Learning the facts of multiplication

Some of the press who are particularly supportive of Michael Gove's reactionary, dreary and unrealistic curriculum proposals deliberately misrepresent those of us who oppose them. We are accused of not wanting to teach children facts. For example, it seems that we don't think it's necessary for children to learn their multiplication tables by heart, and so on.

Let me put the record straight.

Of course, we want children to memorise their multiplication tables. And the more multiplication results they memorise the better. If this is helped by chanting the tables in the traditional way, then that's fine by me. This is precisely what I say in my books! But there is a distinction to be made between learning these results with understanding and learning them by rote. Rote learning usually refers to learning which is not connected and therefore not meaningful to the learner. Results such as multiplication tables can be memorised by rote, but usually the  learning is less secure and it is always less useful and less transferable. In fact, it is well established that using patterns and connections reinforces memorisation.

So, for example, I would want pupils to use the fact that 3 × 4 = 12 to get 3 × 8 = 24 by doubling, and then to get from this to 6 × 8 = 48 by doubling again; and to see the connection between 3 × 7 = 21 and 6 × 7 = 42, and then by adding another 7 to get 7 × 7 = 49.

Then they can explore the simple patterns in the 5-times, 10-times, and 9-times table. They can find which numbers don't turn up as results in any of the tables (apart from their own). They can learn about commutativity and exploit it. They can learn how to use what they know to work out what they don't know.

Then the children need to connect these results with everyday situations, with rectangular arrays, with areas of rectangles, with the informal and formal language of multiplication, with steps along a number line, with corresponding divisions results, and so on. They need to explore questions like: why do many medicines come in packets of 28 tablets?

Yes, memorise the results, but also explore, exploit and enjoy all the connections. In this way the tables begin to make sense. The 'facts' can be learnt with understanding and pleasure.

For calculations with 2- and 3-digit numbers (and larger) we need know no more than the tables up to 10 × 10. But, bizarrely, presumably because they learnt them at school when there were 12 pence in a shilling and measurements of length were done in feet and inches, Mr Gove and his crew are prescribing that tables must be learnt up to 12 × 12.

There's no harm in children learning by heart the 11- and 12-times tables, but it is impossible to justify picking out these tables rather than any others and making it mandatory that children should memorise them. It would be easier to justify learning the 28-times table, so that those who go into healthcare will be good at handling tablets that come in packets of 28. I like to get Year 6 pupils to explore the 37-times table and to discover the delightful pattern in there: 3 × 37 = 111, 6 × 37 = 222, 9 × 37 = 333, and so on.

So, let me repeat! Give them the facts, yes. But give them understanding and enjoyment and purpose in learning as well.

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.