Published in: Primary Mathematics (Mathematics Association). November 2003. Vol. 7. Issue 3. pp. 21 - 25
Research Uncovers Children's Creative
Mathematical Thinking
Maulfry Worthington and Elizabeth Carruthers
Young children are amazingly talented, learning
at rates they will never later exceed - in every area of their development.
Understanding develops within the social and cultural contexts of their family
and community. Their capacity for learning enables them to master the
complexities of their mother tongue (and sometimes one or more additional
languages within their family) in the first few years of life. They can reason,
explain, describe, relate and argue and have a rapidly increasing vocabulary to
enable them to explore and develop these skills. This ability to understand and
use language makes it possible for them to also use marks to stand for something
else - to develop graphical and symbolic languages such as drawing, writing and
written mathematics.
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:
Different methods are often used for the same
calculation and this can lead to useful discussion (p.21), (QCA. Teaching
Written Calculations. The National Numeracy Strategy. 1999b). The recent Primary
Strategy also emphasizes creativity in numeracy and from September 2003 one of
the priority areas for consultants is ‘fostering creativity’ (Collins, 2003).
Creativity and mathematics: children learning, teachers teaching
Conclusion
The extensive research we have carried out points to ways of working with young
children that provide an alternative to worksheets and supports children's
creative thinking in mathematics. Our findings show the extent to which
encouraging more open ways of working are possible. Our evidence is that
children's mathematical graphics helps children become Bi-numerate and supports
significant learning.
Starting points
Understanding, supporting and developing mathematical graphics raises
pedagogical issues. How can teachers tune into children's thinking in this way?
Here are some useful starting points:
When children respond in a variety of ways then
you will know that you have begun to create a culture that encourages children's
individual mathematical thinking.
Creative mathematics
QCA proposes that practitioners need to support children so that they are ‘able
to communicate ideas and feelings, make connections, innovate and solve
problems. It begins with curiosity and involves children in exploration and
experimentation. As the express their creativity, they draw upon their
imagination and originality. They make decisions, take risks and play with
ideas. Children's creativity develops over time and takes time. It is best
facilitated by adults who sensitively support this process and do not dominate
it. If they are to be truly creative, children need the freedom to develop their
own ideas and the support of adults who can help them gain the skills that
enable their creativity to have expression’ (QCA. 2000. p.118).
Children's mathematical graphics provide opportunities to explore and develop
these skills. It is time that we all spoke out as advocates for creativity
within children's early 'written' mathematics.
REFERENCES:
Collins, K. National Director, Primary National Strategy. Primary Strategy: the
Future of Consultant Support.
Hughes, M. (1986) Children and Number: Children's Difficulties in Learning
Mathematics. Oxford: Blackwell.
QCA (1999a) The National Curriculum: Handbook for Primary Teachers in England,
London: HMSO
QCA. (1999b) Teaching Written Calculations. The National Numeracy Strategy.
London: QCA.
QCA. (2000) The Curriculum Guidance for the Foundation Stage. London: DfEE
Talboys, M. Project Director: QCA (2003) 'Leadership for Creativity' Leading
Edge presentation given at NCSL (March. 2003)
Worthington, M. & Carruthers, E. (2003) Children's Mathematics: making marks,
making meaning. London: Paul Chapman Publishing
Elizabeth Carruthers is a Foundation Stage Advisory Teacher: Devon LEA. Maulfry Worthington has recently been appointed as an e-learning facilitator with the National College for School Leadership. They are winners of TACTYC’s national Jenefer Joseph Award for Creative Arts in the Early Years, 2003.