These are some of the challenging questions posed by Helvia Bierhoff's comparison of the ways mathematics is taught to eight year olds in England, Germany and Switzerland (page l). Her ground-breaking study, conducted at the National Institute for Economic and Social Research, underlines some significant differences in the ways primary maths is approached in Britain and gives empirical support to mathematicians like Dr Tony Gardiner who have questioned the undemanding orthodoxies of the national curriculum.
Ms Bierhoff's findings help explain our poor showing in international comparisons of children's mathematical attainments. By the end of their primary schooling, English pupils are one or two years behind the Swiss in arithmetic, although formal schooling does not start until the age of seven in Switzerland.
Continental teachers rely far more on officially-approved textbooks which, in the case of Germany at least, draw heavily on research into the best ways to approach mathematical concepts. Such research, it seems, is much rarer in this country. And whether because the textbooks are consequently less helpful, or because of an expectation that schools and even individual class teachers will devise their own schemes of work rather than depend on a ready-made one, the teaching materials in use are much more variable. In Switzerland and Germany, such do-it-yourself schemes are not permitted. And yet English pupils are much more dependent upon such materials because teaching is more individualised, and less whole-class based.
Continental teaching at this stage gives much greater emphasis to mastery of mental arithmetic rather than formal pencil and paperwork. A grasp of number facts up to 20, so thorough as to represent a reflex response, is the main focus for seven year olds, while in England number is only one of five attainment targets and more advanced concepts are introduced before basic ones have been consolidated.
Our maths teachers and textbooks, it seems, refrain from guiding children towards efficient strategies for mental calculations on the dubious grounds that children's own methods are always preferable. And in contrast to continental teachers, in England the use of concrete aids to understanding such as blocks, cubes or fingers is seen as a valid method in its own right for finding the answers to sums. As a result children remain fixed in laborious counting strategies.
Nowhere is the contrast starker than in the use of electronic calculators. These are hardly ever seen in continental primary schools where children are expected to do sums in their head; in England their use is a requirement of the national curriculum.
Ms Bierhoff's analysis poses fundamental questions which should be addressed in every corner of primary maths teaching. It challenges the received wisdom of what has hitherto been regarded as "good practice" by teachers and their advisers; it calls into question the effectiveness of many textbooks and maths schemes and suggests the maths national curriculum may be seriously flawed.