Recently in the news has been the government’s grant to MEI to develop material for Sixth Form (http://www.bbc.co.uk/news/education-20153731) based on suggestions made by Prof Tim Gowers (http://www.spectator.co.uk/features/8744071/should-alice-marry-bob/). What does not seem to be being reported clearly is that Gowers’ programme (as it says in yesterday’s Sunday Times) are apparently for a new qualification; an alternative to AS to perhaps correspond to Maths Studies in the IB diploma. Gowers’ proposals seem to be for those who have dropped Maths due to disenfranchisement, maybe at the subject’s lack of applicability, but the many news reports are not clear about the impact of this on whatever follows the current version of AS Maths.
“Sir-what use is this in real life?”
If I’m honest, there are few, if any, questions that can come from a pupil that make me shudder more than this one. I am always tempted to answer “None- this is all a colossal international conspiracy constructed purely to specifically waste your time. Actually none of these symbols make the least bit of sense and honestly, I’m making it all up as I go.”
The question, however, is always a valid one and I suspect my impatience for its rare appearance stems from the possibility that it might be my fault- maybe I’ve not already made the application of the Maths already clear or I’ve not gained enough of their trust that I was ultimately heading there.
There is a well established negative perception of Maths in the media as the theoretical wafflings of elbow-patched intellectuals. Journalists, educationalists and politicians can be seen talking of the need to move to the fore the skills pupils need in Today’s Society, in place of allowing Maths teachers to self-indulgently prattle on about theory, proof & derivation.
Previous attempts to address the abstract nature of Maths this have resulted in frivolous practical questions, making exams look utterly ridiculous (Seth is starting a free range farm. The number of ducks on his estate is given by the equation y=x2-4x+6…). This media perspective always greatly annoys me- first of all to assume we can predict the Maths needed for the current generation’s future is optimistic when you look back over the past 30 years, but also it implies we should be teaching only algorithms & tricks. Should we be getting pupils to relentlessly memorise and practise the formula “A=P(1+R/100)n” that many textbooks put in place for finding the amount in a Bank account under compound interest?
Instead if a pupil is well taught how percentage interest works, they can calculate the same total without ever needing any formula, to say nothing of being able to transfer this understanding to similar but non-identical problems, financial or otherwise.
The issue being raised this time is subtly different. One reported implication is that pupils are dropping Maths because of this perceived break with reality and that we can court pupils back if we show them the subject’s transferability by way of the questions we ask them. I suspect, however, that Prof Gower’s original point is that if we are to keep pupils in the study of Maths beyond GCSE, then we need to find ways of developing their open problem solving skills if the established mathematical approaches have not already worked. The open problems posed for consideration he suggests are lovely ones but I wonder which pupils would welcome this style of question under exam conditions? When I showed this material to my bottom set Fifth form groups, they were interested in the problems, but worried about how they’d be assessed.
When I ask prospective pupils who wish to join us in the Sixth form (as well as their contemporaries at King’s), the question “Why do you like Maths?”, by far the most common answer is a variation of “because I enjoy getting it right”.
It is important to note that the techniques Prof Gowers advocates are terrific and valuable and should definitely appear in our teaching. Indeed last week one member of the Maths dept was sharing how they had introduced Hypothesis Testing by having pupils suggest an optimal algorithm for testing practical data against a theoretical model. In doing this the pupils come up with a query over a proposed alternative which neither he, nor I could immediately answer- scary but exciting for all concerned. Transferring this style of learning into exams, however, is risky and the sort of pupil who likes maths but is not excited by any theoretical element could well be freaked out by the lack of structure offered by these problems.
We need to avoid putting the cart before the horse- open problem solving is a crucial skill which we need to develop but should be trying to do so in the lessons. It should be present in an exam, but not make it up entirely. Exams are where pupils show both how good they are at problem solving and also, crucially, which of the key techniques needed for later in their careers (frequently university) they have retained for future use.
The key issue is the abstract nature of Maths. Always finding a practical angle removes some of the joy of playing with questions where uses and solutions are not immediately apparent and undermines the trust that for some topics, these will be revealed ultimately by the Maths teacher, in whom pupils need to have faith. One pupil (in the same U6 Maths class) was asking this week about wider uses of Calculus other than rates of change. Knowing that the reveal of how we get the volume of a sphere to be 4/3 pi r3 is just a few weeks away I replied that he’d have to be patient- an answer he seemed content with.
How could we introduce complex numbers if we had to lead with applications to “real life” when they don’t even exist in the “real” world? [Alert- Maths heavy sentence imminent] All pupils who learn this inherently weird but most beautiful of topics have to trust that it has uses somewhere along the line, but it is only having studied them up FP2 level that you can show how they can be used to solve second order Differential equations whose Auxiliary equations have imaginary roots. Moments like this (solving a “real life” equation but using formulae that apply to numbers that don’t even physically exist), where seemingly disparate corners of Maths suddenly click into each other, make for the most exciting lessons and are not possible if we’re always putting the practical uses of all new Maths front and centre.
Ultimately if all exam questions are wide, open & problem based, we run the risk of isolating and terrifying those who struggle with that familiar feeling in Maths where you haven’t made it to an answer and are not sure what to do next. There was an amusing, if frustrating, article in the Times last Friday which quoted a study by the University of Chicago that had found “evidence” that equated this fear of Maths to that of physical torture (although, ironically, it quoted a sample size of 14 people who had identified themselves as haters of Maths- i.e. rubbish Maths used to rubbish Maths).
Anyone who works with Maths at any level (including teachers) knows this feeling well- the dropping of the stomach and discomfort/irritation/panic at realising that you can’t immediately see the full picture and don’t know exactly what to do next. This is also often why the “What is this for?” question appears in classes- in a frustrated bid to defeat the question morally if not intellectually. The chance to help pupils master this sensation is fundamental to the teaching of Maths, whilst also underlining the subject’s importance.
Forget finding the formulae for success in modern life- getting pupils to understand that having and mastering this feeling is not only normal but necessary in the learning of Maths is what we need to prioritise.
When presented with a problem outside of the classroom, there are rarely a set of memorised steps for any of us to safely follow when we walk in the “Real World”.