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What are
‘children’s mathematical graphics’? |
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We originated the term children's
mathematical graphics (see for example,
Worthington & Carruthers, 2003)
to describe the range of children's own mathematical marks and
representations that we found, through the hundreds of examples we
have collected within our teaching in the birth - 8 year age-range
and from children in their homes.
Young children use a range of visual
representations that may include scribbles, drawings, writing,
iconic marks, invented (personal) symbols and standard symbols. They
use their own mathematical representations to help them think about
and communicate meaning and to explore specific symbols and
calculations: see for example Gallery 1.
Vygotsky referred to written symbols as ‘symbolic’ or ‘cultural
tools’) and mathematics as a subject has been described as ‘really a
matter of problem solving with symbolic tools’ (van Oers, 2001, p.
63).
Since Martin Hughes’s (1986) publication
Children and Number: Difficulties in Learning Mathematics,
there has been a small but growing interest in what has variously
been termed ‘emergent mathematics’, ‘mathematical literacy’;
‘mathematics with reason’ and what we term children’s mathematical
graphics.
Our work on children’s mathematical
graphics grew from our experiences as teachers of young children as
we supported their early, emergent or developmental writing (known
as 'process writing' in New Zealand). A keen desire to answer the
question: ‘what is it you believe you must do deliberately to
support children’s mathematical understanding’ (Gulliver,
1992) continues to guide our research. |
Representation – or recording?
Process – or product?
The emphasis in children’s mathematical graphics is on children’s
own mathematical thinking, meanings and understanding about
all aspects of written mathematics and symbols. The value of
using paper to explore their thinking is that their own
representations support understanding by allowing children
to see some of their emerging understanding of ‘written’
mathematics. In a sense children’s mathematical graphics are their
mental methods – on paper. Using their own mathematical graphics
helps young children to 'translate' between their early informal
marks and the standard symbols and written language of mathematics.
Children need to be free to choose how they will represent their
mathematical thinking that best fits their purpose, the particular
mathematical context or calculation they are exploring, or the
problem they wish to solve.
The emphasis in children’s own
mathematical graphics is on processes of mathematical
thinking (creative thinking,
reasoning, meanings, understanding, problem solving, negotiation and
co-construction of understanding) rather than products
(recording something done practically). Real contexts for their
mathematics will allow children to make greater sense of their
mathematics when they explore their thinking through their own
representations.
Recording what they did following
a practical activity has limited value and involves lower levels of
thinking. Children do not need to record mathematics if they can do
it mentally; neither do they need to record something they have
worked out in a practical context. Recording places the emphasis
on marks and drawings as a product and is a lower level of
cognitive demand (thinking) in mathematics. The difference between
representing mathematical thinking and recording is
one of quality and depth of thinking.
Understanding, supporting and
assessing
Children’s mathematical graphics also have tremendous value for
teachers since they reveal each child’s thinking about all aspects
of written mathematics. Annotated pieces also offer an invaluable tool for
assessment in mathematics when used with the
taxonomy. |
References:
Worthington, M. & Carruthers, E.
(2003) Children’s Mathematics: Marking Marks, Making Meaning,
London: Paul Chapman, Second edition: Carruthers, E. &
Worthington, M. (2006)
Van Oers, B. (2001) ‘Educational
forms of initiation in mathematical culture’, in, Educational
Studies in Mathematics, 46: 59-85.
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