The vital building block of mathematics?

Elizabeth Truss, Education Minister, spoke last week at the North of England Education Conference about the teaching and learning of mathematics and the current Government's plans. She set what she had to say in the context of some laudable assertions about the importance of people being able to think mathematically and logically, and to speak 'the language of mathematics'. She topped and tailed her main points with references to this country's great mathematical heritage, particularly the contribution of Alan Turing.

And then her own logic fell apart – in order to state yet again the government's determined agenda to ensure that all primary school children are drilled in the algorithms of long multiplication and long division. Here's what she said ...

'... the growing importance of maths shows we need to do more to make sure children speak that language too.'

Who could disagree with that? But, look out, here comes the non sequitur. Next sentence ...

'That is why we are redesigning the primary maths curriculum to focus on mastery and fluency of the vital building block of mathematics, which is arithmetic.'

What? How does that follow from the previous sentence? It's clear from what she goes on to say that what she means by 'mastery and fluency of ... arithmetic' is drill and practice in long multiplication and long division. This is total nonsense! Who in the field of mathematics or mathematics education would ever identify being able to do calculations like 49 × 32 using the traditional long multiplication algorithm and 867 ÷ 17 by the long division algorithm as 'the vital building block of mathematics'? Or even a slightly important building block?  How on earth is spending so much of their time mastering and practising long division going to help children to 'speak the language of mathematics'?

Recently on the television programme Countdown, a highly gifted young man studying mathematics at Cambridge took the easiest option in the numbers game, because, as he said, 'I may be studying maths at Cambridge, but I haven't seen a number for two years!' From where do these politicians get the idea that developing mathematics is first and foremost about doing calculations? And that the key to success in mathematics is to be able to do harder and harder calculations? (Or, as Elizabeth Truss calls them, 'sums' – how embarrassing to have an Education minister who misuses the word 'sum' when outlining policy on the teaching of mathematics!)

Does Elizabeth Truss really think that being able to do long division was even the tiniest bit significant in Alan Turing's work? And don't tell me that learning this taught him how to think algorithmically. What he had to be able to do was to devise an algorithm to solve a category of problems, not learn by rote the steps of an algorithm. Teaching children to master by drill recipes for doing calculations is not going to develop their ability to think algorithmically, to solve problems, to break problems down into their component parts, to reason mathematically. Ironically, the algorithms developed by Turing and his successors have led to the redundancy of formal written calculation methods consequent on the availability of the calculator on the mobile phone in the pocket of just about every 11-year-old in the country!

The crunch in the speech is that the Government has devised an underhand way to ensure that primary schools teach their favoured written methods for calculations. In the end-of-Key-Stage-2 national tests for mathematics, children will be awarded method marks for a question involving a calculation if they get the answer wrong only if 'their working shows they were using the most efficient method'! 'The most efficient method' is government-speak for the traditional formal written methods.

What, I wonder will count as 'the most efficient method' for, say, a question that involves finding the product of 49 and 32? Surely, the most efficient way of doing this is to find 32 fifties (easy, because it's the same as 16 hundreds) and then subtract 32: giving 1600 – 32 = 1568. In this case, would a child who tries to use long multiplication and gets it wrong be ineligible for 'method marks', because they have not used 'the most efficient method'? Logic says they should, but then logic is not the strong point of the present government's approach to the primary curriculum.

Pointless algorithms

When I was studying O-level maths many decades ago, I was taught how to find the square root of a number. This was a complicated algorithm which I never understood at the time, and, frankly, I don't really understand now. But I was a willing student with a good memory and I learnt the process and even got a certain amount of satisfaction from practising it until I had mastered it. I was the exception: most of my schoolmates never got the hang of it. Many didn't even bother to try and the experience just added to their perception of maths as a subject that doesn't make much sense.

To my surprise, I find that I can still remember how to do it. You give me any number you like and I can work out its square root to any number of decimal places. I won't demonstrate this skill here – I never like to teach any mathematics without understanding. So, you'll just have to believe me.

When I left school I read mathematics at university and went on to have a career in mathematics education. But not once in all those years have I ever used this skill that I spent so many hours at school mastering. If ever I needed a square root, I used a set of square root tables, a slide rule or, mostly, a calculator. What a pointless waste of time it was learning that meaningless process. How insightful of my friends at school who decided not to bother to learn this mysterious routine!

Now, here's the point of mentioning this.

Is anyone able to convince me that the long division algorithm – which some prominent people (eg Carol and Michael) want to see back in the primary school maths curriculum – does not have exactly the same status today as the algorithm for 'extracting the square root' had when I was a lad? Do we really want to go back to making children spend hours at school trying to learn a procedure they won't understand and – in an age when a calculator is cheaper than a cup of coffee – will probably never need to use?