# We can work it out;Mathematics

Teachers, parents, politicians, governors - and the media - must realise that the "mental arithmetic" that underpins the National Numeracy Project is a different animal to what we all endured in school a few decades ago. In fact, so great are the negative associations with "mental arithmetic" in most people's minds that it might be better to use the more anodyne (and more accurate) terminology of "mental calculation".

The phrase comprises at least two aspects - mental recall and mental strategies. The Dutch use the phrases "working in the head" and "working with the head". The former covers knowing by heart, or being able to work out very quickly, specific number bonds or tables facts. The latter is about using known facts, to work out unknown ones - such as the sum of a pair of two-digit numbers.

Now, it would be nice if we all (including school standards minister Steven Byers, who broadcast that seven 8s are 54 to radio listeners earlier this year) "knew by heart" all our number bonds and tables. But, realistically, this is unachievable. More important is that pupils remember as many facts as they can, and are taught in such a way that they develop the confidence to use them creatively.

The phrase "mental strategies" describes the use of known or easily calculated number facts combined with a range of implicit (or explicit) understandings of the workings of the number system to effect a calculation. Any list of such "workings" should include:

* commutativity (3x4 = 4x3) * associativity (3+4)+5 = 3+(4+5) * distributivity (3x24 = 3x20 + 3x4) * inverse operations (multiplication "undoes" division) * the effect of adding, subtracting or multiplying by 10 * the ease with which numbers from 20 to 100 can be partitioned according to how they are spoken (34 is thirty and four) and how each part can be dealt with separately before the partitioned parts are recombined * the way in which multiplication by 12 is equivalent to multiplication by 6 and then by 2 (or even by 3, 2, and 2).

Implicit in the concept of mental strategies is the idea that, given a specific collection of numbers, you select the most appropriate method for them.

It is also important that mental calculation is not seen as an "add on" to "real" maths - the completion of pages of written calculations with the numbers set out neatly underneath each other. The National Numeracy Project recommends a three-part lesson, but by describing the 10-minute introduction in terms of "mentaloral work" and by separating this from the 3040-minute "main teaching activity", the project may reinforce this misunderstanding.

Most of the mental calculation strategies described in the framework will need to be taught, and the second part of the lesson will be where this teaching should take place.

Mental work can, and sometimes should, involve some written work, completed either while the mental calculation is taking place or as a summative record of the strategy. But if we wish to improve our performance in international surveys, we must delay the teaching of formal written vertical calculations for several years, as in all other European countries.

The Numeracy Task Force consultation document, Numeracy Matters, says the Qualifications and Curriculum Authority is to produce guidance for primary teachers on mental calculation strategies. Will the QCA get it right? The record of its predecessors - the School Curriculum and Assessment Authority and the School Examinations and Assessment Council - is uninspiring. Three examples spring to mind of hurried decisions that subsequently had to be modified.

* In 1991 - year of the first key stage 1 tests - teachers were told to "assess each child's ability to add and subtract by using recall of number facts, not by counting or computation". This flew in the face of the previous decade's research. Most teachers "cheated" by failing to notice any calculating, and the assessment procedures were duly modified the next year.

* A 1995 analysis of key stage 2 tests included one boy's ingenious, idiosyncratic strategy for finding the cost of 18 children visiting a castle at 95 pence each. His method, which gave the right answer, was criticised as "inefficient and time-consuming". This seemed to condemn what many of us thought we were supposed to be developing - thinkers rather than rule followers. An erratum slip later stated that the wrong example had been cited.

* The 1997 key stage 2 pilot mental tests provided general instructions for markers. These stated that any marks on the paper other than the answer were to be regarded as working and would cost a mark. Fortunately some people dismissed this pedantic distinction between mental and written calculation, and instructions for the 1998 tests say children may now write mathematical workings.

Of course, it is easy to be wise after the event. But it would have been possible in each of these cases to have been wise before. Despite Chief Inspector of Schools Chris Woodhead's and Professor James Tooley's criticism of educational research, much in mathematics education is useful, reliable and valid. Consulting this evidence in advance might have saved money and embarrassment.

The QCA has reputedly sent Numeracy Matters to experts as part of the consultation process. But - and speaking as someone who has written more about children's mental calculation strategies than possibly anyone else in the country - you will not see my name on the list.

* Ian Thompson is a lecturer in education at the University of Newcastle upon Tyne

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