1999b). Guidance for teachers in the NNS documents and on in-service courses
encourages teachers to discuss children’s own mental methods with the whole class
and support all the children. This has been one of the main changes in the teaching
of mathematics since the introduction of the National Numeracy Strategy.
The introduction of visual mental images to support children’s own mental facility
has been a vital part of the new approach to the teaching of mental mathematics in
England. Askew declares that while practical work with young children is useful, there
must be an element of ‘in the head’ mathematics: if this is lacking, children may oth-
erwise think that mathematics is always practical. Mental mathematics provides chil-
dren with images that they may explore on paper (Askew, 1998). As Harries and
Spooner suggest, these images link together mental work and work on paper:
it allows the children to operate between mental and written methods rather than
feel that they are progressing through mental methods to written methods. What
the images allow the children to do is to build up the bank of strategies from which
they can choose an appropriate one for the task. (Harries and Spooner, 2000, p. 51)Not only is mental mathematics vital for children’s representations but the way in
which the mental mathematics is taught is similar to the teaching we advocate for• Supporting children’s own mathematical marks
Evidence from our observations of children of 3 to 8 years of age, during a period
of over 12 years, shows that whilst the youngest children may not be as prolific at 4
at making mathematical number symbols, they do draw and represent things in a
variety of ways. As the examples in this book show, when adults really listen and
observe the marks children make, they will see beyond the ‘scribbles’ and under-
stand the child’s intended meaning.
We have analysed almost 700 samples of mathematical graphics. These cover the
entire 3–8-year-old age range of mathematical marks and representations in which they
used their own written methods. They cover all aspects of number and mathematics
from the wider mathematics curriculum. They range from child-initiated marks within
play to adult-directed sessions in which the children also chose what they wanted to
put down on paper. All the samples have come from our own classes or classes in which
we have been invited to teach. Based on this large sample of original children’s marks
from real teaching situations in real classes, our findings are therefore evidence based.
We have grouped examples to show where some clear patterns emerged.Categories of children’s mathematical graphics
We have already shown how children select different forms of graphics to represent their
mathematical thinking, both at different ages and for different mathematical purposes
(see, for example, Chapters 2 and 3). In Chapter 4 we argued that there are some links
between children’s early (emergent) writing and their early mathematical graphics.
As members of the Emergent Mathematics Teachers’ group, one of the important
questions we endeavoured to answer was whether there was also a developmental
pathway in children’s mathematical marks and written methods. For many years the
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