that such tallies are only one of many iconic forms. Scarlett used circles to represent
one group of teddies and squares to represent another (Figure 10.8). Jennifer used stars
to stand for individual beans in a set in her subtraction sum (see Figure 7.5).
When playing a game with two dice, Chloë drew stars, triangles and little lorries
in place of the dots (not illustrated) and Kamrin drew some highly imaginative
‘Tweedle birds’ to represent the individual numbers in his division calculation (see
Chapter 9, Figure 9.5). Since none of the iconic symbols that the children chose were
suggested by us, it is difficult to know what their origin was. Scarlett may have used
circles and squares since they were quicker to draw than separate teddies. These
shapes then possibly suggested something else to her: by adding a few details she
turned the circles into balls and the squares into presents. To adults, drawing balls
and presents, stars, triangles, little lorries or ‘Tweedle birds’ in place of the items or
numerals that were part of their calculations may appear curious. But the type of
iconic marks children choose to make is not important, provided they follow the
one-to-one principle (Gelman and Gallistel, 1978).
Tallies may be one of the earliest forms of written counting with its origins in prac-
tical contexts stemming from holding up fingers as a temporary record of items.
They may have links with traditional oral counting documented by Opie and Opie
(1969). Tallies pre-date the earliest forms of writing in the world (Hughes, 1986).
If they choose to use iconic marks, it is important to encourage children to move
towards choosing increasingly efficient forms for their counting rather than focus-
ing on beautifully embellished drawings.Written
Using words or letter-like marks which are read as words and sentences is common in
our examples and found elsewhere (for example, Hughes, 1986; Pengelly, 1986). In our
culture written communication is evident everywhere and children come to see this as
a meaningful response on paper. As the examples from Matt in Chapter 2 showed, chil-
dren may begin to differentiate marks that carry meaning as words, from those that
represent numerals or are drawings, before they are 4 years old. We collected many
examples of children writing explanations and written methods entirely in words, as
Figure 7.8 shows. In this example John wrote ‘2 grapes, there is (are) two. 4 grapes,
there is (are) four. 6.’ His addition calculation is a form of narrative, relating a sequence
of events or numbers. We explore ‘narratives’ in greater detail in Chapter 7.Symbolic
Children using symbolic forms use standard forms of numerals (for example ‘2’, ‘7’,
‘15’) and gradually begin to incorporate standard (abstract) symbols such as ‘+’
appropriately. In the examples from a group of children who were subtracting beans
(Chapter 7, Figure 7.5), Eleanor decided to use standard numerals, the ‘–’ operant
and the symbol for ‘equals’. Other children in the group chose graphical forms that
were appropriate for them at the time. In the same chapter, Anna (Figure 7.10) used88 Children’s Mathematics8657part 1b.qxd 04/07/2006 18:12 Page 88