I recently, rather grandly, said to a group of students that I had been teaching mathematics since the last millennium, which is true (I qualified in 1999), if not quite as Dr Who as it sounds. I have, then, planned rather a large number of lessons, and for each one, whether a careful hour-long effort as part of my PGCE or for an observation or in a last-minute rush as I make my way upstairs from break duty, I have asked myself three questions – What?, Who? and How?:
- What am I teaching?
- Who am I teaching it to?
- How am I going to teach it?
These three questions have got me a long way and, in varying combinations and guises, are at the heart of all lesson planning. It’s difficult to see how you could call an activity lesson planning if it didn’t cover what and who, and so when time is short it is the "how" that gets neglected. This has particularly become true for me as experience and other responsibilities have increased hand in hand and I have often reached for my standard approach to teaching a topic – activities that are tried and tested, and explanations I’ve used before. It was very different when I was a trainee – the "how" took up a huge portion of my planning time with carefully constructed questions and complex activities. Some of these activities have made it into my regular repertoire (which underlines the importance of experimenting as a trainee – and, indeed, whenever you get extra planning time: the more options in your standard approach the better).
Recently, however, it struck me that I had been undervaluing the “how”: that both versions of me had misunderstood what that third question should really have got me thinking about. The “how” is not primarily a question of what activities the students will be engaged in but what steps they need to follow in order to learn what I’m teaching them. In my training, I’d been so distracted by activity that even later on (when I should have known better) I’d underestimated the value of this thinking. This revelation came to me as I reviewed a mock M3 paper that my further mathematicians had sat a few weeks before they went on study leave. There were two questions at which they had failed horribly and so I sat, looking at their answers, wondering how they could have got it so wrong – wondering what I had got wrong when I was teaching.
The two questions were on circular motion and I knew I’d taught them everything they need to know – I’d gone through the derivation of acceleration as , I’d shown them that angular velocity and linear velocity are related by , I’d done some examples on the board and they’d had a go at enough practice questions. I hadn’t done anything particularly innovative with my “how” but I’d covered the “what” well enough. I had, in fact, been very focused on the “what” – I’d not actually taught circular motion before and it had been one of my dodgiest topics back when I was sitting the A level, so I’d really focused on getting it right. The students had definitely had all the right input but they weren’t giving the right output.
I thought through the steps they needed to follow, the basic skills that come together with the two key circular motion equations to make the topic the beast it is. They need to be able to resolve forces in two directions and so need to be comfortable with vector quantities and with trigonometry. They need to be able to write down the conservation of energy equation and Newton’s second law and substitute into them. They need to be able to solve (rather nasty) simultaneous linear equations to eliminate surplus unknowns. For some of the questions, they need to be able to realise that the particle will come loose when normal reaction or tension are zero and at that point will behave like a projectile. They, therefore, need to be able to solve projectiles questions. If they can do all that (and I knew that they could) then circular motion questions are easy – there really isn’t that much they can ask you.
My 'Eureka!' moment
I pondered a while on the students’ ability to put together all these “basic” skills in the heat of the moment and wondered whether they were overloading their working memory. Because their algebra wasn’t slick enough, the formulae weren’t coming sufficiently fluently (this is certainly an issue but there was more than that). I reviewed the exam questions and confirmed that actually there wasn’t much to them: general circular motion might be a real pig but at A level… At that point I broke off because I realised that it is such a real pig that there are basically only two types of question they can ask you at A level: vertical circles and horizontal circles. In a blinding flash I saw that I knew that limitation, the examiners obviously knew it, but the students didn’t. I’d taught them circular motion in full generality but had never identified for them the important simplifications that make the inaccessible merely horribly complex. I spent a lesson on vertical circles 101 (resolve radially, don’t even think about resolving tangentially but write down conservation of energy instead) and one on horizontal circles (resolve horizontally and vertically – don’t let the placement of the particles on a banked track lure you into parallel and perpendicular). Suddenly it clicked – puzzled looks turned into smiles, muddled working into correct answers: it was as much of a "Eureka!" lesson as I’ve ever had in teaching.
Some will accuse me of compromising my principles and teaching to the test and, in May of Year 13, I’d not feel too embarrassed if that were the case – but I think there’s more to it than that. The truth is that when I decided to teach circular motion (as my “what”), I was, in fact, teaching two simplified situations where the model is easily resolved and understanding that we don’t yet have the tools to apply our equations in full generality is an important part of the answer to “How are they going to learn this?”
How students will be occupied in the lesson is an important part of planning but it isn’t the key “how”. The third question I need to ask myself before walking into a lesson is “how are these students (the “who”) going to fit this new knowledge (the “what”) into their existing understanding?”. It’s not an easy question and it’s not one I have all the answers to yet but at least I know what I’m doing with circular motion (although not in full generality – that really is hard).
James Handscombe is the principal of Harris Westminster Sixth Form and spends as much time teaching mathematics as his other duties will allow