Children\'s Mathematics

young learners. In the literature children are seen as needing to translate between

the informal and formal mathematical languages, though in practical terms there

has been a dearth of guidance on how this might be achieved. However, recent lit-

erature has very much supported the idea of older children using their own written

methods mathematics. For example, Thompson (1997) uses the term ‘personal

written methods’ and Anghileri (2000) writes about ‘invented methods’. There has

been lack of evidence from published examples of younger children’s mathematical

graphics, yet this is where most of the difficulty begins: children’s confusion is

ingrained before they might be given the freedom to try their own written ways in

mathematics. We are arguing that teachers need to understand children’s mathe-

matical marks much earlier than current available literature suggests.

We know the pressures that teachers experience, including tests, inspections and

demands for ‘results’ and ‘standards’. Some of these pressures can be interpreted as

a need for children to achieve abstract symbolism rather than use informal mathe-

matical graphics in the short term.The pressure leads to children being hurried on by

recording in ‘acceptable’ (neat or standardised) forms such as worksheets. It can also

include teachers’ requests that children use a particular form of representation: this

might include standard calculations, or that all children in the class use tallies or a

pictographic form when working on a particular calculation.

In Figure 5.2 this shift is represented as a bridge, with children moving fromtheir

informal mathematics to the standard and increasingly abstract forms of

representing mathematics in school. The implication in such a model is that the

desirable movement is generally of one-way traffic, therefore it will follow that the

most desirable position will be to move children on to abstract mathematics

without a transitional or multicompetentstage.

In Figure 5.3 we provide an alternative model. Using their own mathematical

graphics and constantly moving between their own informal understanding and

abstract mathematical symbolism in an infinite loop ensures cognitive feedback. In

the centre of the model the area represented by a dotted circle shows where

Figure 5.3 Becoming bi-numerate

80 Children’s Mathematics

Early childhood

education setting

Home and

community

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