children’s knowledge of informal, ‘home’ mathematical marks and formal, written
‘school’ mathematics combine. This is similar to the way in which young children
combine their knowledge of first and second languages (Cook, 1992, p. 16) and two
written languages. In their recent publication, Pahl and Rowsell identify a ‘third
space’ in which children explore drawing and writing ‘using home and school liter-
acy’ (2005, p. 66). Furthermore a similar model has also been identified by Anning
and Ring, where children ‘make sense of continuities and discontinuities’ between
home and school practices (2004, p. 7). These studies point to research findings in
areas as diverse as second language learning, bi-literacy and drawings and mathe-
matical graphics and highlight related procedures in their learning as children link
informal and formal and home and school understandings.
This allows concepts to metamorphose: informal marks are gradually transformed
into standard symbolism. Children become what we term ‘bi-numerate’ and like
bilinguals they will come to use these two languages of mathematics fluently. Their
understanding of the second language – the abstract mathematical language – will
develop at a deep level since they will have constructed their own understanding of
the role and function of the symbols themselves. John-Steiner observed that such
‘complex and opposing relationships’ were noted by Vygotsky, who suggested a two-
way interaction between a first and second language (John-Steiner, 1985, p. 368).
The strength of this model is that whilst the most significant development occurs for
children during the Early Years phase, older children can also benefit from using their
personal mathematical graphics – at any stage or for any calculation or problem. Chil-
dren’s own representations appear to help them reflect on problems ‘and in that think-
ing – and reflecting-process ... understanding becomes more complex’ (Brizuela, 2004,
p. 65). Mathematical graphics appear to allow children to bridge the ‘bi-cultural’ divide.
Cook outlines features of second language learning which we believe are the same
in early mathematical graphics, including approximation, invention, re-structuring
and falling back on the first language (Cook, 2001). Reflecting on the observation
that learning mathematics is ‘like an unfamiliar foreign language’ allows us to see
that children’s own mathematical graphics supports children in developing their
multicompetencesand enable them to become bi-numerate.Using blank paper
The findings of our questionnaire (Worthington and Carruthers, 2003a) (see Chapter
1), showed that whilst 79 per cent of the teachers of 3–8-year-old children who
responded used worksheets, 82 per cent of the same teachers also either allowed or
encouraged some use of blank paper for mathematics. At first glance these figures
appear encouraging. The teachers who said they use blank paper provided 476 exam-
ples of the sort of mathematical marks children might make on blank paper, yet of
these almost 85 per cent were either when the teacher told the children what to do
and how to record, or when the teacher produced what was, in effect, a copy of a
worksheet.
Nine per cent of examples referred to the use of blank paper within children’s role-Bridging the gap between home and school mathematics 818657part 1b.qxd 04/07/2006 18:12 Page 81