John Bald: The maths we all need is based on arithmetic.

John Bald is a former Ofsted inspector. He is Vice-President of the Conservative Education Society.

Some years ago, a parent asked me to help a child with the maths SAT. I hesitated, having had trouble with some aspects of maths at school, particularly geometry. Fortunately, an enlightened teacher, the late Bill Broderick, introduced statistics. As these are based on arithmetic, I found them straightforward, managed a respectable O level pass, and benefited later when I had to analyse educational research. The basis of maths in the real world, from government expenditure to the weekly shop, is arithmetic, and Elizabeth Truss and Nick Gibb did well to give it more prominence in primary schools.

As with literacy, progressives avoid points of difficulty rather than tackle them. The crunch is multiplication tables, where they have evolved elaborate and tedious systems to avoid children having to learn them. Again, as with literacy, they make the mistake of starting from the end product and working back, rather than finding the best ways for children to build from their own, varied starting points. To the progressives, notably Professor Jo Boaler, the end product is higher maths, chiefly based on algebra, rather than the maths we all need, which is based on arithmetic.

As with literacy, the result of progressive approaches to maths has been failure and misery. Towards the end, Labour’s maths strategy was producing materials to teach the 2x table to 10- and 11-year-olds, who had grown accustomed to counting in multiples and using their fingers to try to work out where they were. This issue is still with us, and is based on the idea, for which I can find no evidence, that counting in multiples helps with tables, when in practice it makes learning them harder, as the issue is co-ordinating one column that goes up in single digits, while the other goes up in multiples.

The twelve-year-old I wrote about last time assured me a fortnight ago that three twos were eight. He has now learned the 2x and 3x tables, and finding 4x easier. Last week I started work with a 10-year-old, assessed as autistic, who has fewer difficulties with reading, but did not know what tables were.

I started with both by showing how tables enable us to calculate more quickly than we can by counting. We then took the first five items of the 2x table, and practised them until they were known thoroughly and without calculating the answers. This puts less strain on the memory than attempting the whole table at a time, and was all we did on maths in the first session. I then explained that, having reached a zero, the second digit started again with 2. The second five items were learned more quickly, and the third became an easy letdown. It is, incidentally, including 11 and 12x in each table, as annual contracts are usually expressed in terms of monthly charges.

Ensuring that the 2x table was learned thoroughly, before proceeding further, has been the basis of my successful work with tables since that first request. It was published in The School Run, and is still available on this link, free. It is consistent with the brain research of Professor Stanislas Dehaene, head of neuroimaging in France, and described, to his warm approval. In these days of fake science, it is worth noting that I was recommended to him by a Fellow of the Royal Society.

The approach builds the child’s understanding of number from a base that they already know – I’ve not yet had a junior school child who could not count to 20, but if I did, I’d teach that first. One of my pupils was involved in Labour’s belated attempt to tackle problems in literacy and maths by individual tuition. Unfortunately, they issued a prescriptive book that prevented tutors from thinking for themselves, and my pupil, who knew no tables at all, was made to start with 7x, one of the most difficult. They would have been as well trying to build a house by putting on the roof.

The Conservative government made a similar error with its tables check for nine-year-olds. Instead of giving children credit for what they knew, and thus encouraging them to learn more, it used an all-or-nothing approach, which rewarded the most competent, but gave the others no recognition at all. The thinking was that children needed to know the tables without working them out, which is true, but does not take account of stages of learning along the way. It would have been better to have a bronze award for, say, 2,3 4 and 5x, silver for 10, 11 and 12 (which relatively easy), and gold for a full set. As most of the failure takes place with 2x, including this in the award system is crucial if all are to succeed. My usual sequence for learning tables starts with 2,3,4,5, in which the size of the product grows gradually rather than in large jumps. Then 10,11,12, 9,8, which allow application of counting and 2x table. Finally 7 and 6, which for some reason I find trickier than 7.

Civil service procedures are a factor in most errors in maths as well as literacy, but this time they are not due to bias. Ministers rely on their advice, and it is based, not on full evaluation of evidence, but on synthesising the views of organisations and selected individual experts. Senior civil servants are expected to “know” their field, but they cannot know it in the same depth as specialists. The approach is, up to a point, a safeguard against maverick views, but organisations are no less prone to error than individuals, and are themselves subject to domination by cliques of like-minded people. Special advisors had some impact on the dominance of the civil service, but the case of Dominic Cummings shows that they bring problems of their own. In the end, we must rely on Ministers to make the best decisions they can on the basis of the evidence, and both they, and we, have to live with the mistakes.

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