We had been very excited by our early observations, but shorter observations of
schemas are more manageable in a busy classroom. These informal observations
were spontaneous and usually only took two or three minutes. During a period of
nine months we found that the most popular schemas for these 4–6-year-olds in a
school classroom were spiralsand rotation,grids,shapes, connection andtrajectories
(often explored through maps). This list can be contrasted with Arnold’s list of most
frequently observed schemas in one nursery (see pp. 39, 40).Zoë’s schema was dominant for such a long time that some of the children were
drawn into her interest and explored them in their own ways. When the other chil-
dren’s schema interests were mapped (see for example Figure 3.7), it was clear that
only the children who were already interested in aspects of rotation, circles, arcs, tra-
jectories and semicircles chose to explore spirals. These children were at a point
where exploring spirals fitted with their current schemas and added to their under-
standing. At the same time other children’s new perspectives on spirals and rotation
added to Zoë’s understanding by exploring an ever-widening range of behaviours,
resources and opportunities.Children’s representations and marks of their schemas
The children’s current schemas appeared to captivate them and often surprised us by
their endurance. In the middle of a class discussion one day, David put his finger in
his ear and said ‘Oh! Our ears are spirals!’
Several children used construction apparatus to explore spirals. Daniel, 5:1, drew
a circular scribble saying ‘It’s like my stirring wheel’ (that he’d made the previous
day with construction materials). Several weeks later when he was drawing, he
found some blunt pencils and fetching a pencil sharpener said ‘I’ll get my little
stirring thing’.
A group of children in the same class decided to make a ‘spiral road’ with the big
wooden blocks. This resembled a circular maze and drew lots of children to it, to
make their own journeys to the centre and out again. James, 5:4, and Mitchell, 5:2,
cut out paper spirals and stood on the blocks waving them, repeating ‘Spirals and
spirals and spirals’ (Figure 3.4).For over two terms everyone in the class knew that Zoë, 4:7, had a passion for
spirals. She drew them, cut out spirals and talked about them endlessly. Outside
she often walked in a spiral formation in the playground saying ‘I’m winding myself
up’, and then, reversing her direction, announced ‘I’m unwinding myself’. At home
she loved playing with a ‘spirograph’ set on which she made circular patterns with
the help of rotating wheels. One day her mother brought in some Greek cakes
called ‘baklavas’ for the children to share: as she handed the plate round, Zoë
proudly told her friends that ‘they’re spirals’.Mathematical schemas 478657part 1b.qxd 04/07/2006 18:07 Page 47