How might mathematical mark-making influence deep-level learning in the early years?
Mathematical mark making supports children to develop effective learning habits or dispositions in mathematics. In creating mathematical graphics, children develop characteristics of effective learning, defined in England (DfE, 2020) as they:
• ‘are willing to have a go’ as their mark making is voluntary and low stakes
• ‘are involved and concentrating’ showing high levels of involvement and sustaining this concentration on a mathematical activity for longer because of the process involved in mark marking
• ‘have their own ideas’ as they determine the mathematics in the situation and what is important to record
• ‘choose ways to do things’ as they make decisions about what symbols or marks to use and how to arrange these
• ‘find new ways of doing things’ by re-presenting a mathematical situation in a new way and in a way that might be different to how others choose to represent it (including adults). New ways might include more efficient ways.
• ‘enjoy achieving what they set out to do’ with ownership over the process and product, they are experts in their own representations.
This depth of understanding provides a strong foundation for future mathematical learning in three ways. Firstly, children build new knowledge by connecting and adapting existing understanding to deep understanding, where children can use and apply knowledge flexibly, providing a secure and stable basis for future learning. Secondly, deep understanding gained through mark making enables children to work autonomously, secure in the knowledge that they can record their process in a way that is meaningful to them. This allows them to check and track back, promoting confidence and resilience. It helps them to feel like mathematicians, as insiders not outsiders in the world of school mathematics. Thirdly, mathematical mark-making enables adults to access children’s mathematical thinking, in a similar way to manipulatives in that there is a record if thinking that can be shared and discussed. Early valuing of mark making makes children more willing to show their working as they get older. Making mathematical thinking visible in this way enables their teachers to pinpoint errors or difficulties and crucially see how the child has approached a problem, knowing where teaching of more efficient methods is needed.
As children move through school, the balance will shift from informal to formal methods of recording, but informal methods are not redundant. They can be a useful stepping stone to learning a new written algorithm or a supplementary way of either checking an answer or initially making sense of a problem before moving to more formal methods for modelling and solving. Mark making for older children might take the form of jottings: having a space to make such jottings while thinking can reduce the pressure on working memory. It can support organisation of thinking and being systematic, as well as encourage risk-taking to try out methods that the child is less certain about.
The impact on learning
Making marks to represent numbers, shapes or mathematical operations need not always be on paper. Children have increasing opportunities for mark-making with technology; and these have been used for some time in early mathematics curricula and interventions (e.g. Sarama & Clements, 2021). This may facilitate learning, by removing the need for manual dexterity. Price, Jewitt & Crescenzi (2015) observed 2 and 3 year-old children engaging in a free finger-painting activity and a colouring activity, both on paper with physical paint and on a tablet computer. Videos suggested that the tablet limited the number of fingers used, limited the sensory experience of using paint, and resulted in more uniform final compositions. However, it increased speed and continuity, which resulted in more mark-making and different scales of mark-making. Although this study did not look specifically at mathematical activities, it does suggest that traditional and computer-based mark-making activities may have different advantages to offer and may complement each other.
If we accept that all young children are able to think deeply about and be proficient in mathematics, which is the underlying principle at the heart of learning and teaching for mastery, then encouraging and supporting children’s own emerging mathematical graphics is one way of enriching and deepening children’s understanding. If we wish our teaching to build on what children know and can do mathematically, children’s informal mathematical graphics provide a window into their thinking, and one we would do well to take account of.
Acknowledgments:
Children’s work from Janine Davenall, Helen Williams, Rachel Fleming, Sharon Palfreyman
©Early Childhood Mathematics Group 2021
We invite families and practitioners to send us examples of children’s mathematical graphics to accompany this piece. Please add a sentence to explain the context, the age of the children and attach them to an email to us.