of data handling. Our modelling is of a very different nature to the way in which the
term ‘modelling’ is currently used by teachers in England. In the Netherlands, Grave-
meijer (1994) recognised this particular interpretation of models ‘that causes the
formal level of mathematics to become linked to informal strategies’ – as connecting
with the re-invention of RME’ (Heuvel-Panhuizen, 2003, p. 15). Linking the formal to
children’s informal mathematics is at the heart of children’s mathematical graphics.
When to use directmodelling?
Like a real tool-box (full of spanners, chisels and screwdrivers to which new tools are
added from time to time), children will then have an expanding mental resource of
symbols and written methods (symbolic tools) on which to draw and which take
them beyond what they can do now, as in Vygotsky’s ‘zone of proximal develop-
ment’.
We have shown that modelling mathematics at the beginning of a lesson does not
work: in effect, it has not become transferred to their ‘mental toolbox’ of symbolic
tools. In this context you will be providing anexamplethat may be useful on occa-
sions – but which will lead to all the children doing what you have shown. If your
concern is to encourage children to think and to use their own ideas in mathemat-
ics, you will need to provide directmodels outside of mathematics lessons. This will
allow children to add what you have shown to them to their existing mental models,
to help them make their own decisions and choices about the way in which they
represent their thinking. Our research has shown that it is important that direct mod-
ellingtakes place throughout the week and ideally within the context of an authen-
tic purpose (sometimes also for a real person). The distinctions between direct or
Direct modelling
With opportunities for discussions about their mathematical graphics, children
can become aware of their own and others’ use of different marks on paper and
their potential for representing mathematical meaning. However, to extend
children’s mathematical thinking as they move gradually towards using abstract
symbols and explore written methods for calculations, teachers need to directly
model different ways of representing quantities and of calculations.This direct
modelling is of greatest value in order to:
- extend the children’s repertoire by adding to the children’s ‘mental tool-box’,
in order to extend the possible ways in which they may represent their
mathematical thinking - support children in moving towards increasingly efficient ways of using symbols
and calculations - be selective when they approach new mathematical situations or solve
problems.
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