standing has developed through their own mathematical marks and written
methods. Building secure foundations in this way, children’s understanding will
develop at a deep rather than at a superficial level. Education needs to appreciate
that, as Vergnaud noted, ‘conceptual fields’ such as addition ‘develop slowly from 3
to 14 years and beyond’ (cited in Aubrey, 1997b, p. 150).
As Thrumpston stresses, ‘schools can train children to become skilful operators, to
perform well in the short term but this does not develop the network of connections,
symbolic representations and meanings which extends the power of thinking and
hypothesising’ (Thrumpston, 1994, p. 12).The development of children’s mathematical graphics:
becoming bi-numerate
Figure 7.13 represents the whole of children’s development of mathematical graph-
ics, from birth to 8 years. It shows how children build on their earliest explorations
with marks through written number and representations of quantities (see Chapter
6), to explore their own written methods of calculations.
Note that the taxonomy is not strictly hierarchical (see page 131):- children all need to be freely representing quantities that are countedbefore moving
on to early operations in which they count continuously - children need extended periods of time in which to explore symbolsin their own
ways, before they are ready to use standard symbolic operations with small numbers,
with understanding - additional growth in understanding is often indicated when their representations
show that they are combining aspects of two dimensions, for example Figure 7.7c
(p. 120) where Jack combined separating setswith both implicit symbolsand also
exploring symbols - older children may return to explore some of the earlier forms or dimensions (e.g.
using aniconic form as Alison did in Figure 9.12: p. 187), for security and fitness
of purpose in a particular context.
Adrian was sitting in the writing area. He took an envelope and attached six
‘Post-it’ notes to the outside of the envelope. On the first four he wrote a ‘+’ and
on the fifth he wrote a ‘y’ saying ‘yes’. Finally, on the sixth piece of paper he wrote
another ‘+’.When he had completed his marks he touched each one saying ‘No,
no, no, no, yes, no’. A few minutes later Adrian gave the envelope to me, telling me
‘it’s for you’.
He watched carefully as I removed the brass fastener he had used to secure it
and as I was about to remove the little notes from the envelope he said ‘they’re
not kisses’. I laid them out on the envelope as he explained ‘they’re plusses’.130 Children’s Mathematics8657part 2.qxd 04/07/2006 17:21 Page 130