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.

Calculator ban KS2 Maths Test

Elizabeth Truss, Minister for Education, announced this on 9 November: the use of calculators will not be permitted in the mathematics national test taken at the end of Key Stage 2.

The logic behind this is misguided. Problem-solving, not calculations, is at the heart of mathematics and our assessment of the mathematics curriculum should reward this.

Currently there are three papers in this mathematics assessment: a mental mathematics paper, a non-calculator paper, and a calculator paper. So there are plenty of opportunities to assess mental and written calculation methods in two of the three papers, which is where such questions are concentrated.

The calculator paper makes it possible for the test to assess a broader range of important mathematical processes. In applying mathematics to real-life problems, children learn to identify which calculation is required and how to interpret the result of a calculation and to check the constraints of the real-life context. These steps are as important in problem-solving as the calculation itself.

The minister in criticising the current set-up gave as an example of a calculation for which children currently might be allowed to use a calculator the following: £12.50 multiplied by 7. I really do not believe that 'find £12.50 multiplied by 7' has ever been a question in the calculator paper of the KS2 maths test. It might have been a calculation that would have arisen in the context of a real-life problem and the child would have been required to determine that this this was the calculation that was required.

When the child has done this on a calculator, the result displayed would be 87.5. This has to be interpreted as representing £87.50 in the context of the original problem, whatever it was.

A typical problem might be: Dev earns £12.50 a week doing a paper round and works for 7 weeks. At the end of the 7 weeks, how much more does he need to be able to buy a bike costing £95?

An eleven-year-old who can decide what calculations are required - and if they use a calculator can interpret the results displayed - has done some genuine mathematics! Certainly worth a couple of marks in the KS2 test.

This is a sad day for primary school mathematics and a worrying indication of what lies ahead in the proposed curriculum reform.