studies (for example, Hughes, 1986; Walkerdine, 1988) show that ‘while practical
work and “real” contexts need to be chosen carefully ... pupils’ success on a concrete
task should not be taken as an indication of understanding the abstract’ (Askew and
Wiliam, 1995, p. 11). Hall emphasises that the use of concrete materials has not nec-
essarily made the links children need between procedural and conceptual knowledge
(Hall, cited in Maclellan, 2001)..
Thrumpston contends that ‘children need help to form links between formal and
concrete understanding, building on their informal methods of calculations and their
invented symbolism in order to develop, understand and use more formal modes of
representation’ (Thrumpston, 1994, p. 114). These ‘powerful modes’ are, Pound
explains, second-order symbols such as writing and numbers which ‘must be based on
a firm foundation of children’s own invented symbols’ (Pound, 1999, p. 27).
The range of children’s informal methods and symbolism which we document
in this chapter, is supported by the Curriculum Guidance for the Foundation Stage, (QCA,
2000) and The National Numeracy Strategy(DfEE, 1999a). Sharpe argues that this is one
of the strengths of the National Numeracy Strategy (Sharpe, 2000b, p. 108).
Thompson comments that ‘calculation takes place in the mind’ (Thompson, 1997,
p. 98). This is not precisely the same as mental calculations, but whether children (or
adults) use paper for their calculations, thinking takes place. The type of thinking will
not be identical for mental calculations as for those on paper but both will take place
in the mind. Thompson proposes that calculations are only written down:- as a record of mental activity involved
- as a means of support for the individual doing the mathematical thinking
- as a means of communication to others.
These are all valid reasons and ones that we support. However, we return to our
central argument, outlined at the beginning of this chapter and more fully explored
in Chapter 5.
Thompson contends that in England ‘we appear to be obsessed with written work
in mathematics. It is as if no work has been done unless there is a written record to
verify it’ (Thompson, 1997, p. 78). This is not what we propose: what we recommend
includes the type of examples within the pages of this book. But they will not be the
same, for other teachers work in different contexts and children are all different.
Neither do we believe that children should be using paper to explore their mathe-
matical thinking every day – but when it may be appropriate, for the mathematics
and for the child, and within a play-based approach.
When we first started to give children opportunities to explore their own paper
and pencil methods we found it was difficult to make an informed assessment of
their mathematical recordings. Hughes had put his findings in neat categories which
was helpful but we found many other variations (Hughes, 1986). Gifford gave 6-year-
olds a calculation task and asked them to record in their own way. She also found it
was difficult to interpret children’s own mathematical thinking on paper (Gifford,
1990). The confusion over what these marks mean may lead to some teachers aban-
doning the idea of giving children opportunities to make their own mathematicalUnderstanding children’s developing calculations 1078657part 2.qxd 04/07/2006 17:14 Page 107