Research exploring the development of children's early drawing and early writing has opened our eyes to what had been hidden. Understanding this development has helped teachers to recognise and value these early marks, to start from children's personal, informal understanding. When this happens, children respond in often highly creative and challenging ways and frequently exceed expectations. For children's understanding of the written language of mathematics, there is significant value in supporting and building on what they already understand.
Children's difficulties
Changes in mathematics and numeracy teaching in recent years has led to increased confidence in teaching and some gains for children. However for children, understanding of the abstract written language of mathematics continues to cause confusion for many. In is clear that there has been concern about children's difficulties with mathematics for a considerable time. In his significant study Children and Number: Difficulties in Learning Mathematics, Hughes identified the problems children experience in mathematics (1986). He showed how abstract symbols such as '+' and '=' and the abstract language (oral and written) of calculations such as '2 + 3 = 5' made no sense to young children. In his research older children continued to be confused since they had not build firm foundations: the 'school' mathematics was quite unconnected to their informal 'home' mathematics. It is this gulf between children's informal understanding and the formal written mathematics that has been so difficult to bridge.
Mathematical meaning
The problem with early 'written' mathematics is that a great deal of it has not been for real purposes, for real people (for an 'audience') or within a context that children understand. In a true sense it does not mean anything. Written mathematics has largely been produced by adults. The problem with worksheets is that they do not work. Their use has been extensive but has failed to help children better understand abstract written mathematics. This is a very real predicament that will remain unresolved until we are able to help children to bridge the gap. Children's ability to link early marks with meaning and to communicate their thinking through these marks is an important stage - both in becoming writers and mathematicians. Since the publication of Hughes's book, many studies have shown that young children entering school have a great deal of informal mathematical understanding that is largely unrecognised. The key question we endeavoured to answer was whether there was anything we might do deliberately to strengthen the links between children's informal (personal) and formal (abstract) mathematics. Our research led us to an extensive study of children's mathematical graphics.
What are children's mathematical graphics?
We coined this term to describe the full range of marks children make when exploring their mathematical thinking. They fall within the context of what is generally known as 'mark-making' in the Early Years (usually early writing and drawing). We analysed almost 700 examples of children's mathematical graphics from children of 3 - 8 years whom we taught. The samples included very early marks (scribbles); personal or 'invented' numerals; drawings; icons such as dots, drawings or tallies; numerals and writing. We have been astonished by children's capacity for understanding, invention and creativity. It is time to re-evaluate young children's considerable abilities to understand the written language of mathematics.
How can using their own ideas help children understand abstract symbols? Bi-numeracy
Using their own marks and making sense of what they do help children make personal meanings. When children's own mathematical graphics are shared and discussed, this enables children to become 'Bi-numerate'. This allows children to 'translate' between their informal 'home' mathematics and the abstract mathematics of school - to bridge the gap. Being Bi-numerate means that children can exploit their own intuitive marks and come to use and understand standard symbols in appropriate and meaningful ways. Developing personal written methods for calculations is an integral part of this (Worthington & Carruthers. 2003).
Analysing children's development
We analysed the forms (types) of graphics the children chose to use. Hughes's research (1986) had focused on some aspects of young children's early mathematical understanding, represented through their early, informal marks: we developed additional categories of forms based on analysis of our extensive sample. We then traced the development of mathematical graphics - essentially children's growing understanding of the abstract symbols and written calculations - the mathematics itself. Children respond in highly personal ways. The categories or 'dimensions' we developed highlight the children's development which arise within children's earliest explorations with marks and lead to explorations of written numerals in highly individual ways. Young children represent numerals; they represent quantities (of things) that they have not counted and other things that they do count.
Joe's spider
In
the nursery Joe (age 4yrs 3months) was playing with soft toy spiders. He later
chose to draw a picture of a spider (see figure 1). He showed his drawing to the
teacher and said, 'My spider has got eight legs.' He had drawn the spider with
many more than eight legs. Looking at the spider he saw lots of legs and
represented that idea in his drawing. Joe showed a growing awareness of number
and quantity and was able to describe it. He knew that a spider has eight legs
and you can represent that idea in your drawing. Joe has used numbers in his own
meaningful way. In our study we put this form of graphics in the dynamic range.
There is liveliness in Joe's spider. He was uninhibited in his demonstration of
his knowledge of spiders and number. We want to keep this mathematical thinking
as Joe progresses through his education. The mathematics he used is what we have
termed dimension 4a, 'representing quantities that are not counted'. This early
development ensures firm foundations for early addition and subtraction,
explored in ways that make sense to the child. One fascinating area that was
revealed is the way in which children explore symbols in non-standard ways, by
implying or inventing symbols that can be read as though the symbol was there.
This understanding leads to calculations that also include multiplication and
division.Kamrin's Tweedle birds
Kamrin
(5years 7months) was in a reception class. The children had been introduced to
several forms of division and the teacher had modelled the children's ideas of
division. Kamrin chose the number eight to work out if it could be shared
equally without leaving a remainder. Kamrin represented this in his own unique
way (see figure 2). He created the 'tweedle birds', giving each bird four eggs.
He then wrote the numeral eight and a question mark. At first he decided that he
could not share eight equally and put a cross. He checked this and then decided
that he could do so. He scored through his cross and put a tick beside the
cross. Kamrin used a combination of symbolic and iconic forms of graphics. He
used early mathematical operations which we have termed the fifth dimension.
Kamrin's use of his own mathematical graphics had moved on from Joe's since he
not only progressed beyond representing quantities but used his knowledge of
counting to find out about division. He has also skilfully self-corrected.When calculating with larger numbers children may also decide to integrate the 'jottings' they have been taught (such as the use of an empty number line) and sometimes draw on earlier strategies.
Frances and the train
The reception / Y1 class had visited a market town by train. On our return, Aaron remarked: 'I bet there were a million seats on the train!' We talked about how we might find out and after some discussion one child suggested we phone 'the train people'. Aaron phoned and told us there were 75 seats in each carriage and 7 carriages on the train. I used this as a meaningful problem to solve, inviting children to find their own ways of working out how many seats there had been altogether on the train. Ideas and graphical responses were chosen by the children and were highly differentiated. Some chose to draw a single carriage filled with as many seats as they could fit in and others used iconic responses such as squares or dots to represent seats. Two children decided to explore repeated addition by selecting some large scallop shells and placing two wooden beads in each. Frances, 6:1 years, used a range of responses, exploring several different ways including '75' written seven times. She then drew a carriage with 75 seats and decided to re-count (check) what she had done. Finding that she had represented 76 seats she crossed one out.
I was impressed by Frances's ability to represent the 75 seats accurately. Although I did not expect her to multiply 75 x 7, I wondered if she saw any possible next steps towards a solution. Smiling, I remarked 'but there were seven carriages.' For a moment Frances looked puzzled, then burst out excitedly 'the photocopier!' and explained she'd need 'six more'. When the additional six copies were laid on the floor with her original, this was a very powerful representation of repeated addition for all the children. When Frances begins to explore more standard forms of multiplication she will have a deep level of understanding on which to build.
Frances and the train - '75 seats’ (developed to ’75 x 7')
Does creativity have a place in mathematics?
The highly individual ways in which children explore their mathematical thinking on paper helps them make sense of what must appear to them the alien language of written mathematics. Creativity in mathematics education relates to the processes and particularly the personal and thinking skills that children can develop which will support deep levels of learning. It has been described as: ‘the quality of the thinking taking place - a breakthrough in thinking - a creative moment and significant to learning' which can 'initiate powerful and challenging learning experiences and harness the imaginative power of individuals’ (Talboys. 2003). Creativity extends to all aspects of the curriculum, to all aspects of human development - including mathematics. The QCA's guidance for the Foundation stage emphasises: ‘Effective teaching requires practitioners how help children see themselves as mathematicians’ (2000. p. 71). For children to become (young) mathematicians requires creative thinking, an element of risk-taking, imagination and invention - dispositions that are impossible to develop within the confines of a work-sheet or teacher-led written mathematics.
Thinking and creativity - meaning making
Creativity has often been linked to the arts such as painting, music and dance. We need to think wider than this - in every area of the curriculum children can be encouraged to think creatively - to innovate, to take risks and tackle problems in their own ways. The National Curriculum 'Handbook for Teachers' proposes that ‘Creativity is a skill that needs to be promoted across the curriculum. Creative thinking should enable pupils to generate and extend ideas, to suggest hypotheses, to apply imagination and to look for alternative outcomes’. (QCA. 1999a. P.22)
When teachers encourage children to decide how they will 'put something down on paper' and listen to what children say about their marks, they are provided with new insights into children's development. Children need to make sense in their own ways rather than colouring-in ours. And young children are very good at making their own sense through their own marks and symbols, as these examples show. The dominance of worksheets - highlighted by OFSTED as a key issue in a recent report - will never change unless practitioners support children's own ways of exploring their mathematical thinking so that they make strong connections with their own understanding.
Official guidance
Valuing children's own ways of representing their mathematical understanding is supported by a growing body of research. The government are keen to promote more flexible ways of working, and innovation and creativity are encouraged. The Curriculum Guidance for the Foundation Stage (mathematics) emphasises: