John Bald is a former Ofsted inspector. He is Vice-President of the Conservative Education Society.
“We don’t teach arithmetic. We teach mathematics,” said the retired headmaster. “Mathematics is problem solving.” Fine. There is more to life than page after page of sums. Still, as Stanislas Dehaene FRS showed in The Number Sense, mathematics grew historically, as it grows in young children, from counting, and multiplication and division are key steps in the process. Both depend on tables. As Dehaene says, these are an artificial concept, but so is the representation of words by letters, and indeed the rest of mathematics. An example is a negative number – obvious to mathematicians, but a huge adjustment to thinking for a young child. How can something be less than nothing? And how, when multiplication usually produces a greater number, can multiplying something by a fraction make it smaller? A person I met on a training day had their understanding of this in Scotland reinforced by daily beltings, until they gave in and agreed with the teacher.
The last Labour government made such a mess of teaching multiplication that by the end it was producing materials to teach the 2x table to 11-year-olds. It spoiled its provision of individual tuition by hobbling teachers with “guidance” that prevented them from considering the real needs of the child they were working with. A pupil of mine, who did not know the 2x table, was put straight to 7x, no doubt because it is was seen as the hardest. I started with 2x, beginning with the first half of the table. It worked, the approach is here, and anyone is welcome to use it.
The last Conservative government gave tables a specific place and time scale in the National Curriculum. This was a big step forward, with a check for pupils in Year 4, which gave two years to rectify problems before secondary school. Like the phonics check for infants, it has been retained by Labour, and the latest results show 37 per cent of pupils achieving full marks. The check itself, however, has serious weaknesses. It credits only full scores, and so does not distinguish between children who were nearly there, and those who had barely started. When I raised this with ministers, the DfE’s response was that they “had to know all of them.” They do, but this does not tell us how to get them there. It would be hard to find anything more discouraging to a child who had worked hard, but had not quite mastered a full set, to be given no more recognition than those who had learned none at all. Bronze, silver, gold awards would work better, and there is nothing to stop schools from designing their own.
The statistics are confusing. The figure for pupils “completing” the check includes those who did not take it, on the grounds that their schools decided that they could not answer the simplest questions. Many FE students forced to resit GCSE maths are in the same position, usually because they have been taught to count in multiples, a major error in the National Curriculum, which leads to children trying to keep their place with their fingers, and guessing when this does not work. The overall mean score was 22 out of 25, which doesn’t sound too bad until we see that this included the full scores. Take these out, and the average falls to a little over 15 out of 25. An error rate of 40% is far too high for tables to be relied on for calculation, and there is no follow-up at 11 to see how far problems have been tackled.
Tackling this involves using evidence on the early development of the brain. Dehaene argues convincingly that learning involves adjusting our thinking to take account of new material that does not fit our existing understanding. So how do we move from counting to tables?
The example above is a good illustration, and here is another. Two pupils, respectively aged 10 and 13, began with no knowledge of tables at all, the younger – disapplied from the test despite “completing” it, assuring me that 3×2 = 8, and then guessing wildly when informed that this did not work. The elder just guessed. So we began with the first half of the 2x table, to establish the idea that one column moved one unit at a time, and the other rose in multiples. In extreme cases, I’ve initially stopped at 3×2, but this time we managed the first five items, ending in 0, after which the sequence in the second digit started again.
Both pupils needed practice to establish the idea, but we did not move on until it was clear. We continued to reinforce this table by jokey practice – thank you, Michael Rosen, for describing my explanations as “jokey”, which sums them up – while proceeding through the sequence in the link above. ( 2,3,4,5. Then 10, 11, 12. Then 9, 8,7,6.) Each step builds on the understanding established in the previous one, and we do not jump from 2, to 5, to 10, as most schemes do. Instead, we build understanding of number gradually, explaining how we are adjusting our thinking at each stage, and building confidence, which comes from knowing that we are going to get something right, as we know it is based on something we have got right before.
HM Treasury, whose work depends on numbers, has removed tests of literacy and numeracy from its selection process, to promote “diversity”. How far, I wonder, is “diversity” compatible with arithmetic, and what would Katherine Johnson, the black “human computer” honoured by NASA for her work on Apollo 11 have made of this? Some of us thought the woke nonsense was confined to the Home Office. It is not. Johnson showed that intellect has no race or gender. The Treasury believes – I won’t say “thinks” – otherwise.