Calculations: foundations of mathematics?

The myth that the mastery of the processes of written calculations, particularly long multiplication and division, is fundamental to doing mathematics continues to be perpetrated by people with political influence and control of our school curriculum. This myth was exposed for me recently in the experience of helping one of my grandsons prepare for his A-level mathematics examination.

We worked together through loads of questions from past examination papers – pure maths, mechanics and statistics. I made the following observations.

Not once in doing A-level mathematics was he required to do a written calculation, since he always had a calculator to hand. His calculator skills were stunning and showed real mathematical understanding, in terms of processing the steps of a complex calculation in the appropriate order, in selecting the correct function keys and handling brackets, and doing this with speed and accuracy.

More important than written calculation skills were the ability to interpret the calculator answer and checking whether it looked reasonable. Additionally, with a little encouragement from me, he improved markedly in using mental strategies for calculations that could be done more efficiently that way than by resorting to the calculator.

But, I repeat, not once did he use a formal written calculation procedure. Yet, there he was doing advanced level mathematics! If he had had to take his eye off the structure of the problem to do a written calculation it is very likely that he would have lost his grasp on where he was going.

For centuries mathematicians have devised ways of avoiding or reducing the demand of written calculations, simply because they get in the way of the real mathematics and effective problem-solving, and take up too much of your precious time. So, we had Napier's bones, and logarithm tables and slide rules, and so on. Now we have modern technology, so please let's give younger children the chance to use it and start doing real mathematics.

Efficiency of a calculation method

The proposed new curriculum for primary mathematics stresses the importance of children learning what they refer to as the 'efficient' method of doing a particular kind of calculation. That's an intriguing concept. The 'efficiency' of a device is defined as the work achieved divided by the effort put in to achieve it.

On this basis, doing a division like 784 ÷ 18 on a calculator must surely be the most 'efficient' method: very little effort put in and the best chance of the output being the correct answer = high efficiency. For some reason, they seem convinced, though, that long division must be the most efficient method of doing a calculation like this: which involves a great deal of effort put in (so lower efficiency), and a significant chance of getting the wrong answer (which would be zero efficiency!)

OK, so they don't want children to do calculations like this on a calculator; I have no idea why, but they don't. But why would long division be more efficient than, for example, this approach:

10 × 18 is 180, so 20 × 18 is 360, so 40 × 18 is 720 (so far only multiplying by 10 and doubling, which is easy; not much effort expended!)

I now need to get from 720 to 784.

Adding 18 at a time, I get 720, 738, 756, 784.  I needed a further three 18s. The answer is 43.

I recognise real mathematical and creative thinking when 10–11-year-olds share different ways of doing multiplications and divisions, How sad if the new curriculum suppresses this and sends out the incorrect message that there is only one proper way of doing a multiplication or a division.

So, here's a challenge for any reader. Can you come up with 12 ways of calculating 75 × 12?

I'll give you the two most efficient to get you started:

1) You just happen to know your 75 times table because you are an avid watcher of Countdown, so you just write down the answer, 900.

2) Use a calculator.

OK, now find 10 more ways of doing it ...

Bald criticism

In a posting on http://conservativehome.blogs.com/localgovernment/2012/02/teach-tables-and-long-division.html John Bald complains that in my best-selling book Mathematics Explained for Primary Teachers I do not describe or subscribe to long division, and that I do not explain how to teach multiplication tables.

His criticism is based on a quotation from an old edition of Mathematics Explained for Primary Teachers. In the current (4th) edition, published 2010, I do actually outline the steps involved in long division – although I continue to encourage the use of other methods that can be taught with understanding rather than learnt by rote. The fact that John Bald has managed to drill a dyslexic 12-year-old into reproducing the long division algorithm does not undermine my position. When I said that I had been unsuccessful in teaching the method, I was using the word 'teach' in a sense that does not just mean 'instruct' and that is in relation to classes of children in schools, not individual drill-and-practice tuition.

The first obvious question is whether Mr Bald's pupil will still be able to carry out this procedure in a year's time, without spending more valuable learning time continuing to rehearse it with further practice examples at frequent intervals.

The second question is whether the experience will have helped this young man to learn how to learn mathematics in a meaningful way? It is more likely that it will have reinforced a rote-learning mind set in the learner.

The third question is whether it was worth all the effort! I would be pleased about the teacher's success here if he had given the young man something that would be really useful for him. But in fact I feel sorry that this young man has had to spend so much of his time mastering something that is of such little value. Perhaps he can now move on to learning how to extract the square root of a number? Then on to how to calculate the cost in pounds, shillings and pence of quantities measured in hundredweights and stones?And then how to hunt sabre-tooth tigers?

What is it about long division that gets some people, particularly non-mathematicians like Bald, so heated? Do they not want children to have every opportunity to learn with understanding? Do they really think that learning to reproduce this one particular algorithm is the pinnacle of achievement in primary school mathematics? Do they really want children to spend so much of their time in primary schools mastering a technique that is not required in any of the questions in the end-of-Key-Stage 2 National Tests (SATs) for mathematics? Is there not enough mathematical material much more interesting and helpful and meaningful to teach anyway? There are 183 pages in Mathematics Explained for Primary Teachers on understanding number and calculations!

John Bald does concede that I advocate children being taught to memorise the multiplication tables, although again I stress that they do this with an emphasis on understanding the relationships involve – and I give examples of how this can be done. But Mathematics Explained for Primary Teachers is not a book that sets out principally to tell people how to teach. Although it contains numerous teaching and learning points, the main focus is on helping primary teachers themselves to understand the mathematical concepts and principles that underpin what they teach. The huge sales of the book and the continued positive feedback from teacher-trainees suggest that they find this approach really helps them to feel more confident in their teaching of a subject about which many of them had previously felt insecure. They tell me that to their surprise they discover that mathematics is a subject that can be understood, and that it is is not just about memorising meaningless rules and recipes for doing various kinds of questions.

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?