children’s different mathematical graphics at the end of either a group or class lesson
means that peer-modelling extends what the teacher modelled: for the children it
may also help to reinforce the fact that there are many ways of representing mathe-
matics. Significantly, it will also confirm that the teacher really does value the per-
sonal sense individuals make through their chosen ways of mathematical graphics.Figure 10.9a and b Integrating teacher’s visual modelsModelling mathematical symbols and signs
Graves argues that modelling opportunities ‘are infinite’ and that within the
teacher’s modelling concepts are built (Graves, 1983). In mathematics modelling
ways of recording will include specific symbols used in context and will allow for dis-
cussion about alternative ways of representing the same meaning. We have shown
how modelling mathematics allows children to choose from a variety of ways of rep-
resenting meaning: these may include standard, abstract symbols when appropriate.
By doing this the teacher can help children make links between their own (non-
standard) marks and symbols (their first mathematical language) and standard math-
ematics (their second language).
The samples of childen’s representations of subtracting beans in Chapter 6 show
the range of representation within a group of children and the different levels of
symbol use that the children appropriated: they had incorporated aspects of mathe-
matics previously modelled, (direct, indirect and peer-modelling), such as hands and
standard symbols, and some had taken Barney’s new use of arrows and built this into
their representation of subtracting beans. As we have seen, a positive classroom
culture can encourage them to draw on a range of models for their own purposes so
that children will do this with confidence. Heuvel-Panhuizen proposes that to help
children move between their informal and formal levels, ‘models have to shift from212 Children’s Mathematics10.9a 10.9b8657part 2.qxd 04/07/2006 17:40 Page 212