tematic ways’. Therefore, young children not only have to make sense of individual
symbols but need to understand their role within a system, whether, for example,
these are letters within a written word, musical notation, or a mathematical sign or
numeral within a written calculation.
The many studies of young children’s early writing development suggest some
ways in which teachers support the growth of understanding (see Chapter 4). Gins-
burg lists three principles of written symbolism in mathematics:- Children’s understanding of written symbolism generally lags behind their infor-
mal arithmetic. - Children interpret written symbolism in terms of what they already know.
- Good teaching attempts to foster connections between the child’s informal knowl-
edge and the abstract and arbitrary system of symbolism (Ginsburg, 1977, pp.
119–20).
Determining ways to foster these connections has been a challenge for teachers but,
as Hughes (1986) and others have observed, a failure to do this is likely to be where
many of children’s difficulties with mathematics lie. Supporting children’s early
writing development is problematic for some teachers and it appears that introduc-
ing abstract symbolism of mathematics is more so. As Hiebert (1984, p. 501)
observes, ‘even though teachers illustrate the symbols and operations with pictures
and objects, many children still have trouble establishing important links’. Vygotsky
emphasised that – as the examples of Matt’s explorations with different marks
demonstrate (Figures 2.1 and 2.6) – there is a ‘critical moment in going from simple
mark-making on paper to the use of pencil-marks as signs that depict or mean some-
thing’ (Vygotsky, 1978, p. 286). For parents and teachers of young children, wit-
nessing such ‘critical moments’ is an enormous thrill and a privilege.
Gardner argues that ‘given a sufficiently rich environment, many a five year-old
is already sensitive to different genres within a symbol system’. Therefore, rather
than viewing young children’s early writing and their mathematical graphics from
a deficit perspective, appreciation of their understanding can be seen as a stunning
achievement. It is, Gardner observes: ‘hardly an exaggeration then, to say that the
five or six year-old is a fully symbolic creature – an individual who has the “first
draft mastery” of the major symbolic systems in her culture. The child can “read”
and “write” in these systems’ (Gardner, 1997 p. 22).
The way we set down mathematical symbols can cause confusion for young chil-
dren, for example the numerals 6 and 9 may appear the same to children since one
is the inverse of the other in appearance. Place value causes problems, for example
‘2’ is different from the two in ‘25’; they mean different things. The subtraction sign
and the equals sign are similar as are the multiplication and the addition signs. Some
letters and numbers such as ‘6’ and ‘b’ also look similar. To further complicate
matters, children are also learning about two symbol systems at the same time,
writing and mathematics (DeLoache, 1991). It could also be argued that the writing
system makes more sense to children. When given the choice some children prefer
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