The mixed-ability Year 7 class at Hampstead School in London study three circles on cards labelled A, B and C that their teacher, Margaret Tetty, has placed on the classroom walls. "Which is the biggest?" she asks briskly. The pupils argue briefly before Ms Tetty takes a vote and asks why they cannot agree. One pupil says: "B looks biggest, but it is nearest so we don't know." Another says: "Sometimes the eyes play tricks." This prompts Ms Tetty to ask: "So how do we make sure?" She gets suggestions on lifting, or tracing the circles to place them on top of each other, or measuring them.
The approach used in Cognitive Acceleration in Mathematics Education (CAME) is spreading fast. While nearly half of all secondary schools use Thinking Science lessons from its sister project Cognitive Acceleration in Science Education (CASE), the number of CAME schools - those using Thinking Maths - has grown from 10 to about 400 in three years .
Cognitive Acceleration (CA) is nothing to do with the notion of "acceleration", of more able students' learning or pressing younger pupils into advanced formal work and GCSE exams, something which CA researchers think is counter-productive. CA focuses on better teaching through teacher attention to divers stimuli and better structuring of lessons: also termed accelerated learning.
CAME is an idea whose time is at hand. Emphasis on thinking is a national priority. Education Secretary David Blunkett has officially endorsed the CAME approach and the Department for Education and Employment is piloting it across school subjects as part of the key stage 3 initiative on transforming teaching and learning. Qualification and Curriculum Authority officials, National Numeracy Strategy consultants and regional officers also see it as part of the "next step" in the numeracy strategy. Investigation, problem-solving and understanding concepts are central to CA.
Back in our Year 7 class, the conversation develops: "What do we actually measure?" leads to "How do we know where the diameter is?" and then "What about comparing fixed pipes that you cannot measure across?" Finally, Ms Tetty asks key questions: "Is it always true that the larger the diameter the larger the circumference?" and, "If we know the diameter can we also know the circumference?" "What method can give us a result?" "If we use strings rather than tapes, can we find out how many times the diameter is the circumference?" Pupils start working with strings around paper cups and plates, around their heads and arms and the wastepaper bin. They cut the strings and compare lengths across and around circular objects.
Most discover the relationship to be "3 and a bit". Writing that on the board, Ms Tetty asks a girl to demonstrate, then asks if the "bit" is the same length for different size circles. The class agrees that the "bit" is not the same size for all circles but also relates to size of the diameter. They set out to find whether it is about a quarter of the diameter, an eighth, or another fraction of it, and to show that with strings.
About 30 minutes have passed with the pupils working in pairs and as a whole class with near full engagement, without yet putting pen to paper.
One group is working ahead of th class, so those pupils are to compare the circumference of the circle with the perimeters of two shapes: an enclosing square and an in-drawn hexagon. Ms Tetty's class of high-ability Year 9 pupils complete the same work in 10 minutes.
Now the whole class together visually compares the circumference with the perimeter of an enclosing square (four times the diameter) and of an in-drawn regular hexagon (three times the diameter). They discuss, with drawings, the notion of the circle being the same as a regular polygon with infinite number of sides. Now they are intuitively and visually approaching the area of the circle in relation to the radius, or "half-diameter". To help them, Ms Tetty draws three small squares on the radius of the circle. They cover three-quarters of it, with three corner pieces outside extra.
She asks: "Would the three corner pieces cover the fourth quarter-circle?" A boy demonstrates that "you need three small squares and a bit" to cover the circle. She responds, "So how do we write mathematically that three-and-a-bit of the squares-on-the-radius" are needed to cover the circle fully? The next step is for pupils to plot the two functions of the circumference and the area on the same graph. That would lead them to explore the shapes of the graphs, meaning of intersections, and the continuity of functions.
The Year 7 and Year 9 classes were both using the same CAME lesson on the circle, with the teacher matching the rate of conceptual exploration to pupils' understanding. Several of the exemplar CAME activities involve delving into the thinking underpinning non-linear graphs, algebraic manipulations, probability, correlation and trigonometry. Many teachers find this intriguing - a novelty for lower secondary maths work. It develops them as well as their pupils.
The CAME approach is an integration of two child development theories: the cognitive development psychology of Piaget with the social psychology of Vygotsky. First, the learning task is structured to match the range of thinking levels in the class. Then conducting the lesson in class allows contributions at different levels, always guided by the teacher towards higher-order concepts. Prizes for all!
But CAME is no wishy-washy feel-good approach. Without challenging structured tasks, "good teaching" may not be really beneficial. Yet good activities without interactive teaching may be wasted, too. To these must be added a culture of learning. Teachers and students explore ideas together and motivation is high. Exam and test results of pupils who have had CAME lessons have confirmed that this approach works.
From the first, GCSE and KS3 results have been higher. Improvements are due to the thinking activities themselves, but also to the way that the teachers develop as teachers by teaching them. It is in this interactive relationship between classroom activity and professional development that CAME is likely to make its greatest contribution to maths education.
Mundher Adhami is CAME principal researcher at Kings College London 'Thinking Maths' by Michael Shayer, Mundher Adhami and David Johnson is published by Heinemann, pound;75.CAME Professional Development programmes are run by the Centre for the Advancement of Thinking at Kings College London School of Education. Tel: 0207 848 3134E-mail: Julie.Hypher@kcl.ac.uk