these kinds of stories can be useful for mental mathematics because they help chil-
dren quickly visualise a calculation in their heads. However, they are of limited value
in terms of children’s own representation since they tend to spend more time
drawing pictures than focusing on the mathematics.
Ginsberg argues that ‘young children are engaged in the spontaneous learning
of economical strategies for counting. As children develop, many of their
activities tend towards economy and efficiency’ (Ginsberg, 1989, p. 20). These are
termed by Court as ‘stages of abstraction’ (Court, 1925, cited in Hebbeler, 1981, p.
153). Our study has shown that this tendency is evident in young children’s early
graphical mathematics: children move through different forms of recording, using
a variety of strategies to help them calculate and discarding the forms and strategies
they had previously used. However, when they move on to a more demanding level
of calculations they often return to earlier strategies for reassurance. The strategies
that children choose to use cluster into some common processes although there are
numerous variations and often unexpected and highly individual responses, some
of which are invented strategies for solving problems (Groen and Resnick, 1977;
Leder, 1989). Significantly, Askew and Wiliam argue that whilst studies such as
Aubrey’s (1994a) and Munn’s (1994) concentrated on mathematical content
knowledge, they paid ‘little attention to children’s ability to solve problems
through choosing and using appropriate mathematics. Competence involves
not just the knowing of content, but also its application’ (Askew and Wiliam,
1995, p. 7).Separating sets
Children use a range of strategies to show that the two amounts are distinctly sepa-
rate. They do this in a variety of ways including:- grouping the two sets of items to be added either on opposite sides (to the left and
right) of their paper, or by leaving a space between them - separating the sets with words
- putting a vertical line between sets
- putting an arrow or a personal symbol between the sets.
Once again it appears that when children use these strategies they are using skills
with which they are already familiar as they move into new ways of working. We doBritney, 6:0, has drawn three distinct bowls of strawberries with different
amounts of strawberries in each to be added (see Figure 7.6). She appears to
have combined counting continuouslyand addition (of items in three sets) when at
the foot of the page she wrote:
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +Understanding children’s developing calculations 1178657part 2.qxd 04/07/2006 17:17 Page 117