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.

Fear of mathematics 1

On Wednesday this week, I spoke to an enthusiastic group of primary school mathematics leaders on an Every Child Counts conference at Edge Hill University in Ormskirk. I had been invited to speak about 'Fear of mathematics'.

One of the main points I made was about the relationship between mathematics anxiety and a rote-learning mind set. There is research evidence (see a study by Karen Newstead, 1995, 1998) that children aged 9 to 11 years are more likely to develop mathematics anxiety if they are taught by traditional methods that rely on memorizing and drill of calculation processes without a focus on understanding.

It seems that teaching that promotes reliance on rote-learning in mathematics generates more anxiety about the subject. The learner gets the idea that there is only one right way of doing any mathematics question, worries about which process to use for which problem, gets different processes confused and cannot cope with anything that is not immediately recognisable as requiring a standard, routine response.

But this is a viscious circle, because mathematics anxiety seems to reinforce reliance on rote-learning. The anxious learner is more likely to have given up expecting to make sense of mathematics and seeks only to try to remember what you have to do for certain kinds of questions. Anxiety inhibits learning with understanding.

The key seems to be to teach mathematics well! In other words, to teach in a way that promotes a meaningful learning mind set, that encourages children to make sense of mathematics, by making connections between language, symbols, practical and real-life situations and pictures. I have written about this extensively in Chapter 3 of Mathematics Explained for Primary Teachers, 4th edition: Learning How to Learn Mathematics.