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Head cases

Mental arithmetic isn't about memorising answers, it's about thinking of methods of calculation. Jenny Houssart on why it's useful to know how others arrive at the same answer

ave you noticed that what we are supposed to be in the Nineties is flexible? You can find this word in job adverts, sandwiched between "enthusiastic" and "ability to run the football team an advantage". The word flexible is in the mathematics national curriculum, too, with flexible methods of calculation mentioned at both key stages 1 and 2. Consideration of what it might mean in this context was a major part of a recent Buckinghamshire County in-service training day.

When the subject of mental calculation was raised, it became clear that adults work things out in ways often contrary to the methods they were taught. "Mental arithmetic" is not about knowing the times-tables, as is often popularly supposed, but about figuring out calculations in your head.

The example 26 + 27 inspired people on the course to use a surprising number of different methods. Doing the calculation was followed by the important but sometimes difficult step of explaining how they did it. The method "add one to the number of cards in a pack" was greeted with admiration and surprise.

The national curriculum encourages pupils not only to develop and try to explain their own methods but also to record in ways which relate to their mental work.

When children are encouraged to reflect, share and discuss the way they do sums in their heads, they will not only be surprised that other children do things differently, but they will go on to think about new ways to solve problems themselves.

Mental calculation does lend itself to a diversity of methods but it is perhaps harder to think of flexible methods when it comes to many standard text or workbooks. Therefore, one of the INSET sessions started with the question "What can you do with a page of sums?" One suggestion was a consideration of the offending page before embarking on it, asking such questions as "Which ones can I do in my head?" and "What will I use for the others?" Another, more unusual, idea is to ask the question, "which is your favourite sum?" Classroom activities can be based around one calculation. For example, children can be invited to alter the question, which sounds delightfully subversive. This can be done by swapping the digits, perhaps with the aim of making a larger or smaller answer. Another approach to a page of sums, of course, is to tear it up (to separate the calculations, not to prepare it for the bin.) The "sums" can now be sorted into sets. For instance, children can sort them into two sets by means of a question and a simple yesno tree diagram.

I now know what to do with the box of surplus SATs under my desk. I can tear them up, sort them out, alter the questions and pick my favourites.


If you don't have a box of surplus SATs, you could try some of the following on any page of calculations whether found lurking in an old book or written for the purpose. For some of the activities the calculations can be written on small pieces of card, although children can always write their own.

Choosing methods

Ask the children which of the "sums" they can do in their heads. They may wish to ring the ones they have chosen or put them in a pile. Encourage them to recognise cases where working in their heads can be a much easier option than using standard written methods, see left.

Explaining methods

Pick a suitable calculation and ask the children to work out the answer in their heads. Get them to try and explain how they did it. Collect as many methods as possible and encourage the children to listen to the way others have solved it. Some may have methods which help with particular types of calculation, such as adding 1 or 10 or 9 or multiplying by 5 or 11.

Picking favourites

Ask everyone to pick their favourite from a page of sums. Initially this may cause laughter and perhaps debate about what you mean. They may pick the easy ones, those with a pattern to them or those which lend themselves to quick methods.

You can ask them why some sums are easier than others. Often this has to do with recognising patterns. You can then suggest looking for patterns in other examples.

Favourites can also include those which look hard but turn out to be a little easier. I remember a child who took great delight in working out 17.5 per cent of anything using a method she was very proud of (find a tenth, find half of that, find half again, add the three numbers together).

Sorting sums

Start with a question and arrows and ask the children to sort the "sums" accordingly.

Children can move on to selecting their own question. They will soon realise that you don't have to work out every calculation and the activity involves a lot of approximation. A final step can be to look at someone else's sorted "sums" and guess how they were done.

Altering the question

You could start by setting out the chosen question using number cards. Children can draw round the cards, then alter the calculation by moving cards into different positions.Example: Can they make a sum with A higher answer than the original?

A lower answer?

An odd answer?

An even answer?

An answer ending in zero?

Jenny Houssart is lecturer in education at Nene College, Northampton

3x2 5+7 2+1 11x3 100 1.25 -99 x4

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