Children\'s Mathematics

a “model of” a particular situation to a “model for” all kinds of other, but equiva-

lent situations’ (Heuvel-Panhuizen, 2001, p. 52).

In terms of writing stories, Graves argues that modelling is especially useful to

explore whatyou have chosen to do and why (Graves, 1983). When a teacher

models the use of the standard sign for ‘take away’ or subtraction, it is helpful if the

children can see not only what is being written as in the example (‘–’), but also why.

Provided the classroom culture supports co-construction of meaning, those who are

ready to relate the abstract symbol to their own ways of representing ‘take away’

will be moving towards the use of more standard forms.

Modelling has a role in what is termed ‘progressive mathematization’ which dis-

tinguishes the Dutch ‘Realistic Mathematics’ or REM approach. This, Beishuizen

argues, is important ‘in the development of abstract thinking on different levels’

(Beishuizen, 2001, p. 130). Whilst we do not claim to be using the REM approach,

supporting children’s mathematics through their early marks and own written

methods may share some similarities in its process with Freudenthal’s principles.

We argue that co-constructing and negotiating meaning together are supported by

a range of increasingly abstract models. These provide children with ‘guided’ oppor-

tunities ‘to “reinvent” mathematics by doing it within a process of “progressive

mathematization”’ (Anghileri, 2001a, p. 34).

Modelling: symbolic tools for children’s ‘mental tool-boxes’

We have shown how from their earliest marks (and other multi-modal representa-

tions) children gradually develop a range of ways of representing their thinking. They

become increasingly selective and begin to make conscious decisions about the type

and layout of these marks and in mathematical graphics we have termed these forms

(see Chapter 6). The forms do not need to be directly taught since they arise from the

range of marks children intuitively make but some will be used within the teacher’s

models. However, as they mature it is clearly the teacher’s responsibility to also model

standard symbols and possible ways of representing, as children gradually move

towards the standard, abstract, written language of mathematics and calculations.

The way in which we use directmodels is not to directly teach children how they

should represent their mathematics, but to explore possible ways of representing

mathematical thinking (e.g. drawings, symbols, iconic representations, various forms

of representing data) and written methods of mathematics; to provide what

Streefland also refers to as ‘models of’mathematics so that children can use them in

the future as a ‘model for’ their mathematics (Streefland, 1993). Heuvel-Panhuizen

emphasises that models ofmathematics within meaningful contexts ‘can fulfil a

bridging function between the informal and formal level’ to a modelfora particular

piece of mathematics in which the child is engaged (Heuvel-Panhuizen, 2003, p. 14).

Using an indirectform of modelling mathematics within play (see Figure 10.10

below) appears similar in some respects to modelling within the Dutch Developmental

Education curriculum (for example, see Oers, 2003; Oers and Warkdekker 1999),

although we go further with modelling written methods for calculation and features

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