Picture a mathematics classroom where 14-year-old students are sitting in groups of three; they lean forward, gesticulate, talk, listen. Each group has a loop of string, rulers, angle measurers, calculators. They are moving the string around, stretching it into different shapes, measuring, jotting numbers on pieces of paper. The teacher moves round, listening to the discussions, responding to questions, asking questions.
The students have been asked to find out the largest possible area which their particular loop of string can enclose and present a report about their findings to the class.
The movements of hands, direction of eyes and urgency in voices indicate active involvement in thinking and learning. These small group discussions are useful for forming concepts, trying out ideas, feeling ways forward with a task, and checking understanding of mathematical skills.
The Russian psychologist Vygostsky stressed the importance of language, produced through social interaction, in the development of consciousness (Thoughts and Language, MIT 1962). His view was that thoughts are formulated and formalised almost exclusively through the use of language. Discussions in the classroom give the students that opportunity to formulate thoughts and develop mathematical concepts. One of the students, Sheetal, says: "It's easier to understand from the others, like Nita, she explains it at my level . . . it's easier to talk to them."
The students complete their exploratory work on the area enclosed by the string. They now have to prepare and present a report to the rest of the class. They have sugar paper, graph paper, marker pens and the use of the chalk board. They share out the writing up part of the report: there is more interaction between them as they consult on words to use, axes of graphs, spellings. They consult the teacher frequently: "How can we put this?" "Is there a word for . . .?"
This preparation for a formal presentation pushes the students into clarifying their thoughts and ideas further. The language becomes more structured as they think through the links they will make between the stages of their work. They discover a need for mathematical terms to put across their ideas.
As the presentations are made, one group at a time, the students go through their rehearsed talk, pointing at the sugar paper, drawing on the board to make their points. The others listen attentively.
Some of the listeners ask questions: "How did you know if all the angles were the same?" "How did you know how accurate your areas were?" The presenters answer, sometimes conferring, sometimes looking to the teacher for advice.
The activity has enhanced pupils' language skills and social skills, as well as improving their mathematical understanding. As one student says after one presentation: "As I was explaining it to the others, it explained itself inside my head."
Annie Gammon is head of mathematics at Sarah Bonnell School, East London