dren to move towards the abstract. There is also more of a link between children’s
mental methods and their own written methods.
The transition between mental methods and standard written algorithms is a dif-
ficult one because, it is argued, ‘the mental approach to mathematics is almost dia-
metrically opposed to the written conventions’ (Holloway, 1997, p. 27).
Unfortunately the videos and the materials produced for teachers of this phase by
the DfES (2002b) focus on the teaching of mathematics in the narrowest sense. Play
especially is misunderstood in these materials: for example, a teacher-directed
cooking activity is described as play. Children’s own methods are encouraged in the
National Numeracy Strategy (DfEE, 1999) but there is little guidance on how to
support children’s own marks. The focus of this book addresses this difficulty.The tins game
It has been well documented that the key to children’s understanding of formal
mathematics is to support them so that they make the transition from their own
informal home mathematics to formally based school mathematics. As early as 1977
Ginsburg argued that the gulf between children’s invented strategies and school-
taught, formal written procedures was a very likely reason that children had diffi-
culty with school mathematics. Hiebert (1984) had a similar thrust, arguing that
making connections between the formal school mathematics and the children’s own
mathematics was imperative, if children were to become mathematical thinkers
rather than mindless followers of mechanical rituals.
In 1986, Hughes’s research heightened awareness of the gap between home and
school mathematics. The literature that followed this publication confirmed the
belief that the answer to children’s difficulties might lie in the role of children’s own
recording and mark-making (Williams, 1997).
Much of the literature has concentrated on explaining Hughes’s ‘tins game’
(Montague-Smith, 1997; Pound, 1999; Vandersteen, 2002). Some researchers have
duplicated the tins game to find out if it brings the same results in different
circumstances (for example, Munn, in Thompson, 1997). Yet this published
research and these texts have failed to show children’s own mathematical marks in
circumstances other than the tins game or variations of the game (see Gifford,
1990). Atkinson’s book included teachers’ stories of children’s own mathematics:
this was the only reference we found that rooted the practice of children’s own
mathematics in real classroom situations with a variety of tasks which were
significantly different to the tins game (Atkinson, 1992). Most of the research in
this area has been of the clinical task type which claims a ‘human sense’ approach:
however, we argue that in order to understand children’s own marks then the
research must make ‘child sense’ (Carruthers, 1997a).
Our study is based on evidence from our own teaching. This enabled us to see a
wide variety of children’s mathematical graphics in a full range of mathematical con-
texts, with children throughout the 3–8 years age range. The variety of examples we
gathered during a number of years helped us to develop our teaching. As we did so76 Children’s Mathematics8657part 1b.qxd 04/07/2006 18:11 Page 76