After weeks of writing and talking to teachers about behaviour management, I am back (with relief) to posting on my favourite topic: Maths teaching. That is no doubt the effect of meeting two highly inspirational people (one Maths teacher and one Headmaster) within a few days.
I would like to start this post with a seemingly remote comment which was kindly sent to me by a reader concerning the Maths ability pyramid, pointing out that the pyramid does explain ‘what we do in Maths and why we do it’, but not HOW we do it.
Which is absolutely true: the HOW question is extremely important and it will require several posts to develop this (inexhaustible) point. One could even say that, for Maths students, it’s the ‘HOW we do Maths’ that makes a huge difference.
However, we have to be more specific: which students we are talking about ? Indeed, the ‘HOW we do Maths’ question is quite different whether we are talking about primary schools, secondary schools or post-GCSE education ?
In this post, I would like to focus on secondary school students, because that is where the erosion of Maths ability and performance is highest. It is no big secret that many kids entering secondary schools with a fairly good Maths foundation lose interest in the subject, accumulate gaps in their Maths knowledge and finally end up with a lower GCSE grade than they would have got in Year 7…
So, of course, the big question is: how can we avoid that ?
Leaving aside behaviour management issues (which we really shouldn’t, because they actually play a major role in this erosion), the answer is very much in the ‘HOW we do Maths’.
There again, the Maths ability pyramid helps us to understand the situation. If you think of exams, i.e. Maths GCSE, then you are focusing on the second layer of the pyramid: skills. If, as a teacher or a student, you are focusing on the skills layer only, then it’s skills for skills’ sake; or skills for GCSE’s sake, which is pretty much the same in the end. And that, of course, can be pretty boring.
A quick look at Maths GCSE papers will only confirm this. There is nothing particularly creative about GCSE questions. That is not their purpose, anyway. GCSE questions are standardized and repetitive chunks designed to test a number of basic Maths competences. Period.
In other words, it’s not Maths exams that are wrong, it’s the way Maths exams are considered and used as a base for Maths teaching.
The same could be said of Maths levels in secondary schools. I am not saying levels are absurd and shouldn’t be used. There is ‘some truth’ in levels: on the whole, a level 5 student ‘performs better’ than a level 4 student. But to regard it as a measure of a student’s ability in Maths is wrong. It would be like looking at the Maths ability pyramid and saying that only the middle layer (skills) is real. What levels measure is not students’ ability in Maths, but their response to the way they have been taught Maths.
Does that mean that skills shouldn’t be practised ? Of course, they should. Does that mean that massed practice should be banned ? No, it shouldn’t, but there is now enough available evidence to manage practice in a much more efficient and subtle way (see Daniel Willingham’s works — particularly his book Why don’t students like school ? ). And more importantly, to subordinate skills practice to the more exciting dimensions of Maths.
There are many parameters that make a difference between one Maths lesson and another, between one Maths teacher and another. It’s a complex alchemy and nobody has the absolute right answers. But there is one parameter that is fundamentally important: time.
Ultimately, at the secondary school stage, it’s the way you allocate time to the various Maths activities that reflects your vision of Maths teaching.
Here’s the key message: the shape of the time pyramid should be very different from the Maths ability pyramid. Ideally, it should be pretty much upside down: spending more time on problem-solving, then skills, then explicitly fundamental abilities (I do say ‘explicitly’, because skills and problem-solving constantly draws on fundamental abilities anyway).
This inversion of the Maths pyramid puts exams in the right perspective. To suggest that Maths GCSE should be treated in an off-hand manner might seem a little exaggerated or iconoclastic, but at least it helps understand that Maths competences required for GCSE papers are only a ‘chemical residue’ of a much wider (and happier) Maths learning experience. In other words, the point of learning Maths is not to pass Maths exams, and to consider it this way boosts student performance at Maths exams considerably.
A few years ago, Sir Ken Robinson made a number of videos and written contributions trying to get his key message across: schools tend to kill creativity. At the time, this was rather badly received by the education establishment. I think his message was widely misunderstood.
Here, we have an example of what I think Sir Ken really meant. Creativity is not a lofty objective for idealists. Creativity doesn’t mean there shouldn’t be repetitive practice and exams. Opportunities for creativity exist in tiny little moments of classroom life when pupils are given an appropriate challenge. And THAT is precisely one of the many benefits of problem-solving in Maths. This is a particular kind of creativity, among many others. More about that later…