of wanting an answer or, even worse, of having any old answer, that classrooms are full of learners like Peggy.
There is a chimera in all this, the notion of ‘scaffolding’, almost universally accepted as good. If we mean by this replying to the learner’s question with a series of less complex questions that will bring the learner to the answer for which the teacher hoped, then that is almost as bad as simply telling the learner the answer, and explaining why, to you, it makes sense, or can be deduced from what you know. It hasn’t helped the learner in any intrinsic way, just given them something they might remember, but more likely not. This can be easily seen in geometry questions relating to angles. The learner is stuck, can’t see a way to the final ‘answer’, so the teacher takes them through the steps she saw - ‘what is this angle, and why?’, ‘so what is this one..?’. The learner is hardly thinking, but bringing forth small snippets of knowledge. Sometimes it is expedient to do so, as in the case of Dave:
‘Sir, how many inches in a foot?’
‘Dave, how tall are you?’
‘Five feet eleven.’
‘So how tall will you be when you grow another inch?’
‘Six foot. What are you on about...!’
Nope, David hadn’t yet joined the thinking gang, and I’m not sure if he ever did.
So what are my strategies? Get the learners working for themselves, thinking for themselves, and sharing in their groups so they take control of their own learning. Keep a well stocked cupboard so you have a physical resource for almost any question, even it is only a piece of square paper. Line the walls with posters with plenty of descriptions, definitions, examples, and learners’ work covering a variety of things. Learn to keep your mouth shut. And learn to not go bothering the learners by going from table asking if anyone has any questions.
Colin Billett has recently retired after teaching in schools and colleges across the Midlands.