increasing range of graphics to represent number operations and some may be gen-
erated by individuals. It is important to encourage children to see the connections
between the ways in which they represent their own ideas and the ways in which
other people choose to do so. Egan writes that ‘while we are encouraging children to
be makers and shapers of sounds and meanings we will also give them many exam-
ples of other people’s shapes’ (Egan, 1988, p. 12).
The variety of graphical responses that children choose also reflects their personal
mental methods and intuitive strategies developed from counting all. We explore
this in the following section on the development of children’s early calculations on
paper.Counting continuously
In our study of young children’s early calculations on paper we collected many
examples of children’s calculations through counting. Here we explore the extent to
which the verbal and practical strategies for counting we have discussed are evident
within children’s early mathematical graphics (addition and subtraction).
We use the term ‘counting continuously’to describe this stage of children’s early rep-
resentations of calculation (addition and subtraction) strategies. Several studies have
shown that young children can carry out simple additions and subtractions (with
objects and verbally) and that they do this by using counting strategies: the most
common strategy is to ‘count all’ or to count the final number of items (Carpenter
and Moser, 1984; Fuson and Hall, 1983). Since the ‘counting-all’ strategy is not one
that children are taught, Hughes suggests that we can infer that this is a self-taught
strategy (Hughes, 1986, p. 35).
When young children are given a worksheet with two sets of items to add, they
count the first set and then continue to count the second set. This is misleading for
teachers because although such a page will be termed ‘addition’: children use it to
count. As we show, counting is a valid and important stage in developing under-
standing of addition, but is not itself addition. However if the child chooses to repre-
sent addition like this (see Figure 7.1) then they have begun to understand the
separation of the two quantities and are developing a sense of addition by combining
the two sets. This is an important distinction and also a good reason to give children
opportunities to use their own methods of recording. The teacher can then under-
stand where the children are in terms of addition as opposed to the worksheet model.
One of the features of the early development concepts that Fuson detailed is when
children use their counting to answer the question ‘How many?’: by doing this they
have begun to integrate counting and cardinality (Fuson, 1988). In addition to
counting items (pictures or icons) continuously, in our study we found children
often represented the objects to be counted as numerals, whereby each number in
the sequence represented one object. In using numbers themselves as countable
objects, this gives children a ‘flexible means of solving addition and subtraction’
(Fuson, 1988, cited in Munn, 1997, p. 11).Understanding children’s developing calculations 1098657part 2.qxd 04/07/2006 17:14 Page 109