# Both sides of the classroom door

How should lessons use the common sense of everyday maths? Hugh Burkhardt reviews a collection of writing from current research into teaching approaches

There is something in this splendid collection for anybody seriously interested in maths education - and quite a lot for most of us. The editors have brought together an eclectic collection of distinguished authors, covering many aspects of mainstream research on learning mathematics (despite the title, research on teaching is thinly represented). There is research on the important effects of attitudes, relationships and social environments as well as on cognitive aspects of learning. Each chapter describes a specific approach in a clear style suitable for a wide -though unspecified - audience. The authors mostly balance descriptions of their own work with outlines of what others have done. The mix of fascinating experimental description with theoretical interpretation is well judged. It is worth remembering, as always, that our theoretical models of learning are rather weak - illuminative but rarely a sufficient, reliable basis for prediction or design.

The first of the four main sections, Mathematics and Intelligence, has just two chapters. Gerard Vergnaud outlines his rather deep ideas on concept learning, with illustrations from symmetry in geometry and from additive and multiplicative structures in arithmetic.

Terezinha Nunes is much more down to earth on mediated action - the idea that tools, from concrete objects such as a blind person's stick to abstract symbolic systems such as number or algebra, are an integral part of the way we think and must be included in any study of learning. Nunes gives many fascinating illustrations from her own and others' research, comparing children's use of maths in school and in everyday life around the world.

Section II, The Development of Mathematical Understandings, begins with a fine review by Peter Bryant of the work on early years' learning of number. The excellent chapters that follow are careful and detailed introductions to the research evidence on some key areas: "word problems", proportional reasoning, and algebra and functions.

One thing troubles me - the talk of "understanding" as if it were a Boolean (Yes or No) variable. These chapters describe the many aspects of understanding of each topic, which develop at different rates. It would be better to recognise that nobody ever understands anything completely - it is a matter of how well, and for what purpose. (This view was reinforced by my coming to this book from a two-week break doing research in theoretical physics, where we had agonising confusions over a basic question that we had all "understood" for years. Researchers have a lot in common with pupils.) Section III, Social and Cultural Influences on Mathematics Learning, brings out their importance in both the classroom and everyday life in various countries. The didactic contract in most classrooms - that is, the beliefs both of teachers and of pupils as to what should go on - discourages the synthesis of "home maths" and school maths. The challenge of linking children's everyday mathematical understanding with more formal school maths is one of the principal themes of the book. Rightly so - 20 years ago, my inaugural lecture at Nottingham ended: "Our aim must be to help children to make their mathematics part of their common sense." This goal is now widely accepted but remains elusive. Despite the rhetoric on the importance of "applications", both high and low achievers in maths still seem to agree on one thing - when you go through the maths classroom door, you leave your common sense outside.

Why is this problem so challenging? Maths is inherently abstract: that is its power. A simple example - you count different things in the same way. However, we all find it hard to use maths to model practical situations, seeing how one situation resembles another so that we can use our limited mathematical tool kit to help solve diverse practical problems. When did you last find algebra useful? This flexibility in application is central, not a minor add-on, to learning maths - yet Attainment Target 1 is only 20 per cent of the national curriculum, and less in practice.

Section IV, Constructing Knowledge in the Classroom, returns to school and the relationship between the concrete and the abstract. Gravemeijer and Streefland from the Freudenthal Institute describe work on teaching implications of trying to build this essential linkage, both in general and specifically in teaching fractions.

Richard Noss discusses evidence on some contributions that the computer can make in bridging this gap and, through its different "personality" (my phrase), on modifying relationships in the classroom. The didactic contract and how it might be modified is again the focus of Regine Douady's final chapter.

Douady introduces the term "didactic engineering", which seems a mistake. The term engineering should surely be reserved for imaginative design and careful systematic development of practical solutions, with thorough, sensitive testing in realistic circumstances. The Japanese, and some others, are beginning to use educational engineering in this sense. As we move forward with a Government which will "back what works" (from one that thought that it knew what works) education needs much more good engineering to develop what works. In that enterprise, the knowledge described in this book is important infrastructure.

What do all these insights mean for teaching and the teacher? They will sharpen awareness of what may be going on in each pupil's head, and of the complex elements and connections that every topic in maths involves. Is it helpful to offer these extra challenges to teachers, who have only one minute per pupil per lesson to observe, discuss, diagnose and assist? It could be, by making it clear that success requires that pupils learn to detect and correct their own misconceptions, to "debug" their own procedures (a result from a body of research on teaching not much represented here). This implies that a key teacher role should be to develop these high-level skills, rather than to tackle, one by one, the many facets and misconceptions in each concept area. To achieve that, teachers will need support, both in the classroom and in continuing professional development; if it is to work, such support needs to be well researched and well engineered.

Professor Hugh Burkhardt is based at the Shell Centre for Mathematical Education, University of Nottingham and at the University of California, Berkeley

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