Why value children’s graphical representations and how do they assist children on their mathematical journey?
Personal recording is important in mathematics at all ages and stages, as it helps us organise or track our thoughts and thus contributes to our mathematical thinking and our ability to solve problems. It is also a means of communicating mathematically with others. The foundations of both these roles for mathematical mark-making are established when children are young, by adults encouraging them to make jottings alongside their play with practical apparatus, and as we respond positively and with curiosity to children’s attempts to sense-make on paper.
In the early stages, children make deliberate marks as they enjoy experimenting with different tools, surfaces and media. They explore the cause and effect of their actions, trialling different actions to create a range of marks.
For very young children, this can be indistinguishable from emergent writing in that the meaning that the child associates with their marks may be quantities, shape, pattern, words, actions or images. As mark making develops, the distinction between which marks the child intends to represent words and those which have a more mathematical meaning becomes clearer to the adults who know them well and ‘marks’ at the later stages include symbols and drawings.
These early marks are often referred to as ‘mathematical graphics’ (a term popularised by Carruthers & Worthington, 2003). In mathematical mark making the number of marks can be significant as well as their shape, position and orientation, as shown by the example in figure 1. The marks act as a record to ‘hold’ thoughts as we engage in mathematics. This is as true for adults as for children. The process is therefore crucial. A final record of 4 balls in a basket may show four lines, for example, but the process might have been two then one more then another. This mode of mathematical communication is owned and made sense of by the child.
Sometimes mathematical mark-making is not approached as integral to the task, but rather as an end point, as a record of the activity or even the object of the mathematics (e.g. completing a page of ‘sums’) and this position paper explains how informal mark-making can be used whilst familiarising children with numerals and the generally understood code of mathematics (e.g. introducing and writing numerals) to enhance and deepen children’s mathematical development. Informal mathematical mark-making is a key way of children externalising their own internal meaning making to share with families, practitioners and other children. It is crucial to early mathematical development and invaluable to adults in understanding the child’s mathematical thinking. Thus this is a two-way process. It is to do with communication and relationships. Children are apprenticed into the way numbers are culturally represented, as well as developing a repertoire of ways of expressing themselves. Number symbols and signs are positively useful because they are generally understood and help us to communicate mathematically.
Here is the Educational Programme for Mathematics from the Statutory EYFS Framework in England (DfE, 2020):
Developing a strong grounding in number is essential so that all children develop the necessary building blocks to excel mathematically. Children should be able to count confidently, develop a deep understanding of the numbers to 10, the relationships between them and the patterns within those numbers. By providing frequent and varied opportunities to build and apply this understanding – such as using manipulatives, including small pebbles and tens frames for organising counting – children will develop a secure base of knowledge and vocabulary from which mastery of mathematics is built. (DfE 2020:10)
Whilst the terms ‘mathematical graphics’ or ‘mark-making’ are not included in the Educational Programme, these are an important part of the ‘frequent and varied opportunities to build and apply this understanding’. This is apparent in the nonstatutory guidance, where mathematical mark-making features in both non-statutory EYFS guidance documents (DfE 2021, EE 2021). Birth to Five Matters (EE 2021) in particular, points out the connection between mathematical mark making and mathematical thinking:
Range 3– Children should freely explore how they represent their mathematical thinking through gesture, talk, manipulation of objects and their graphical signs and representations, supported by access to graphic tools in their pretend play (p46 Learning and Development)
Range 4– Encourage children to use marks to represent their mathematical ideas in role play (p97)
Range 5– Model and encourage counting and representing numbers within role play, e.g. making a telephone call using a list of numbers. Value children’s own mathematical representations within their pretend play. Encourage children to use their fingers to show an amount e.g. when asking another child to share resources, to show on their fingers how many they need.
Range 6-Talk to children about the marks and signs they use to represent and communicate their thinking. As appropriate, model and discuss informal and standard ways (e.g. using arrows, plus and minus signs). Begin to model calculations in mathematical stories and number rhymes and in real contexts, using a range of ways of representing (e.g. five-frames). Use both informal and standard ways to record these, including tallies and symbols. Discuss children’s own graphical strategies to solve problems, using some vocabulary of addition and subtraction.
‘Development Matters’ (DfE 2021) also values mathematical mark-making:
3 & 4-year-olds will be learning to: Experiment with their own symbols and marks as well as numerals. Examples – Encourage children in their own ways of recording (for example) how many balls they managed to throw through the hoop. Provide numerals nearby for reference (p51).
Maths in the early years involves practical experiences, and it is these which children want to represent, manipulatives, structured or unstructured, play an important part of these experiences (Griffiths, Back and Gifford 2016). Manipulating equipment and toys helps to organise our personal ideas in a way that makes sense to ourselves. How might we link the use of manipulatives, as in figure 2, with graphical representations?