Children\'s Mathematics

As their thinking develops, children appear to progressively filter out everything but

what is necessary to them at the time.

Early development of mathematical meaning

In analysing almost 700 examples it became evident that, as they make marks on

paper, the children’s mathematical thinking and understanding supports their

meaning. In turn, as their marks and representations are co-constructed and negoti-

ated with others, this extends their ideas – not only about the form of their marks

but also about the mathematics. As we analysed the children’s mathematical marks,

we could see a development of both the children’s marks and of their meanings.

The development of children’s mathematical graphics

From their early play and marks, through counting and their own written methods

that children choose to use, we have identified five dimensions of the development

of mathematical graphics. These dimensions span the period from 3 year olds in the

home and nursery, through to children of 7 and 8 years old in school.

• multi-modal explorations and exploration with marks

• Early written numerals

• Numerals as labels

• Representations of quantities and counting

• The development of early written number, quantities and counting

For a taxonomy of children’s development, see p.131.

It is important to emphasise that this taxonomy is not strictly hierarchical and

therefore the various dimensions should not be seen as ‘stages’ through which all

children pass, nor should they be directly taught. However, there are several features

which are significant for teachers:

• all infants move from gesture, movement andspeechto make their own early explo-
  rations with marks


• children will all need to be freely representing quantities that are countedbefore
  moving on to early operations in which they count continuously.

Multi-modality

Making marks on a surface, for example with fingers or a pen, has a history. These

arise from the infant’s gestures that both precede and accompany a child’s first marks

(Trevarthen, cited in Matthews, 1999; Vygotsky, 1983). Children may be using ‘their

own body actions and actions performed upon visual media to express emotion’

(Matthews, 1999, p. 20). And, as Kress has documented, there are multi-modal ways

of making meaning ‘before writing’ (Kress, 1997). Children’s mathematical marks are

only one of the ways they use marks to communicate and carry meaning.

In Chapter 3, the observations of Aaron’s pattern of schemas indicate the rich and

diverse ways in which he made meaning. As Athey revealed in her important study

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