Children\'s Mathematics

representations. In this chapter we are therefore continuing to look carefully at chil-

dren’s marks and their meanings.

The fifth dimension: written calculations

For an overview of the development of children’s mathematical graphics, see Figure

7.13. (p. 131).

In analysing examples of children’s calculations a pattern of development

appeared which we have grouped into categories:

• Counting continuously

• Separating sets

• Exploring symbols

• Standard symbolic calculations with small numbers

• Calculations with larger numbers supported by jottings

Before they are ready to understand calculations, children represent quantities that are

counted. In their own written methods of calculations, there is great diversity in their

chosen approaches before they begin to explore standard calculations with small numbers.

Some children also re-visit a more familiar way of working, for example Harriet’s

drawings of people in Figure 9.13 and Alison’s use of iconic marks in Figure 9.11. By

the time they are calculating with larger numbers supported by jottings, children will have

developed a deep understanding of the abstract symbols and written methods and

can apply them in a wide range of novel contexts such as problem solving.

Representations of early operations

Beginning with counting

Before they begin formal schooling, research has shown that young children ‘have

some understanding of addition and subtraction with small numbers’ (Baroody, 1987,

p. 144). Carpenter and Moser (1984) found that young children first use counting

strategies to solve simple (oral and practical) word problems involving addition and

subtraction. They identified the following levels of strategies: counting all; counting

on from the first numbers; counting on from the larger number; using known facts

such as number bonds they know by heart and using derived number facts such as

doubles to calculate what they do not know (Thompson, 1995). Orton argues that

whilst schools focus on combining and separating sets as an introduction to addition

and subtraction, children’s preference for counting persists (Orton, 1992, p. 145).

Studies of 5-year olds’ calculation strategies have highlighted the range of proce-

dures the children used that they had not previously been taught (Carpenter and

Moser, 1984; Groen and Resnick, 1985). Such studies indicate that children ‘tend to

work towards more efficient strategies when given the opportunity to solve a variety

of problems’ (Nunes and Bryant, 1996, p. 60). Children do explore and use an ever-

108 Children’s Mathematics

8657part 2.qxd 04/07/2006 17:14 Page 108