they were short studies, the researchers were unable to analyse the development of
children’s mathematical graphics (Gifford, 1990; Pengelly, 1986).Practical activities
From our questionnaire, teachers revealed that when they do give children oppor-
tunities to record mathematics most use worksheets (see Chapter 1). Many teachers
did make some use of blank paper, though most gave examples in which they told
the children how to record or provided outlines or directions for them to follow. The
exception to this is that some teachers did say that they made blank paper available,
only 16 per cent of teachers in our study referred to children making their own
marks, recording in their own way, choosing how to record or using their own jot-
tings (see Chapter 1). This figure includes those teachers who referred to children
making marks through their role-play. The picture is bleaker than these figures
appear to show, when we analysed the type of marks children made and what the
teachers did with the paper on which they were written (see Chapter 5).
Teachers tend to be over-reliant on practical activities and miss out giving children
opportunities to make their own mathematical marks. Many Early Years mathemat-
ical books written for teachers emphasise practical activities. For example, Lewis
emphasises practical recording with materials and mathematics equipment, and sug-
gests that children’s own recording should be accepted. However, she does not give
any examples or explain how teachers might support children in this way (Lewis,
1996). Threfall argues that ‘the complete absence of sums in the Early Years is the
only real alternative to concentrating on them. There is no viable middle way’, and
proposes practical alternatives (Threfall, 1992, p. 16). However, based on the evi-
dence we put forward in this book we propose that there is a strong alternative.
Practical activities are proposed as the solution that help children understand the
abstract nature of mathematics. However, whilst we believe that practical activities
are important, on their own they will not help the child come to use standard algo-
rithms with understanding. Neither will they help the child understand the nature
and role of abstract mathematical symbols.
As we have argued in Chapter 5, in order to help children translate from their
natural, informal (home) mathematics to the later abstract symbolism of standard
school mathematics, teachers need to support children’s own mathematical graph-
ics. Supporting their mathematical thinking by co-constructing and negotiating
meaning helps children make connections at a deep level. It is through their math-
ematical graphics that children become bi-numerate.Mental methods
Askew discusses the fact that it is not helpful to ask when a mental calculation
becomes a paper and pencil method. He asserts ‘any method involves mental
activity’ (Askew, 2001, p. 13). In England the National Numeracy Strategy (NNS) has
stressed the importance of encouraging the teaching of mental methods (DfEE,Making sense of children’s mathematical graphics 858657part 1b.qxd 04/07/2006 18:12 Page 85