Tuesday, 6 September 2016

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.


  1. How do you know he isn't superbly good at mathematics now because his written methods were so good on entering secondary school that he didn't have to waste weeks relearning the skills before every new topic? I'm not saying that the written methods are the cause of his greatness really, just puzzled by your leaps of logic.

  2. I was making the point that in doing A-level mathematics you do not need formal written calculation skills. So there is no justification for the argument that these are fundamental skills in mathematics. I heard a Cambridge maths graduate comment recently that in 3 years reading maths at Cambridge he had never once done a calculation. You might want to argue that written calculation skills are useful in other real-life contexts than doing proper mathematics, but in a technological age such as ours you will lose the argument! The only context in which they are actually useful is in doing the Key Stage 2 National test for mathematics, thanks to our misguided National Curriculum. How sad.

    1. Wow. I had never considered this. I'm using your book mathematics explained for primary teachers in my PGDE primary maths degree. What a help it has been. Thanks!

  3. I agree with Mrs brown. Fluency with number comes from being comfortable and able to calculate without aids them you move on.