Children's mathematical graphics is a term used to describe when children choose to use their own marks and representations to explore and communicate their mathematical thinking. This term is based on our analysis of more than 700 examples, in which children from two to seven years old explored their mathematical thinking.
These marks can include scribbles, drawings, writing, tally-type marks, invented and standard symbols including numerals, as children make their own personal mathematical meanings.
When children choose their own symbols, graphics and layouts to help their mathematical thinking, they understand what they are doing because they have explored their own thinking processes. These processes link with the early learning goal: 'Use developing mathematical ideas and methods to solve practical problems' (Practical Guidance for the EYFS, page 76). What is important to understand is that children's own problem-solving lies at the heart of their mathematical graphics.
EFFECTIVE LEARNING
Sir Peter Williams' influential Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools stresses:
- 'effective early years pedagogy values and supports children's own mathematical graphics'
- there needs to be 'a culture with a significant focus on mathematical mark-making in line with early writing through, for example, role play and the use of popular mathematical mark signage in the environment'
- the importance of 'a learning environment that encourages children to choose to use their own mathematical graphics to support their mathematical thinking and processes' (DCSF, 2008b).
However, many schoolteachers ask children to 'record' their mathematics only after the class is familiar with a commercially available set of number rods. Such recording activities fail to support and extend children's thinking, as they often involve the children in simply copying what they have already done, and so require only low-level thinking.
Children's mathematical graphics, by contrast, is about children using graphicacy to support their thinking about their mathematics, and so encourages high-level thinking.
Making meanings
The recent publication of Mark Making Matters (DCSF) has raised the profile of young children's ability to use marks, symbols and drawings to make and communicate their personal meanings and ideas.
At the same time, focusing on 'mark-making' raises important questions about how teachers and practitioners might best understand and support children's explorations.
Imaginative play
Children's mathematical graphics originate in the meanings they explore, make and communicate in their imaginative (symbolic) play - meanings that are vitally important to children's cognitive development.
The relationship between imaginative play, meaning-making and writing was first identified by Vygotsky, who showed how children use gestures, actions, speech and resources as signs to 'mean' something.
Studies of 'meaning-making' are known as 'semiotics', and more recently, Gunther Kress has shown how children explore their ideas as they make meanings with 'lots of different stuff'. An example of this was when in one nursery, Jemima handed the practitioner a flat stone to which she had added grass and gravel, explaining, 'Here's your dinner.'
Pretend play contexts such as role play and 'super-hero' play provide especially rich opportunities as children use resources they find to stand for something else. Although at first children's meanings may not always be transparent, when sensitive adults 'tune in' to children's imaginative play, children's complex thinking becomes clear. Tuning in and valuing children's meanings help adults understand and further support children's thinking.
Junk modelling
Junk modelling and 'cut-outs' also offer rich possibilities for children to explore and communicate meanings. For example, in one nursery a group of boys chose to play 'paper calculators' over the course of two terms.
They made marks on pages of small notebooks, tearing off each page as another calculator while their ideas about technology and video games developed.
Their talk was very rich as they pretended to operate their gadgets and shared and negotiated meanings about Batman and other super-heroes, about numbers on calculators and about the buttons they used to control their games and calculators (see figure 2).
Free play
Playing with meanings in freely chosen contexts allows children to understand that marks and symbols can also carry meanings. For example, in their drawings children often notice the zig-zag shapes made by cutting with pinking shears, and name them 'crocodiles'.
Children's ability to make and attach personal meanings to marks also provides a significant window into children's drawings and emergent writing and is recognised by the EYFS. In the examples given here, the children used different media and resources to create their 'signs'.
This research places new emphasis on young children as powerful meaning-makers, exploring their thinking and meanings in ways that make personal sense as they come to understand the abstract language of 'written' mathematics.
Elizabeth Carruthers is head teacher of Redcliffe Children's Centre and Maintained Nursery in Bristol and Maulfry Worthington is engaged in research for her doctorate (Free University, Amsterdam). They are co-founders of the Children's Mathematics Network
EXPLORING SYMBOLS: OBSERVATIONS FROM NURSERY AND RECEPTION
Finn (aged three and three quarters) was very interested in his age, because he wanted to be four, like his friends.
He discussed it with his nursery teacher, his friends and then at home one day with his mum. Next day he came into nursery and told his friends and teacher that he was 'three and three quarters'.
He said 'I'm not three. I'm three and three quarters ... Look, this is how you write three and three quarters' (figure 3). Then he said, 'This is how you write three and a half' (figure 4).
This was Finn's personal symbol for something extremely important to him - his age - and so provided a real context for his mathematical development. He was exploring the concept of being nearly four and attaching symbols to that meaning.
The dialogue between his teacher and friends was vital to his thinking. This continued over several weeks and the children made many more examples of their own fractions.
Keeping scores
In their outdoor play area, some reception children had chosen to take turns to throw a ball into a net and count their goals.
One child decided to keep a count of his scores on paper on a nearby easel, and wrote his name to show that he had scored one goal (figure 1).
Others joined in, representing their scores by writing the quantity they had, such as '3', '4', '5' or '1', and some wrote tallies.
Next, Ellie wrote her name twice, to signify 'two goals scored'. Rakeem (five years and one month) had his own ideas and drew a Christmas tree in a pot with a circle around it, to show the single goal he had just scored. Jody (four years and 11 months) was intrigued by Rakeem's idea and when she scored two goals also drew a Christmas tree, but this time, drew two pots beneath it as her personal sign for 'two goals' (figure 5).
Having freedom to decide how they would represent their goals meant that the children chose ways that were most meaningful for them. They created signs to solve their own problems (that is, representing their scores), using ways that made personal, mathematical sense.
Rakeem's single Christmas tree became a peer model for Jody, who adapted this sign in a novel way to represent 'two goals'. Soon their teacher joined the children and together they discussed the meanings of their various signs.
Potential to explore
These play episodes reveal the considerable potential of young children to explore their mathematical thinking, meanings and understanding through graphicacy.
As one nursery teacher recently remarked, 'Children's mathematical graphics has influenced everything we do - and also the way in which we view children's play.'
MORE INFORMATION
- Carruthers, E & Worthington, M (2006) Children's Mathematics: Making Marks, Making Meanings. Sage Publications, (2nd Ed)
- Children's Mathematics Network, www.childrens-mathematics.net
- DCSF (2008) Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools. http://publications.teachernet.gov.uk
- DCSF (2008) Mark Making Matters: Young children making meaning in all areas of learning and development, http://nationalstrategies.standards.dcsf. gov.uk
- DCSF (2009) Children Thinking Mathematically: Problem Solving Reasoning and Numeracy in the EYFS (due to be published in late autumn 2009)
- Kress, G (1997) Before Writing: Re-thinking the paths to literacy. Routledge
With thanks to the children and their teachers: Sarah Ryan at Barnsole Infants' School, Gillingham, Medway and Emma Higgins and Carole Keane at Redcliffe Children's Centre, Bristol
STARTING POINTS
To support children's mathematical graphics, start by:
- providing an environment where papers and pens are easily accessible
- valuing children's mathematical graphics
- observing and annotating children's mathematical graphics - this allows adults to uncover sensitively children's thinking and meanings
- modelling mathematical graphics indirectly - it is important for children to see different ways to write down mathematical thinking
- discussing the children's mathematical graphics with a colleague or a group of interested practitioners - this will give you ideas and support your thinking. Join a local children's mathematics network group or start your own. Visit http://www.childrens-mathematics.net/cmnetwork_groups.htm.
