Implicit symbols
Other children show that they have an understanding of ‘+’ or ‘=’, but have not rep-
resented the symbols: the marks they make, or the arrangement of their calculation,
show that the symbol is impliedand that they understand the calculation in their
head. At this stage children may ‘read’ their calculation as though to include
written features that are absent: speech is therefore used ‘as a means to make
explicit the implicit dynamic aspects of the children’s intended meaning’ (Oers,
1997, p. 244). We believe that this represents a highly significant point in children’s
developing understanding. Examples of implicit symbolsinclude Figure 7.7c and all
on p. 124.Code switching
This term is used in second-language learning and originated from teaching method-
ologists (Cook, 2001). In terms of spoken language, code switching occurs when a
speaker switches from one language to another in mid-sentence (for example, from
her native language of English, to French as in the spoken statement: ‘J’ai mangé du
fish and chips aujourd’hui’.). This has been observed in studies of the speech of bi-
lingual children (for example, Drury, 2000 and Murshad, 2002). Significantly exam-
ples of code-switching within the writing of young bi-lingual children have also
been found (Mor-Sommerfield, 2002). Our research findings in children’s own
written mathematics therefore are supported by published research of both adults’
and children’s learning of a second (spoken) language and children learning to writeJack is exploring abstract symbols in a different way. He has drawn two separate
sets of grapes, leaving a gap that allows this to be read as ‘4 and3’. Gifford (1990)
provides a similar example. Following this he confirmed the amounts to be added
by writing the numerals and then drew a line between these and the final ‘7’ (his
answer).The line functions as an equals sign for Jack (see ‘Exploring symbols’,
below).Louisa, 5:1, like Britney, was adding strawberries (Figure 7.7a). She has combined
pictorial and written graphical forms with symbolic (the numeral ‘6’).There is
another narrative that reads ‘2 and four more all together 6’. She uses words as
she experiments with the role and function of symbols. Scarlett, 5:6, (Figure 7.7b)
and Jack, 5:3, (Figure 7.7c) were adding grapes. Scarlett’s calculation is similar to
Louisa’s although she has added a numeral next to each group of grapes she has
drawn. She concluded ‘and there’s 7 now’.Understanding children’s developing calculations 1198657part 2.qxd 04/07/2006 17:17 Page 119