standard symbols (numerals and ‘+’) for adding together the dots on the two dice she
was throwing. This example contrasts with that of Amelie, almost a year younger,
who represented the amount on each dice without adding, in a highly dynamic and
personal way (Figure 10.3).
As we discuss in Chapter 7, before children reach the stage of using plus, minus
and equals symbols, they often make their own sense of such abstract symbols in a
variety of intuitive and individual ways.Forms and structures
There appears to be a strong relationship between these formsof mathematical graph-
ics and the structures of marks which Matthews has identified in children’s early draw-
ings (Matthews, 1999). Matthews’ work is a departure from previous studies of
children’s visual representations in his attention to detail when exploring and
analysing children’s actions and marks from their beginnings in infants’ gestures.
Matthews strongly supports the premise that children’s marks are not haphazard scrib-
bles but are products of a systematic investigation. He observes children’s actions and
mark-making through what he terms first, secondand third generation structures and
shows how these structures are found in subsequent drawings and symbols.
Matthews’s first generation structuresrefer to horizontal and vertical arcsand marks
made by push and pull movements, generally using large arm actions. Matthews
describes his own children from six months to two years using these structures.
Examples of these include Matt’s marks (see Figures 2.1; 2.6a and 2.6b).
The second generation structuresare continuous rotation, demarcated line-endings, travel-
ling zigzags, continuous linesand seriated displacements in time and space. Molly (Figure
6.3a) used second generation structures such as demarcated line endings, seriated dis-
placements, and travelling zigzagsthat bear a similarity to the physical action of writing.
In third generational structures,the child organises and transforms first and second
generational structures. Third generation structures are closure, inside/outside relations,
core and radial, parallelism, collinearity (lying in the same straight line), angular attach-
ments, right-angular structures, and U shapes on baseline. These structures support all
visual representations. These three ‘generations’ of structures support all the common
forms of graphical marks that we have identified. The combination of third generational
structuresis increasingly evident in children’s more complex representations (see for
example Figure 6.3b) where Alex has used third generation structures of closure, angular
attachments, core and radial, outside/insideand U shapes on a baseline. He has a sense of
the code of written numbers and his own representations help him explore the way in
which this ‘works’. In Figure 9.5a Kamrin’s ‘Tweedle birds’ combine core and radialand
closurestructures, U shapes on a baselineand angular attachments. Alison’s use of closure
(Figure 9.11) to manage space on her paper is interesting. Alex and Alison’s numerals
are a synthesis of all the generational structures that they have mastered and show that
the complexity of producing standard written letters and numerals is a considerable
achievement for young children, (see also pp. 61–2, for the relationship between chil-
dren’s written letters and numerals and Matthews’s generational structures).Making sense of children’s mathematical graphics 898657part 1b.qxd 04/07/2006 18:12 Page 89