Two digits to pen and paper work
As someone who has written extensively on the importance of counting in the development of number understanding in young children, and also on the educational uses that can be made of calculators in the teaching of number concepts, I read her discussion paper avidly.
The research constitutes a fascinating, detailed and thorough comparison of English primary mathematics textbooks with those used in Germany and Switzerland. What it does not do is provide evidence that standards of performance of nine-year olds in this country are lower than those in European countries.
None of the newspapers made the point that the "evidence" they quoted was gleaned from informal (rather than systematic) and brief (rather than longitudinal) observations of lessons in German and Swiss schools by a group of teachers, maths advisers, and Institute researchers.
The actual research which suggested that English nine-year-olds fared worse than those in 12 other countries in international tests was carried out as part of the International Assessment of Progress exercise in 1991.
However, if English children's performance in whole number arithmetic really is a year or so behind that of their European counterparts, we must try to ascertain the causes. Personally, I think the real problem has little or nothing to do with fingers or calculators and a lot to do with the relative emphasis we place on mental and written calculations.
Of the 42 pages in this document only one is devoted to the use of fingers and counting strategies, and only two discuss calculators. If we look at the remaining 39 pages of Bierhoff's paper, which compare primary school textbooks, what comes over forcibly is succinctly summed up in the following quotation: "Mental calculation is regarded on the Continent as a priority, to the exclusion of formal pencil and paper calculations until the age of nine. " Almost every page illustrates how much Continental educators stress the importance of the development of mental calculation.
There appears to have been more research into the teaching of mental arithmetic in Germany than in Britain or America, and Continental textbooks tend to approach the topic in a very structured manner. Specific mental strategies are broken down into carefully-graded steps and are formally taught and practised to such an extent that nine-year olds are expected to be able to perform two-digit calculations in their head.
Such is the emphasis on mental arithmetic that children are not taught written methods of calculation for two-digit numbers, and are only introduced to these methods when working with three-digit numbers.
I am sure that the English obsession with written methods and, in particular, with vertical written methods for young children, has had a substantial negative influence on children's calculation performance. In most primary maths schemes two-digit calculations are almost always presented in vertical format, and many such schemes also introduce single-digit calculations in a similar format Q the purpose of which I have never understood.
There is also substantial evidence that children prefer to use mental calculation strategies that operate on numbers by following the natural way in which they are spoken Q from left to right, working with the tens before the units. Most children functioning at this level would add 43 and 28 either as "40 and 20 is 60 . . . 3 and 8 is 11 . . . so it's 71" or as "43 and 20 is 63 . . . and another 7 makes 70 . . . so it's 71." Few would work from right to left, adding the units first as is demanded by the standard algorithm for addition; even fewer would resort of their own accord to "carrying tens".
It is this disparity between normal mental calculation methods and the execution of standard algorithms that is the root cause of many of the errors in two-digit calculations that are made by young children in this country. The Continental philosophy of emphasising the development of mental calculation at the expense of written methods needs to be integrated into our own approach to the teaching of whole number calculation.
This will no doubt necessitate a paradigm shift on the part of many of those involved in mathematics education: teachers, head teachers, parents, advisers, inspectors and, not least, compilers of SAT questions.
However, if we can succeed in playing down the importance of written calculation procedures, and playing up the importance of mental methods, it will prove to be well worth the trouble.
Ian Thompson is a lecturer in the School of Education at the University of Newcastle upon Tyne