An end to one of America's most bitter "education wars" is in sight, and it may have implications for Britain's schools. In September, the National Council of Teachers of Mathematics (NCTM) issued recommendations for teaching the subject to American schoolchildren that emphasised the importance of algebra in primary schools.
This apparently innocuous report represented a U-turn by the NCTM. In 1989, it had sparked off the "maths wars" with a set of guidelines for teachers that emphasised "open-ended" solutions to problems, encouraging children to develop their own techniques.
The 1989 report strongly influenced school textbooks and the authors of US state maths curricula. The result was a generation of underachieving schoolchildren and angry parents, many of whom petitioned their local schools to drop what they called this new "fuzzy maths."
Universities became concerned by undergraduates' poor numeracy—a fifth of first-year students now require remedial maths teaching. Then, in 2003, the Trends in International Mathematics and Science Study (Timss), which compares the mathematical abilities of schoolchildren around the world, ranked 8th-grade (age 13-14) American students 15th. The list was topped by Singapore, ahead of South Korea, Hong Kong, Taiwan and Japan. England and Scotland were 18th and 19th respectively. The question below, taken from Timss 2003, was answered correctly by 69 per cent of Russian children—who follow a similar curriculum to children in Singapore—but only 48 per cent of Americans.
The strong showing of Asian countries in the Timss study influenced the latest NCTM report. The council's president, Francis Fennell, has said that the new guidelines will move US mathematics teaching closer to the Singapore model. Pre-algebraic concepts will be introduced as early as first grade. To make time, there will be less focus on data analysis and shape and space.
The Singapore model of teaching derives from the postwar Russian curriculum, devised by working mathematicians who emphasised the teaching of algebra in upper elementary grades. China adopted it in 1955 and it subsequently reached Singapore. American maths teaching, on the other hand—along with that of most of western Europe—is based on the work of Jean Piaget, the Swiss developmental psychologist, and in particular his theory of cognitive development. (The cold war prevented the eastward spread of Piaget's ideas.)
Piaget argued that children learn and develop by passing through a series of four cognitive "stages." He found that it is not until children reach the last of these stages—the "formal operational stage"—that they are able to reason symbolically about permutations and combinations of figures. The educationalists who followed Piaget therefore ensured that children are not introduced to algebra before early adolescence. Piaget did not claim to be a mathematician, but that didn't stop him from endorsing the pedagogical work of his followers.
The problem is that this work was based on faulty premises. Piaget developed his stages framework by studying children diagnosed with learning difficulties. He aimed to ensure that the child followed an identical framework to his by "spontaneous operational development;" he was not concerned with making actual comprehension any easier.
The Timss results on the mathematical abilities of Russian and Asian schoolchildren repeat findings that were known to Piaget in the 1950s. They indicate that children can cope quite readily with abstract concepts before early adolescence. Denying young children algebraic concepts and notation, as Piaget and his disciples mandate, does irreparable damage: the curriculum is reduced to a shopping list of loosely connected topics—fractions, long multiplication, division. No wonder US maths teaching is described by critics as "a mile wide and an inch deep."
The US has no national curriculum: the NCTM guidelines are there for individual states to take up or not as they see fit. But given the powerful influence of the 1989 report, as well as the strength of the backlash against "fuzzy maths," there is every reason to believe the new recommendations will be followed. Some of the strongest proponents for the change come from California, where highly numerate immigrant engineers and technicians from Asia are unwilling to let the system fail their children.
What about here in Britain? For the last three years, I have been working with ten British schools, state and private, providing booster mathematical lessons that aim to put into practice the ideas of the educationalist Caleb Gattegno, the major figure in mathematics teaching innovation in the 20th century. Along with Georges Cuisenaire, a Belgian teacher, Gattegno revolutionised the teaching of mathematics by the use of his textbooks and Cuisenaire's coloured rods. Using the rods to teach algebra first—before arithmetic—they found that children were able to grasp difficult ideas earlier than Piaget predicted. When this approach was first introduced to British schools in the 1950s and 1960s, 100,000 textbooks and sets of rods were sold to parents, schools and teacher training colleges. Nevertheless, the innovation failed to cross the chasm from early adoption to general use, and was almost lost.
Our Stanford Tizard project aims to reintroduce early algebra to British maths teaching. It has been able to reproduce a systematic quartile gain in performance by children at key stage 1 over their peers by teaching algebra before number. Now, with politicians ceaselessly talking up the "knowledge economy" and also warning us about the competitive threat we face from the rising powers of Asia, what better move could there be than to replace an outdated model of mathematical learning that holds back our children? America has come to its senses. How long before our own mandarins wake up?
More information can be found at http://parents.sociality.tv/BETT07/SW65
