Vygotski, the pre-war educational psychologist, wrote that "too much stress on visual and concrete thinking smothers the small beginnings of abstract thinking". According to Helvia Bierhoff, author of Laying the Foundations of Numeracy : A comparison of primary-school textbooks in Britain, Germany and Switzerland, British methods of teaching maths in primary schools develop children's fingers - as they count on them and tap into calculators - but Continental methods develop their minds.
There are several strands to the Continental approach detailed in Ms Bierhoff's report and most of them are being tried out in a pilot project in Barking and Dagenham. Sig Prais, professor of economics at City University, has worked with the LEA and with funding from the Gatsby Foundation to introduce thrice-yearly injections of Continental maths teaching into six east London schools. Eighteen months into the project, results are highly encouraging in a borough which a few years ago had some of the worst standards in the country.
Curriculum and teaching methods on the Continent might at first appear to hark back to the "bad old days" of chalk and talk. In fact, they are a demonstration of how whole-group teaching can be whole-group learning.
Every state in Switzerland and Germany has a prescribed textbook for each year. The curriculum is far narrower than in England and Wales: number is almost exclusively concentrated on for the first two years of school (corresponding to our Years 2 and 3).
In the first year of teaching the aim is to get children's number bonds up to 20 to the standard of "reflex response"; in the second year to extend that competency up to 100. All children are expected to achieve this and to reach minimum standards each successive year. On the whole they do so by Year 5, two years ahead of English children. Those who don't repeat a year.
Teaching strategies are aimed at getting the whole group to reach a prescribed level of competence. Professor Prais wants to see if the higher general standard reached and the higher level of economic success which prevails in the Continental countries are related, and if raising overall achievement in maths in a deprived area of Britain will eventually feed into raised economic performance.
Not only is the Continental curriculum more narrowly focused on number (80 per cent of lessons), it also covers far fewer topics, eight or ten a year as opposed to two to three times that number in England. Consolidation is much more heavily stressed. Continental textbooks allow for up to six times as much practice and consolidation as English schemes. Conversely, topics, once learned, are not revisited on the Continent, whereas English schemes always incorporate revision. This suggests, says Ms Bierhoff, an acceptance that consolidation of concepts will be shaky is built into English maths education.
In her small survey, Ms Bierhoff found that English teachers, when asked what they would look for in a maths scheme, highlighted coverage of the national curriculum, suggestions for activities and a fun quotient. Continental teachers chose adequacy in enabling pupils to gain a thorough understanding of mathematical concepts and mastery of mathematical skills. Must these considerations be opposed?
They are not in the mind of Graham Last, senior primary inspector at Barking and Dagenham and co-ordinator of the Gatsby maths scheme: "What is important is that the lessons are a joint venture in which all participate."
The style of Continental maths teaching is the opposite of the individual child with a worksheet and pencil, and his or her hand stuck in the air. In each 45-minute lesson, 30 minutes is whole-class oral work and 15 minutes is written. Written algorithms are not seen as teaching the concepts but as a means of recording. Thus, the class begins with the teacher explaining and possibly demonstrating concepts. This includes, for example, strategies for mental arithmetic, such as going to the nearest 10 when adding. Then the children are encouraged to practise the concepts, both through games and activities (bingo and brainteasers are equally popular) and by answering problems set by the teacher and by each other. A child chosen to answer must show the working out. Other children can help if they see a mistake.
As well as self-confidence, there are spin-offs into the rest of the curriculum, says a deputy headteacher involved in the scheme. "They have better listening skills since they don't sit for long periods with their hands up, but know they can be called on at any time."
Co-operation is the key to what Mr Last calls "an enriched form of oral work". The ethos in Swiss and German schools has nothing in common with the kind of humiliation older readers may associate with class mental arithmetic. "In many lessons," Mr Last says, "we have never seen a child take pleasure in another's mistake. Nor have we even seen or heard a teacher shout. The Swiss concentrate on getting the atmosphere right so that the class works as a community. "
Oral work also serves to develop concentration and memory. Even in Continental kindergarten, exercises are done to develop memory. Later on, this is developed: in maths one of the means is "chain sums". Try this, at the speed of speech: 64 V 8 + 7 V 5 x 9 + 5 V 4 1 (answer below). It foxed a majority of Barking and Dagenham's education committee and fostered their enthusiasm for the Gatsby scheme. Mr Last says children in the pilot schools are able to concentrate without distraction for 50 minutes, articulate confidently a problem and its solution in mathematical terms, and know what it is they have learned in each lesson.
The second part of the lesson consists of consolidating exercises. This is the period in which more able children can be extended with differentiated work on the same topic and the less able can be supported. It is also when widespread misunderstanding can be picked up, incorporated in the oral review which ends the lesson and, if needs be, carried forward into the next lesson. Children participate in the summing-up as they have participated in the discussion. They take consolidating exercises home.
An essential part of all of this is accuracy and precision. When comparing children's written work, Mr Last was struck by how rigorously Continental children kept their working in columns and squares. This attention to meticulous setting out goes along with avoidance of calculators, which are not used on the Continent until secondary level (13-14 years). Both Ms Bierhoff and Mr Last agree that use of calculators can promote mental laziness.
Laying the Foundations of Numeracy is published by the National Institute of Economic and Social Research