Developing mathematical graphics: emergent recording

What does emergent mathematical recording look like and how might we support its development?

Research is clear that understanding formal representations of mathematics is not easy for young learners, but in the words of Martin Hughes:

“… we can help children build meaningful links between the world of written symbols and the world of concrete reality.” (Hughes, 1986: 134)

The work of Martin Hughes in the 1980s brought the importance of informal recording for young learners into focus for us as educators (Hughes, 1986). Hughes researched how 3–7-year-olds recorded quantities and the operations of subtraction and addition. He found that children who had been taught both standard numerals and addition and subtraction symbols and equations often failed to use these when asked to show on paper a task that involved a representing quantity or a change in quantity. Hughes’ famous ‘box task’ and ‘tins game’ required young children to record the number of bricks hidden inside a box. He classified children’s responses under the following four headings:

Figure 3 shows some children’s responses to a version of Hughes’ task where they have been invited to record the number of beans hidden under a pot. As you can see there are a range of responses to this invitation, which we can organise under Hughes’ headings:

Figure 3. Children’s recording of the number of beans under a pot

Hughes’ work inspired a number of studies into children’s numerical representations. For example, Ewers-Rogers (2002) gave English, Japanese and Swedish children Hughes’ box task and obtained similar results, across the cultures, to those of Hughes. She also gave 4 year olds a task that involved writing a note to a milkman to indicate how many milk cartons were wanted. Although the task seems more socially relevant, it appears this was even more difficult for the children. About half of the responses were idiosyncratic without obvious numerical relevance, demonstrating that each task brings its own challenges. Maybe these children by being asked to “write a note” did not consider numerals as part of this communication?

Mathematical ideas need to make personal sense. Children’s own graphics help them to understand the written language of mathematics and how it can be used (DCSF, 2008). Mathematical graphics offer a conceptual link between practical exploration and symbolic representation. Encouraging children to talk about their recording allows them to rehearse their ideas and provides practitioners with an insight into individuals’ mathematical understanding. Fosnot and Dolk (2001) suggest that the representations that children make on a 2D platform are not a copy of what they see but a representation of “their interactions with the object.” (2001:78). The lines drawn communicate what is known about an object physically, translated to a 2-dimensional status. So, a tree drawn as several touching circles indicates the experience of walking around it, touching the bark, feeling the shade of the leaves above. A child draws a clock with lines and dots saying, ‘tick-tock’ as he does so representing the sound experienced. In order to understand the child’s expression of their interactions with an object, we therefore need to observe closely. We need to watch and listen as children create graphics and ask them to interpret them for us. Children, as the experts in their own thinking, are able to help us as adults to understand their mathematical thinking through their graphics.

Children’s recordings are a means of communication, where emergent mathematical thinking is expressed. As shown by the work of Hughes (1986), we can see how children use marks to represent quantities. Linked to practical, problem-solving contexts such as keeping scores in a game or sharing out an amount of gold for pirates, these reveal children’s emerging ideas of applying number and calculation and allow practitioners to support their understanding of standardised symbols. They are permanent records of their thinking and as such can be used to encourage children to reflect on mathematical ideas and new information.

In Figure 4, a Reception child has represented ‘take away’ as the action of a hand:

“A hand was drawn to symbolise the action of both cars moving away. When prompted, numerals were confidently written for the original and final amounts (2 reversed and zero).” (Davenall, 2016)

The dynamic element of removing is very evident in this piece of work, something the formal symbols do not convey by themselves.

Talking with children about their graphics often leads to gesture and action as children explain how they combined two quantities, for example. This is particularly powerful where the dramatic action of a story holds significant value for children leading to heightened emotion and energy in the children’s explanations. Contexts that are meaningful for the child (part of their world) are powerful. Story contexts can add meaning for the child and are a powerful context for children’s mathematics which they express through their graphics.

Figure 5. How many hens are left when the naughty fox sneaks in and eats some?

The examples in figure 5 are Reception children’s recording of simple subtractions that were set in a context where the fox was stealing hens. In this example, children were constructing and representing stories based on a real incident, not just finding an answer to a given problem and this of course would add to the rich detail. They were given fluffy chicks to collect and draw, together with a toy fox to enact their story. The children represented their number of chickens and chose the number eaten as well as finding ‘How many were left?’. Here we see the subtraction represented by crossing out, an arrow and lines for each hen taken away, as well as the subtraction symbol used conventionally. The children have conveyed so much more than the numerical relationship. They have represented the process (as they perceive it) and the outcome. This is meaningful for the children. Subtraction is not a task with a correct answer but a dynamic situation. Conceptual understanding is developed through the focus on process and these children demonstrate their understanding of subtraction through their graphics.

The problem in figure 6 is how to share six biscuits fairly between three children. The children had been given the biscuits to move around and figure 5 shows three children’s solutions. It is clear that all three children were able to successfully solve the problem but the graphics show us so much more than this. They show the splitting of 6 into 4 and 2 as well as 2, 2 and 2. They also show us the action of sharing in division and concepts of fair sharing with equal piles or numbers of lines.

Whilst many of the examples in this paper relate to number, mathematical development is supported by mark making in all areas of mathematics. Recording a journey or a route, for example, is rich in communicating children’s mathematical thinking (for examples of early map-making, see our spatial reasoning guidance). Children’s understanding of shape and perspective is often extended and revealed through their mathematical graphics. If we look at the biscuits in figure 6, we see different ways of communicating a pile of two biscuits (where one is stacked upon the other).

Figure 8: Constructions with small figures: 4 year olds

Children’s recording of their block play is fascinating in showing how they perceive units of blocks within their construction and how they represent their three-dimensional model in two dimensions (figure 8).