Education: Don't panic] Sums really can be fun: Colin Hughes reports on methods that may help children and teachers to relax over their arithmetic

DO THIS sum in your head: 7 x 8. Now pause for a moment before reading on, and ask yourself how you answered that question.

Maybe you think your way of arriving at the answer is the obvious, indeed, perhaps, the only way? Wrong.

This is how six able pupils, aged 14 or 15, set about answering the question:

'I just knew it.'

'Well, I know 6 eights are 48, so I just added on 8.'

'I did my eights (on her fingers) - 8, 16, 24, 32, 40, 48, 56.'

'8 eights are 64, so it must be 56.'

'2 sevens are 14; 2 fourteens are 28; 2 twenty-eights are 56.'

'10 sevens are 70, take 14, so it's 56.'

Mental arithmetic is one of the most valuable mathematical tools in anyone's working life. Any decent maths teachers knows that; and the national curriculum emphasises its importance.

But mental arithmetic has been neglected in many schools until recently. According to Mental Methods in Mathematics: A First Resort, a paper recently published by the Mathematical Society, 'speed tests' were common up until the early Sixties. Teachers would fire questions at pupils, recording oral answers against the clock, and then award marks out of 50, or 100. School reports would often award scores to pupils in 'mental' and 'mechanical' arithmetic, almost as if they were separate subjects.

Teachers turned against that kind of instruction. In part, they dropped it because they felt it alarmed many children, and reinforced feelings of panic about numbers. Also, whole-class mental arithmetic lessons proved difficult in primary classes with a wide range of ability. And teachers were switching, anyway, from whole-class methods, to individualised maths schemes.

Unfortunately, however, ways of developing mental arithmetic skills were not often built in to the new teaching methods. It is still not clear to many teachers - particularly in primary schools - how mental arithmetic can be improved. And that could be very disappointing for pupils, many of whom enjoy mental arithmetic, almost as a 'game' in itself.

The Mathematical Society's aim is to encourage mental calculation as a 'first resort' method. Children, it suggests, should be given time to think, in an uncompetitive atmosphere, with a high value placed on their own ideas, encouraging alternative methods, independent thinking, and a clear understanding of the methods they use.

The starting point is to understand how mental arithmetic works. Some basic knowledge is obviously essential. The second child calculating 7 x 8 above simply knew that 6 x 8 was 48, without calculating: the only calculation was the simple addition of another 8 to arrive at the answer.

Then, as the Mathematical Society's paper argues, a great deal of flexibility helps. Quick ways of solving problems will vary considerably, depending on the problem - even when the problems look quite similar.

For example, this is one way you could calculate 7 x 15 in your head:

7 x 30 is 210 (7 x 3 with a 0 on the end).

Half of 210 is 105.

7 x 15 equals 105.

But the apparently similar sum 7 x 19 could be approached very differently:

7 x 20 equals 140.

140 minus 7 equals 133.

7 x 15 equals 133.

The team of maths teachers who put together Mental Methods point out that the development of mental arithmetic should not be undermined by the increasing use of calculators: on the contrary, they say, there are all kinds of ways in which opportunities arise for using mental methods together with calculators. For example:

Supposing you are asked to find two consecutive numbers which multiply to give 1,406, using a calculator. One obvious way is to use a simple 'trial and improvement' method, mixing mental arithmetic and the calculator.

So, 30 x 30 equals 900, and 40 x 40 equals 1,600 - sums that most people can carry out mentally quite easily. Since the answer lies between 900 and 1,600, the two consecutive numbers lie between 30 and 40.

You could now use the calculator quite efficiently to try the options between 30 and 40. You could even speed up that process by noticing that 1,406 ends in a six: the only consecutive numbers that will multiply to produce a 6 at the end are 2 and 3, or 7 and 8. So the only two pairs worth trying on your calculator are, in fact, 32 x 33, and 37 x 38 (which is the correct answer).

The Mathematical Society's booklet is aimed at teachers. But much of its advice could just as usefully be offered to parents who want to help their children with homework. One particular recommendation is to 'speak out' the mental method you are using. It helps children to increase their repertoire of techniques, by seeing for themselves the kind of techniques they are using.

In fact, as the booklet suggests, teachers can usefully try a little 'speaking out' of mental methods themselves. This, for example, is how six different teachers calculated 60 per cent of pounds 40:

'10 per cent of pounds 40 is pounds 4, so 6 x 4 equals pounds 24.'

'It's got to be something to do with 6 x 4 equals 24, and 60 per cent is more than half - so, pounds 24.'

'60/100 x 40/1 equals pounds 24.'

'0.6 x 40, which equals pounds 24 - there must be two figures before the decimal point.'

'60 per cent is 3/5. 1/5 of pounds 40 is pounds 8, so 3 x 8 equals pounds 24.'

'50 per cent is pounds 20 and 10 per cent is pounds 4. That makes pounds 24.'

As the Mathematical Society team points out: if six teachers can come up with six different methods, teachers can clearly learn alternative methods from their pupils, and their pupils learn alternative methods from each other.

'Mental Methods in Mathematics: A First Resort' is available from The Mathematical Association, 259 London Road, Leicester, LE2 3BE. For more information telephone 0533 703877.