Dispatches from the maths war

In a piece for Prospect last year, I reported on developments on the US front of the "maths wars" (an ongoing conflict over how maths should be taught in schools) and suggested that these may have implications for the 40 per cent of children who leave British primary schools without adequate maths or English.

My report was based on the work of schools and children in the Stanford Tizard project. My fellow researchers and I are re-evaluating a math curriculum developed by the math educator Caleb Gattegno, the founder of the UK Association of Mathematics Teachers. For a time it seemed that Labour's new regime would take the opportunity to replace our outmoded model of maths learning. In this article we look at what has gone wrong, why, and what we might yet do about it.



Last July, Ed Balls set up a Review of Maths Teaching in Early Years and Primary Schools. He asked its chairman, Peter Williams, two key questions:

a) What is the most effective design and sequencing of the maths curriculum?

b) What subject knowledge should we require of primary teachers?

The conventional primary maths sequence introduces kids to new operators one at a time over six years: sequence and order, plus, minus, multiply, divide, fractions as operators and as magnitudes. It ends with algebra. Numbers grow in size as the years progress.

In an interim report, open to further evidence until tomorrow, Williams asserts "there is a clear and logical evolution in the primary curriculum from number and counting eventually to more complex and abstract concepts in mathematics."

We owe this "clear and logical" progression to Jean Piaget, a pioneer of standardised testing. His pre-WWII experiments became the foundation for the 1970s Chelsea College Maths Concepts Project (CSMS). CSMS delivered personnel, test data and curriculum frameworks for the national curriculum, and its international offshoot—the National Foundation for Educational Research (now owned by private equity). Piaget similarly supplied the US K-8 approach to math education (his US followers call themselves constructivists).

Williams concluded his interim report by saying "this conventional approach has much to offer and no recommendations for fundamental change to the sequence in which math is delivered are being made at this stage.'' He recommends that government should invest in a "primary maths" specialist for each of the 1,500 schools with lowest performing children. The aim is to ''recover'' 30,000 children's maths.

Piaget was interested in studying how groups develop. He said that he "had no interest in the individual." Why, then, attempt to "recover" individual children's with fossilised Swiss maths? Mathematicians and software developers now know much more than Piaget did about how people think mathematically, how state curricula work, and how kids learn maths. Williams should be far more ambitious in his report, and should pay attention in particular to two key points.

First, we now know that, to the child, Piaget's conventional progression is neither clear nor logical. It is the root cause of children's failure. Kathryn Sullivan, head of a US math education policy centre, refers to a three-foot pile of reports that show that "our approach to education in the United States is by and large operating under what we can see now is a very flawed model of development.'' According to Sullivan, the US curriculum persists in its present form because of a forgotten premise of the National Defence Education Act (1958). After Sputnik, the goal of elementary math education was to identify and fast track scientific talent, with the rest forgotten. Much the same is true in Britain (albeit it was the needs of the grammar schools that set the goal).

Second, thanks to Caleb Gattegno, Piaget's sometime collaborator in the first international commission for the improvement of math education, we now know how to get maths right. In the 1950s, Gattegno showed us how to teach in 18 months the material that the state still takes five years to cover. He did this by teaching mathematics as a language, with meaning expressed in spoken, diagrammatic, symbolic and written form. His system exploits children's inherent ability to disambiguate names to distinguish numerals from numbers. Children first learn to recite (counting is difficult). Then, both teachers and children use permutations and combinations of coloured rods to help develop their intuition for abstract data types, so that they can think mathematically.

Peter Williams has said that he welcomes further input on the maths sequence. In Don't Send us Back, Sir, we present the evidence for reform. Our key observation is that we now have a modern mathematics culture—one that has created abstract data types, Unix and the command line. Education needs to start from here. We believe the Williams report should include three immediate steps:

a) Require QCA and schools to give formative assessment feedback from SATs directly to parents by using Gattegno's 55 concept unfolding. This will guide an individual pupil's study and drive the necessary migration of the curriculum sequence.

b) Don't hide away kids' work in cupboards. Reward teachers, young people and the QCA for publishing it. Math websites giving answers to text book questions are big business in the US. Children are capable of supplying both questions and answers and ranking them for effectiveness. Why not harness their energy to a reform process?

c) Abandon the recommended "primary math specialists" small business opportunity. This initiative is too little, too late and doesn't scale. Instead require every primary teacher who teaches math to requalify over three years by mentoring just one young person. For an adult Gattegno's text books only take a few hours work. The state should give them to every teacher and parent on starting school so that they can anticipate what is to follow.