How to Teach Maths
...if maths exams didn't exist
04 February 2014
This week's Documentary of the Week on Radio 4 goes out on Wednesday at 9pm.  It is entitled, rather provocatively, 'How to Teach Maths'.  Thousands of maths teachers will be tempted to tune in, many of them perhaps a little jaundiced as they wait to find out what the latest fashionable 'theory' is in this long debated and highly political topic (about which just about every parent, teacher, teenager and other person I have not already mentioned has a point of view). Educationalists and Daily Mail journalists are possibly sharpening their knives already, because there are no right answers here.  It depends: on the teachers, the learners, and the objective.
The programme is presented by Alex Bellos, author of the excellent book 'Alex's Adventures in Numberland' and a good choice to tackle this hot potato, because he is viewing this from outside the education establishment.  I am one of the interviewees on the programme as co-author (with Prof Mike Askew) of the book 'Maths for Mums and Dads'.
A half hour programme is of course not nearly enough to cover all the nuances of how maths should be taught.  For a start, there's differentiating between MATHS and ARITHMETIC.  Arithmetic is just a small (albeit vital) part of maths, but when maths comes up in everyday discussion, it's usually arithmetic that people are talking about.  Most of what I chatted to Alex about centred on arithmetic, because parents are (rightly) bemused by the new terminology and new techniques that crop up in primary school. Partitioning, Number Lines and the Grid Method, anyone?  And then there is Chunking, the 'new' method of doing long division (which, like the other techniques, isn't new at all, people have been doing chunking for centuries, it just didn't have a name).
This blog is my chance to present a slightly broader view of How Maths should be Taught than I gave on the programme, where I had a narrow brief.
And actually the answer to How Should Maths be Taught? depends enormously on what you are learning maths FOR.  The main reason why children are taught maths at the moment isn't to train them up for skills they'll need in their adult life. Nor is it to introduce them to a beautiful subject that enables them to discover patterns and to develop skills in logic, deduction and problem solving. Those are by-products that many children emerge with (thanks goodness).  Because actually the MAIN reason why most children learn maths and why teachers teach maths is so that those children can pass maths exams.  I don't mean to sound cynical, but I think that's the way it is for the vast majority of children.  And perhaps it has to be this way, because if exams didn't exist, how would we know if children were learning the vital maths skills that society needs them to have as adults?
The problem is that passing exams is only a crude approximation to learning maths and developing the skills that this country needs.
So in a fantasy world where exams didn't exist, and where teachers wanted to share the passion in their subject and children/teenagers were hungry to learn, how would maths be taught?
Well, on top of a solid grounding in arithmetic, fractions and ratios (which I believe is VITAL), the way I would want maths to be taught would be through discovery and problem-solving.  And a lot of that problem-solving would be connected to everyday life, because the standard and completely understandable teenager lament about maths is: "When am I ever going to need this?" (quadratic equation formula...etc).
Here's just one example of the sort of problem-solving I have in mind:
Where is the best place to aim on a dartboard?
On the face of it, this is just an arithmetic question.  Of course you should aim at treble 20, as that is where the most points are.  But wait, if you miss the 20 segment, the neighbouring segments are 1 and 5, a trap into which many an over-confident darts player has fallen.  So where you should aim on the dartboard depends on how good you are - in other words, it depends on the probability that the dart goes where you want it to go.  If you're really good, aim at treble 20.  But if your darts are iffy, like mine, they will tend to drift into neighbouring segments, in which case treble 14 is probably a better place to aim, because its neighbours are 11 and 9, OK scores.  And if you are really hopeless at darts, you should aim somewhere else: Bullseye.  Why? Because that gives you the maximum chance of at least hitting the board.
This real life problem is one that many people can identify with and it seamlessly gets even non-mathematicians (whatever that means) thinking mathematically.  It involves statistics and geometry, two maths topics that are usually kept in separate silos.
The trouble with the dartboard question is that it's very hard to set an exam that fairly measures the sort of abilities that are being developed here.  And of course you can't swat up on the technique, because the next problem you're given won't be about darts at all.  So although there's now far more emphasis on problem-solving in school maths, the sort of problems you get in exams are inevitably a poor imitation of real-life maths problems.
Fortunately, despite the horrendous press they get, this country has many, many fine maths teachers who somehow manage to juggle the tick-box constraints of exam-passing with the joyous and much more important world of discovering real maths.  The maths that employers actually want in their future employees.  And the maths that teenagers find stimulating and actually enjoy.  I met some of these teachers in Poole and Bradford this week, but I assure you they are to be found all over the country. And those teachers are gold dust.
Now I've heard the programme, a couple of thoughts.
(1) Nice to have such a positive presentation of maths.
(2) The programme may have given the impression that there's an "either/or" with primary school maths:  EITHER we use these new methods like grid method with kids enjoying maths, OR we push the kids harder to learn their tables younger. Actually, I think we need both.  I do think that chunking/grid method are good approaches for learning arithmetic.  However, while they are good for building understanding, these methods are much slower and messier than traditional methods.  So when a child is confident with grid multiplication, they should be encouraged to move on to long multiplication (which is actually the same thing, but more concise).  Liz Truss talked about children learning their tables earlier.  I do think many children could be stretched more in arithmetic, and learning your tables up to 12s by the age of ten is comfortably within the grasp of a significant portion of primary children. And knowing your tables is an important and pretty fundamental skill.