Making Sense of Mathematical Graphics: The Development of
Understanding Abstract Symbolism.
European Early Childhood Education Research Journal, Volume 13 No. 1 2005
Elizabeth Carruthers & Maulfry Worthington
Children’s Mathematics Network
In this paper we develop our theory of Bi-numeracy and show the importance of children’s own invented symbolism.
Most studies to date have concentrated on the analysis of children’s number representations in clinically set-up tasks (Hughes, 1986; Sinclair, 1988; Munn, 1994). These studies have added to our knowledge of and understanding of children’s mathematical marks. Our research differs in that we based our study in children’s homes, nursery and classroom contexts. Rather than being clinical researchers our role has been that of participant observer, based on ethnographic research and grounded theory.
We have analysed almost 700 examples of mathematical graphics. These cover all aspects of number and mathematics from the wider mathematics curriculum. They range from child-initiated marks within play to adult-directed sessions in which the children also chose what they wanted to put down on paper. All the samples have come from our own classes or those in which we have been invited to teach. Based on this large sample of original children’s marks from authentic teaching situations, our findings are therefore evidence based. We have grouped examples to show where some clear patterns emerged and developed categories that will support teachers’ assessment of children’s developing mathematical understanding.
We will show the range of these mathematical marks from early play exploration to later written calculations. Our central thesis is that there is a huge need to more openly support children’s thinking and mathematical thinking in particular.
Nous développons dans cet article, notre théorie de la « Bi-numératie » et démontrons l’importance du symbolisme inventé par les enfants eux-mêmes.
Jusqu’à présent, la plupart des études se sont concentrées sur l’analyse des représentations numériques des enfants, dans des tests réalisés en clinique (Hughes, 1986 ; Sinclair, 1988 ; Munn, 1994). Ces études ont permis d’améliorer notre connaissance et notre compréhension des signes mathématiques des enfants. Notre recherche est différente dans le sens où nous avons réalisé notre étude aux domiciles des enfants, ainsi que dans des crèches et des écoles. Plutôt que d’agir en tant que chercheurs cliniques, nous avons joué un rôle d’observateur participant, qui effectue une recherche ethnographique et obtient une théorie ancrée.
Nous avons analysé presque 700 exemples de graphiques mathématiques. Ceux-ci couvrent tous les aspects des chiffres et des mathématiques du programme scolaire mathématiques le plus large. Ces graphiques vont des signes émis par un enfant lors de jeux, à des session dirigées par un adulte, où les enfants choissaient aussi ce qu’ils voulainet noter. Tous les échantillons viennent do nos propres classes ou de celles dans lesquelles nous avon été invités à enseigner. Par conséquent, sur las base de ce vaste échantillon de signes originaux d’enfants, provenant de situations authentiques d’enseignement, nos résultats sont basés sur des données probantes. Nous avons groupé les examples afin d’indiquer l’émergence de modèles distincts et nous avons développé des catégories, qui appuieront l’évaluation des enseingant concernants la compréhension mathématique en développement des enfants.
Nous indiquerons l’éventail de ces signes mathématiques, de l’exploration dès les jeux de la petite enfance, aux calculs écrits ultérieurs. Notre thèse centrale repose sur le fait qu’il y a un besoin important de soutenir plus ouvertement la réflexion des enfants et en particulier la réflexion mathématique.
En este papel desarrollaremos nuestra teoría de Bi-numeración y demostraremos la importancia de los simbolismos inventados por los niños.
La mayoría de los estudios hasta el presente se han concentrado en el análisis de las representaciones numéricas de los niños en tareas clínicamente preparadas (Hughes, 1986; Sinclair, 1988; Munn, 1994). Estos estudios han aumentado nuestro conocimiento y comprensión de las marcas matemáticas de los niños. Nuestra investigación difiere en que basamos nuestro estudio en el contexto de los hogares de los niños, guarderías y aulas. En vez de ser investigadores clínicos nuestro rol ha sido participar como observadores y está, basado en la investigación etnográfica y en la teórica aceptada.
Hemos analizado más de 700 ejemplos de gráficos matemáticos. Esto cubre todos los aspectos numéricos y matemáticos del más amplio currículo matemático. Con un rango desde marcas iniciadas por niños en un juego hasta sesiones dirigidas por adultos en las que los niños también decidieron lo que querían poner en el papel. Todas estas muestras provienen de nuestras clases o de aquellas en las que se nos ha invitado a enseñar. Basado en esta amplia muestra de marcas originales de niños en situaciones autenticas de enseñanza, nuestros resultados están, por tanto basados en la evidencia. Hemos agrupado los ejemplos para mostrar donde emergen patrones claros y se desarrollan categorías de apoyo para la evaluación de los profesores del desarrollo de la comprensión matemática de los niños.
Demostraremos el rango de estas marcas matemáticas desde las exploraciones de los primeros juegos hasta los cálculos escritos más adelantes. Nuestra tesis central es que hay una enorme necesidad de apoyar más abiertamente los pensamientos de los niños y en particular sus pensamientos matemáticos.
Mit dieser Arbeit möchten wir unsere Theorie der „Bi-Numeracy“ (auf zweifache Art von Kindern gezeigte numerische Fertigkeiten) und die Wichtigkeit eines Symbolismus vorstellen, der von Kindern selbst entwickelt wurde.
Bis heute haben sich die meisten Untersuchungen schwerpunktmäßig mit der Analyse von Zahlendarstellungen durch Kinder in klinisch durchgeführten Versuchen (Hughes, 1986; Sinclair, 1988; Munn, 1994) befasst. Diese Studien haben dazu beigetragen, unsere Kenntnisse und unser Verständnis mathematischer Zeichen, wie sie von Kindern angefertigt wurden, zu erweitern. Im Unterschied zu diesen bisherigen Vorgehensweisen fanden unsere Untersuchungen und Beobachtungen bei Kindern zuhause, sowie in Kindergärten und Schulen statt. Unsere Rolle in diesem Zusammenhang war nicht so sehr die eines klinischen Forschers als vielmehr die des teilnehmenden Beobachters, auf der Grundlage ethnographischer Untersuchungen und begründeter Theorien.
Wir analysierten insgesamt fast 700 Beispiele mathematischer Darstellungen. Dazu gehören alle Aspekte von Zahlen und Mathematik aus einem erweiterten mathematischen Lehrplan. Das Spektrum reicht dabei von Zeichen, die die Kinder selbst in Spielsituationen initiiert haben, bis hin zu Arbeiten unter der Anleitung von Erwachsenen, bei denen die Kinder auch selbst bestimmten, was sie zu Papier bringen wollten. Alle aufgezeigten Beispiele stammen entweder aus unseren eigenen Klassen oder aus Klassen, in denen wir zum Unterrichten eingeladen wurden. Aufgrund der umfangreichen Beispiele von Originalzeichen und Darstellungen, wie sie von den Kindern in authentischen Lehr- und Lernsituationen erstellt wurden, sind unsere Ergebnisse auf Beweise gestützt. Wir haben Beispiele in Gruppen zusammengefügt, um aufzuzeigen, inwiefern dabei klare Strukturen erkennbar sind, und haben Kategorien entwickelt, um es für Lehrer leichter zu machen, das sich entwickelnde mathematische Verständnis von Kindern zu bewerten.
Wir zeigen den Bereich dieser mathematischen Zeichen von frühen Anwendungen in Spielsituationen bis hin zu späteren schriftlichen Berechnungen. Als zentrale These unserer Arbeiten stellt sich heraus, dass nach wie vor ein großer Bedarf darin besteht, die Denkweise von Kindern, insbesondere ihre mathematische Denkweise noch offensichtlicher zu unterstützen.
Key words - mark-making; mathematics; bi-numeracy; abstract symbolism.
Our own work (Worthington and Carruthers, 2003a) highlights the difficulties young children experience when they move from home to the increasingly abstract symbolism of school mathematics. In becoming bi-numerate children move between their own ‘free range’ intuitive mathematical graphics and standard forms and methods of mathematics they meet in school. Our evidence shows that Early Years teachers often fail to recognise children’s own mathematical graphics which subsequently means they do not support these graphics. We argue that it is crucial that teachers recognise and develop children’s own ‘written’ mathematics, because in doing so they will help children translate between their own informal marks and later abstract symbolism. But what do these graphics look like? When we discussed children’s marks with a group of Early Years settings from different geographical regions in England, many of the teachers discussed the point that the children’s marks that they now saw as mathematical, they would previously have thought of as early writing (Carruthers & Worthington, 2003a). Therefore in this study, we attempt to uncover what children’s mathematical graphics look like and the possible development.
So far there has been little research on children’s own informal marks in mathematics and most studies have concentrated on the analysis of children’s number representations in clinically set up tasks. For example, a major study of this kind is Hughes’s study of pre-school children’s representations of small quantities (1986).
The research on which this paper is based, is significant in highlighting the ability of very young children to choose and use their own marks to represent quantity (see Worthington & Carruthers, 2003a) and explored a wide range of original and important questions concerning the teaching and learning of written mathematics for children between the ages of 3 – 8 years that we researched over a period of twelve years.
The focus of this paper highlights many original findings concerning young children’s creative mathematical development as they represent their thinking on paper. It is also the first instance that a taxonomy of children’s ’written’ mathematical graphics has been developed.
Young children’s first marks - sometimes referred to as ‘scribbles’- are a major development in a child’s early steps towards multi-dimensional representations of her world. Malchiodi recognises that a child’s first scribbles symbolise a ‘developmental landmark’, since they now can make connections with their actions on paper to the world around them (Malchiodi, 1998). Gardner (1980) found that children also name their scribbles which may imply that they attach meaning to these marks: this again is a major step for children because not only have they chosen and made actions within the paper but they have, in some cases, expressed their meaning. Malchiodi suggests that if children are giving meaning to their scribbles then they may be moving forward in their development of representational images. Before the advent of speech many infants ‘form in visual media a powerful expressive and communicative language’ which is not recognised by many as significant (Matthews, 1999, p.29). Indeed, many of the studies of early drawings refer to scribbles in an almost scientific way and they address the scribbles as something that are useful for later drawing, (for example Burt, cited in Selleck, 1997; Kellog, 1969). Fein (1997) elaborated what she calls the ‘visual vocabulary’ to describe children’s early marks, whilst Engel (1995) focuses on descriptions that stress meaning.
In his study of the development of children’s art, Matthews observes that in almost all studies of children’s drawing, a wide range of different marks are labelled ‘scribbling’. These ‘scribbles are’, he proposes, ‘products of a systematic investigation, rather than haphazard actions, of the expressive and representational potential of visual media’ (Matthews, 1999, p.19). Scribbles are not careless accidents without worth but have significance for the child. ‘At every phase in the development of the symbolic systems used by the child are legitimate powerful systems capable of capturing the kinds of information the child feels is essential’, (Matthews, 1999, p.32). After visiting the nurseries of Emilio Reggio in Italy, Selleck argues that ‘scribbles’ is a derogatory term for young children’s art’ (Selleck, 1997).
We can conclude from the research therefore, that there is a strong case that children’s first marks on paper cannot be dismissed as a generic ‘scribbles’ stage, because children are expressing in form and content; identities, structures, symbols, events, objects and meaning of their worlds.
Studies of mathematical marks
We have already mentioned the importance of Hughes’s (1986) work. In an attempt to find out the responses of how children (aged three to six years) made marks on paper when representing quantities, he devised a task for children asking them to ‘put something on paper’ to show how many bricks were in a tin. Hughes found that children used a variety of responses that he was able to categorise into forms of graphics. He also discovered within this study that older children when given the choice would not readily use standard symbols when representing operations. Hughes stated that children find it very hard to represent the operations of addition and subtraction and some would rather show these operations by representing a quantity. He puts forward that this may explain why children’s understanding of these symbols (addition and subtraction) does not go beyond the context in which they are taught. It was surprising to us that not one of the older children (of 7 and 8 years of age) in Hughes’s study used conventional additional and subtraction signs, whereas in many of our samples we found that many children had chosen to use standard operator signs, including some as young as five years of age.
Further work developed from Hughes study (1986) does not go beyond replications of the tins game (above) or explanations of the tins game to show that children can record their own marks (Montague-Smith, 1997; Pound, 1999). Two smaller studies basing their work on the tins game (Munn, in Thompson, 1997 and Vandersteen, 2002) validate much that was discovered in Hughes’s study. Munn however suggests that the children in her study used the pictures they drew as a memory aid and not as a communication of quantity. Vandersteen (2002) disagrees with this and suggests that this could be seen as a point where these children are realising that their pictures are communicating and where links between the concrete and abstract are beginning to be made. Williams (1997) would agree that the answer to children’s difficulties with formal mathematical symbols might lie in developing children’s own mark-making.
Atkinson (1992), whilst not addressing children’s marks directly, provided a range of teachers’ stories showing that children are capable and competent of producing a range of their own mathematical marks within real classroom contexts. In two separate investigations, both Gifford (1990) and Pengelly (1986) set up single case studies, and, in doing so they identified the richness of children’s own methods in real teaching situations. However both of these studies were of one task, in one class and, because they were short studies, the researchers were unable to analyse the development of children’s own mathematical graphics. In our own extensive study of children’s own mathematical graphics in natural contexts (Worthington and Carruthers, 2003a), we were able to identify children’s own graphics in nurseries, schools and at home. Because we were the teachers in authentic teaching situations, within our own study we also analysed how to support and assess these marks.
In summary, recent research and literature on children’s marks have identified the importance of understanding very young children’s representations (see Matthews, 1999). It has also been identified, especially in clinical tasks, that children can and do use their own mathematical marks if given the choice. Furthermore, there is some evidence that children’s ability to produce their own marks can happen in a much broader context (see Gifford, 1990 and Pengelly, 1986). From the findings of our larger evidence-based study (Worthington and Carruthers, 2003a) we agree that supporting children’s own mathematical graphics is both possible and realistic in classroom practice. From our research with teachers throughout England, we have also shown that others can develop their understanding and practice to support children’s mathematical graphics (Worthington, 2004; Worthington & Carruthers, 2003b).
In this paper we report on one part of our extensive study (Worthington and Carruthers, 2003a). The aim of this study was to analyse children’s own graphics - not in clinical research tasks - but in real classrooms and homes. The graphics analysed came from a range of settings which included home, nursery classes, nursery schools and Foundation and key stage one classes in state schools, and covered the three to eight years age range. The settings were in rural, suburban and inner city areas. We did not choose any particular group of children and the samples analysed came from a diversity of cultural backgrounds and included children with special needs. We collected over 700 samples to analyse.
Methods and Approaches
This is a qualitative research study based on naturalistic enquiry methods which adopt an inductive approach. It is an ethnographic study (Hoshmand, 1989) which attempts to study a group in their naturalistic surroundings, in this case, children at home and in school. The study also draws upon phenomenology inquiry (Polkinghorne, 1989) which attempts to describe and elucidate human experience.
The context of the samples
The samples of the children’s graphics were taken from our own teaching situations over a period of 12 years and in the family. They came from different teaching contexts from direct teaching to children’s free play.
We were influenced by the developmental literacy movement (e.g., Hall, 1987; Cambourne; 1988, Goodman, 1986) and their emphasis on environments that are alive with print and that see children’s attempts at their own writing in a positive way.
Some of the features of the direct teaching were:
· open questions
· interactive style of teaching
· providing models from the teacher and the children (see Worthington and Carruthers, 2003a)
· blank paper was always provided
Some of the free play features were:
- mathematical equipment
- a variety of writing paper and writing utensils
- free access to all children
- mathematical equipment for use when needed
- a variety of writing materials
- forms and graphs
Features of the environment of the classroom both inside and out include the provision of writing and graphics materials that were always readily available. We ensured that numbers and mathematics occurred in purposeful contexts (for example children’s own graphics on the walls, notice-boards).
The children’s graphics in the home were their own through choice, sometimes in play, sometimes in spontaneous writing/drawing periods.
We recognise that the pedagogy is significant in supporting children’s meaning making through their own graphics. However, the pedagogy is complex to describe and would require greater detail that space permits in this paper, in which our focus is on the children’s mathematical graphics themselves. We intend to focus on the pedagogy of children’s mathematics in a future paper which will enable us to explore this in the detail that befits this important process. Examples of open mathematical tasks are elaborated in Worthington & Carruthers, 2003a.
How did we assess the children’s marks?
We used what Torrance (2001), refers to as ‘divergent assessment’ as a basis of uncovering children’s thinking. This kind of assessment aims to discover what the learner knows, understands and can do and which Berry, (2003) argues, ‘is more in line with current theories of learning, which advocates learner-centredness’ (p.5). Divergent assessment focuses on ‘aspects of learners’ work which yield insights into their current understanding - and on prompting metacognition’ (Torrance & Pryor, 1998, p. 153). Characteristics of this kind of assessment are:
· A socio-cultural view of learning
· An intention to teach in the zone of proximal development (Vygotsky, 1986)
· This acknowledges children’s partial knowledge (Athey,1990)
· A view of assessment as accomplished jointly by teacher, child and family
Teaching, learning and assessment are intertwined in the model we used. As teachers we refined our teaching as time went on, to become more in tune with the children’s own graphics. Stenhouse (1980) compares good teaching to artists who learn through the critical practice of their art: exploration and interpretation lead to revision and adjustment of idea and of practice. Teaching is complex and it is not easy to give an exact description. However it was essential that we used a teaching model to yield much broader and more realistic data which we felt teachers could use in their classrooms; if we did otherwise we may have gone down the route of replicating yet another ‘tins game’.
Analysing the Data
We used the constant comparative method (Glaser and Strauss, 1967) to analyse the samples. We coded the data into as many themes as possible and over a time hypothesised and gradually refined these themes as definite categories as each one emerged. This was similar to grounded theory, used by Cambourne (1988) of the developmental literacy movement.
To validate the research we used inter-rater reliability to judge the samples and the extent to which we agreed.
We carried out a pilot to determine the level of reliability between the two of us. Taking a ten per cent random sample of the total number of samples, we divided them equally between us and individually assigned each example to a category. We had previously developed categories for both the forms (appearance of the marks) and the dimensions (the mathematics) and these provided our initial categories. We then exchanged this set of samples, repeated our analysis and calculated the percentage of examples on which we had agreed. With some refinement to the categories or dimensions, we then analysed the entire 700 samples.
To ascertain inter-rater reliability for the total sample of this categorical data, we calculated the number of agreements divided by the total number of samples. We were in agreement in 97.4 percent of instances.
Inter-rater reliability was the main system we used. We recognise that there are many instances in which inter-rater reliability may be improved through use of multiple raters. Nevertheless, in this instance it was vital that those rating the examples also knew and understood the context in which they were made. For this reason we acted as raters for coding the data (samples of graphics).
We also used the technique called ‘saturation’ (Glaser and Strauss, 1967) where we repeatedly tested the data in an attempt to modify or falsify it. This involved continuous discussion and questioning about the mathematics and the children’s meanings. This combination of methods of analysis ensured greater reliability.
Since the data yielded a wealth of evidence, we can only give an overview in this paper: (for a more detailed account of this study please see Worthington and Carruthers, 2003a). As teachers, we were very familiar with children’s development in writing but as we looked at the 700 samples of mathematical graphics and searched for patterns, we constantly revisited our hypothesis, recognising that mathematical graphics are more diverse. This is perhaps so because children are not moving to common forms of a written language such as English or Greek but because the different areas of mathematics often suggest quite different graphical approaches. For example, in representing data young children may use pictures or ticks and move towards increasingly clear layouts. In early written addition, for example, children usually begin with continuous counting before separating sets.
First it is important to show the forms of the graphics children in our study were using. Secondly, we show that the mathematics they chose to use within these forms, (which we categorised and termed dimensions), is even more significant as this indicates the children’s mathematical thinking.
Common forms of graphical marks
In analysing our examples of children’s marks and written methods we have identified five common forms of graphics, including three of Hughes’s categories: the categories ‘dynamic’ and ‘written’ are two new forms that we identified from our analysis.
We use this term to describe marks that are lively and suggestive of action.
Such graphics are ‘characterised by change or activity (and) full of energy and
new ideas’ (Pearsall, 2001). We categorised Charlotte’s ‘hundred and pounds’
(see figure 1) and Amelie’s dice game (figure 2) in this way since both
pieces have a freshness and spontaneity.
|Figure 1 : Charlotte: 4 years 2 months||Figure 2 : Amelie: 4 years 5 months|
We have used Hughes’s definition, ‘that the children should be trying to represent something of the appearance of what is in front of them’ (Hughes, 1986, p.57). For example, in figure 3 Karl was representing the tables that he had just counted in the classroom. In another example, Britney drew the strawberries that were on plates in front of her, which she subsequently used to add, and then ate.
|Figure 3 : Karl: 4 years 9 months|
These marks are based on one mark for one item when counting. We found that children whose marks are iconic use discrete marks of their own devising (often be reflected in the popular use of tallies that are taught in some schools). However, when children choose ‘marks of their own devising’ (Hughes, 1986, p.58), we find that tallies are only one of the many iconic forms. Scarlett used circles to represent one group of teddies and squares to represent another (figure 4). Since none of the iconic symbols children chose were suggested by us, it is difficult to know what their origin is. Scarlett may have used circles and squares since they were quicker to draw than separate teddy bears. By adding a few details she turned the circles into balls and the squares into presents. Other children chose, for example, hearts, lorries and spirals many of which may be their current schema (Athey, 1990). The type of iconic marks children make are important to them and if they choose their own instead of a teacher imposed, standard tally mark, then they have understood the concept and applied it in their own chosen way. The important point is they have applied the one to one principle (Gelman and Gallistel, 1978).
Figure 4 : Scarlett: 4 years 9 months
Within our teaching we would encourage children to move towards more efficient forms for their count rather than focus on beautifully embellished drawings. Some children chose to move from detailed drawings to quicker methods within one mathematics session: this is because they may have shifted their focus to the mathematics.
Using words or letter-like marks which are read as words and sentences are common in our examples and are also documented in Pengelly’s (1986) and Hughes’s (1986) work. In our culture written communication is evident everywhere and children come to see this as a meaningful response on paper. We collected some examples of children’s written explanations and written methods entirely in words, as figure 5 shows. In this example John wrote ‘2 grapes’, there is (are) two. 4 grapes, there is (are) four. 6’. His addition calculation is a form of narrative, relating a sequence of events or numbers.
Figure 5 : John: 5 years 5 months
Children using symbolic forms use standard forms of numerals (for example ‘2’ ‘18’). We found that children begin to incorporate standard (abstract) symbols such as the addition and subtraction signs gradually into their mathematics. At first it may appear in dynamic form as in Amelie’s dice game (figure 2); then we found that some children used a combination of forms of graphics with the standard operation in symbolic form (see Francesca, figure 6).
Figure 6 : Francesca: 6 years 1 month
Idiosyncratic - or meaningful?
We have found that the categories Hughes (1986) developed to be a helpful starting point to understand and assess children’s graphical forms of mathematics. Hughes’s study helped many teachers recognise that children could use their own marks to represent numerals that they could then read. For teachers who understand and support early ‘emergent’ writing this has resonance. However, after careful consideration, we decided not to use Hughes’s ‘idiosyncratic’ category.
The term idiosyncratic was used by Hughes when the researchers ‘were unable to discover in the children’s representations any regularities which we could relate to the number of objects present’. Many of the idiosyncratic marks in Hughes’s study, in which children’s response was to ‘cover the paper with scribble’ could have perhaps related directly to the number of bricks the children counted (see Hughes, 1986, p.57). However, it appears that the children had not been asked to explain their marks. In this clinical method of interviewing young children there is a flaw, in that young children may not wish to respond to a stranger. They might have responded more openly to a teacher or some other key adult in their life.
We argue that these idiosyncratic responses are significant and need to be understood by Early Years teachers in order that they can support children’s early mathematical communication. It is easy to disregard scribbles and what appear to be idiosyncratic responses, if we are unable to readily understand their meaning. As experienced Early Years teachers we expect children’s early marks and symbols to carry meaning for the child. Whilst we can only conjecture about the possible meanings of some marks, we do believe that young children’s marks carry meaning.
The Transitional Period
We found that children use a combination of two forms of graphical marks, for example iconic and symbolic, when they are in a transitional period. It appears that when they do this they are moving from their familiar marks towards new marks, although they are not yet ready to dispense with non-essential elements. As their thinking develops, children appear to progressively filter out everything but what is necessary to them at the time. Children also return to less developed graphical forms when they find the mathematics presented more challenging: they draw on previous knowledge and forms with which they feel more confident For example Alison, (7 years 2 months) used iconic forms to calculate the 99 times table (see figure 7). This transitional period may be very important as children move towards the abstract forms of mathematics.
Figure 7 : Alison: 7 years 3 months
Hughes concentrated his study mainly on the forms of graphics children used. He also explored children’s strategies for representing addition and subtraction in two tasks with older children (from 5 to 9 years of age) which show a range of strategies they chose to use. Hughes emphasised that they regarded some aspects of the children’s responses to the research tasks as ‘unsuccessful’ (p. 74). However in our own research, we have looked closely at two of the features that concerned Hughes – when children put only the answer and the fact that not one child in their study used conventional operator signs (discussed earlier in this paper). From our research we have been able to show how the full range of children’s responses can provide us with important feedback about the children’s thinking. We see such features as significant (but not unsuccessful) aspects of children’s mathematical development which play a positive role in their development.
We argue that whilst the forms of graphics are significant, the mathematics children choose to use is even more enlightening and more complex. In the next section of this paper we explain our findings in relation to the dimensions of mathematical graphics, and discuss the taxonomy of children’s development of mathematical understanding, through their graphics.
The Dimensions of Mathematical Graphics
From their early play and marks, through counting and their own written methods that children choose to use, we have identified five dimensions of mathematical graphics. The taxonomy of children’s mathematical graphics that we developed from our research is original and based on our analysis of the 700 examples we collected and our observations of children over a period of twelve years. Since most ‘written’ mathematics that children do in Early Years settings is either on published worksheets, or on blank paper at the direction of the teacher (Worthington & Carruthers, 2003b) we argue that the written responses children make fail to show or support their mathematical meaning making. In order to understand and support children’s development, it is important that teachers understand and recognise this significant early development: this taxonomy therefore provides a valuable tool for teachers, practitioners and educators in assessing children’s own mathematical symbolism.
These dimensions span the period from 3 year olds in the home and nursery, through to children of 7 and 8 years old in school. They are:
1. Early play with objects and explorations with marks
2. Early written numerals
3. Numerals as labels
4. Representations of quantity
5. Early operations: development of children’s own written methods
These are not necessarily hierarchical but we found that the first four dimensions precede the fifth.
The first dimension: early play with objects and explorations with marks
Making marks on a surface for example, with a pen with fingers using any available media has a history. They arise from the infant’s gestures and both precede and accompany a child’s first marks (Vygotsky, 1983; Trevarthen, cited in Matthews, 1999). Children may be using ‘their own body actions and actions performed upon visual media to express emotion’ (Matthews, 1999, p.20). For example one of us observed a baby using her finger in her custard bowl to make marks at the bottom of her bowl: as Kress has documented, there are multi-modal ways of making meaning ‘before writing’ (Kress, 1997) and in this instance custard provided the medium. Children’s mathematical marks are only one of the ways they use marks to communicate and carry meaning.
One particular way that we found children exploring with marks is through their schemas (Athey, 1990). For example, Naomi(aged 4 years 10 months) used clay to explore several concepts. She rolled out the clay into cylinders, exploring length, width and perimeter of the table: she was engaged in this self-chosen activity for 25 minutes. As Athey revealed in her important study of schemas, action, thought and marks are interrelated (Athey, 1990).
Children in the study combined their mark making on paper with action, often transforming what they had done.
In the nursery, I had set out a selection of baskets with corks, shells, coins and fir cones. Nearby I put small bags, different sized paper bags and a variety of small boxes. This example of Cody (3 years and 5 months) is a transitional piece, linking play with objects and the marks he made on paper.
As the children filled and emptied containers they used mathematical language ‘inside’, ‘full’, empty, enough’ ‘lots’ and ‘more’. Cody picked up several items and decided to draw round them on paper. As I watched, he then carefully screwed up his paper and then put the paper inside one of the paper bags which already held some of the bottle tops and fir cones which he had drawn round. The paper with its marks appeared to have almost become an object in its own right, Cody’s marks ‘re- presenting’ the three objects in the bag.
Kress points out that there is a strong ‘dynamic inter-relationship between available resources’ - in this case the pen, fir-cones and bottle tops - ‘and the makers’ shifting interest – ‘while it is on the page I can do mental things with it ....when it is off the page I can do physical things with it’ (Kress, 1997, p.22).
It is this fluidity that is important as children move from drawing, to modelling to clay, to small world play or to bricks, often keeping the same theme and exploring all the range of ‘set’ subject boundaries that they will meet in school. One of the areas that we found they explored is mathematics.
The second dimension: early written numerals
Children referred to their marks as numbers and began to explore ways of writing numerals. Some children used personal symbols that related to standard numerals. We found that children’s perception of numerals and letters are of symbols that mean something, first differentiated in a general sense, ‘this is my writing’; ‘this is a number’.
Young children’s marks gradually develop into something more specific when they name certain marks as numerals. At this stage their marks are not recognisable as numerals but may have number-like qualities. This development is similar to the beginning of young children’s early writing (Clay, 1975). Children also mix letters and numerals when they are writing. They appear to see all their marks as symbols for communication and at this early stage their marks for letters may be undifferentiated from their marks for numerals.
An example of this dimension is Molly aged 3 years 11 months (figure 8) who made separate marks which are also number-like. Molly referred to her marks as numbers ‘7, 6 and number 8’.
Figure 8 : Molly: 3 years 11 months
Our evidence shows that in this dimension, children make very personal responses that communicate meaning which is very similar to children’s development in writing (Clay, 1975). Children know that:
Children use their current knowledge to make numerical marks and the following features we found are common:
The third dimension: numerals as labels
Young children are immersed in print as symbols and labels in their environment, in the home, on television and from their community. Children often attend to these labels and are interested in how they are used; they can write in contexts that make sense to them (Ewers-Rogers and Cowan, 1996). Children look at the function of written numerals in a social sense and by the time children have entered formal schooling they have also sorted out the different meanings of numbers (Sophian, 1995). Their personal numbers can still remain at the forefront of their minds and in some cases this can lead to confusion. For example, in teaching children who have just entered school, we sometimes initiate counting round the class. Often we do not get beyond ‘four’ as children say ‘but I’m four, that’s my number!’. They are sometimes puzzled because they interpret the number ‘four’ in the count as a description of themselves rather than part of the set.
In our samples of children’s written numerals we have evidence of their use of number in the environment as symbols or labels. This is significant because they have chosen to use numerals in different contexts: they know about these contexts and how to use the numerals within them. This shows the breadth of their understanding of the function of the numbers and their confidence of committing this to paper. Knowing and talking about these numbers is different from writing them.
When children choose to write these contextual numbers, they have converted what they have read and understood into a standard symbolic language. For example, Matthew (3 years 9 months), drew one of his favourite storybook engines, ‘James the Red Engine’ and ‘5’ which was the number of the engine (see figure 9). Matthew was very interested in numbers, not only on trains but in bus numbers and the destination of buses.
|Figure 9 : Matthew: 3 years 9 months|
The fourth dimension: representation of quantities and counting
4a. Representing quantities that are not counted
Children may represent quantity without counting. For example we can see this in figure 1, where Charlotte (4 years 2 months) depicted a quantity of ‘hundreds and pounds’ and in figure 10, of Joe’s (3 years 11 months) drawing of a spider where he showed us his sense of many legs on a spider. He said that his spider had eight legs and represented this by his drawing: Joe did not count the legs and he drew more than eight but he had a general idea. This is what Clay (1975), in her study of children’s writing called a gross approximation. We reflected that this dimension might be very much linked to the dynamic form of graphics mentioned earlier on in this paper (see graphic forms). Gardner comments on the ‘unschooled mind’ and the energetic, imaginative minds of children before they enter school (Gardner, 1993). These dynamic forms of representing quantities that are not counted are full of the risk-taking of the unrestricted learner.
Figure 10 : Joe: 3 years 11 months
4b. Representing quantities that are counted
During the same period that children represent quantities that they have not counted, we found that they also may begin to count the marks or items they have represented and represent items they have counted. When playing a dice game Amelie (figure 2) counted the dots she made on her paper each time (to represent the dots on the dice). The children in the study also represented and counted things that they could not see. For example one child drew raindrops and counted them (although it was not raining) and another child drew the pets she had at home when she was at school.
Children appear to represent items or numerals in a line, usually horizontally, before they count them. Occasionally when first representing items to count, children set them out vertically: it seems possible to us that the orientation of their paper at the moment that they begin to represent may influence such decisions. In children’s organisation of objects to count (but not represented) they also seem eventually to organise their count into horizontal lay outs (Gelman and Gallistell, 1978).
It appears that these four dimensions of early mathematical graphics provide children with rich foundations from which to develop early written calculations (the fifth dimension). From what we have observed they are able to make use of these skills and understanding to explore early written calculations in ways that are meaningful to them.
A difficulty of the study was trying to find classes that had an open culture conducive to children making their own marks, in addition to our own. In Foundation classes in schools with children aged 4 and 5 years, children were more receptive to ‘having a go’ and putting their own thoughts down on paper in the autumn term. The reason for this may be that they had not been restricted by the school culture of worksheets or by teachers expecting correct and neat layouts.
5. The fifth dimension: written calculations
We have shown how children’s understanding develops through their thinking about numbers as they explore their ideas on paper. From our research it was clear that young children are able to also develop their understanding of algorithms, through their own representations. In analysing examples of children’s calculations a pattern of development appeared which we have put into five categories. This development arises out of and builds on children’s earlier mathematical graphics that focus on number. Whilst previous research has shown how children calculate orally (and mentally) and with objects, this research illuminates for the first time young children’s development of mathematical meaning though their personal choice and co-construction of mathematical graphics. Importantly, the taxonomy shows the relationship and approximate order as their personal mathematical understanding develops and the way in which they move towards the use of standard algorithms. In this paper we focus on the strategies that children choose to use or invent, for addition and subtraction.
5a: Counting continuously
5b: Separating sets
5c: Exploring symbols
5d: Standard symbolic calculations with small numbers
5e. Mental methods supported by jottings
5a. Counting continuously
In our study of young children’s early calculations on paper we collected many examples of children’s calculations through counting. We use the term ‘counting continuously’ to describe this stage of children’s early representations of calculation (addition and subtraction strategies). Several studies have shown that children can carry out simple additions and subtractions (with objects and verbally) and that they do this by using counting strategies. The most common strategy is to ‘count-all’ or to count the final number of items (Carpenter and Moser, 1984; Fuson and Hall, 1983). Since the counting-all strategy is not one that children are taught, Hughes suggests that we can infer that this is a self- taught strategy (Hughes, 1986, p.35).
When young children are given a worksheet with two sets of items to add, they count the first set and then continue to count the second set. This is misleading for teachers, because such a page will be termed addition but children use it to count. As we show, counting is a valid and important stage in developing understanding of addition, but it is not itself addition. An example of continuous counting is Alison (5 years 1 month) who counted the children in her group and each child’s toy, to work out how many would be eating at the class ‘breakfast café’. She had represented the ‘7 children and 7 toys’ as a continuous count of fourteen numbers. Although she had originally written a string of numbers to 17, when self-checking she discovered this and used brackets to show she did not need the additional numbers.
Some features of counting continuously:
Subtraction seems to require more steps than addition. These are the features we observed in their graphics using counting continuously.
In removing some items in order to subtract the children may:
Children understand the need to count everything to arrive at a total then count those that remain after they have removed some. They show they have taken away some items by:
· rubbing out items or numerals they have represented
· crossing out items or numerals
· circling items or numerals to be ‘taken away’
· using arrows - either to show which ones have been removed, or to show
the direction of ‘taking away’
· sometimes showing the action of taking away, often drawing a hand
removing some items or numerals
· putting the total that remains after subtraction.
At this point we recognised that children often represent their graphic calculations as narratives. There is a sense that children are recounting a story, providing a strong sense of introduction, sequenced narrative and conclusion. The operants ‘+’ and ‘-’ could be said to act as ‘verbs’ and the numerals or objects as ‘nouns’. But this is not a ‘number story’ in the traditional sense, as for example when teachers ask children to make up a ‘story’ for the calculation 3 + 8.
These narratives appear similar to drawings described by van Oers - either representing imaginary or real situations - or, in a mathematical context, abstract or concrete ‘they are symbolically representing a narrative’. To ensure that all the features of the narratives are understood, children use talk: some may also sometimes add writing to qualify their representations. In his research on the way in which children use speech as an ‘explanatory function’ to help adults understand the meanings of their drawings, van Oers argues that most of the children’s explanations ‘refer to the dynamic aspects of the situation (what really happens), which they apparently feel is not clearly indicated by the drawing alone’ (van Oers, 1997, p.242 and 244). These findings are relevant to young children’s written calculations: in addition to verbally explaining, children often use symbols of their own devising to indicate their action - usually something they have done in a concrete situation. These symbols include drawings of hands holding an item or numeral (implying adding or literally ‘taking away’), and arrows pointing away from the calculation to signify removal or subtraction: we term this ‘narrative action’. For example in figure 11 William (5 years 7 months), drew the total amount of books in two piles that he wanted to add and then drew his hand linked to the one book he wanted to keep to read.
Figure 11: William: 5 years 7 months
The ‘Melting Pot’
At this stage we can see a wide variety of different representations which we refer to as a ‘melting pot’. Carpenter and Moser (1984) studied the range of strategies children use to add and subtract and found that a variety of informal counting persist through the primary years. Children’s choices of ways to represent operations do not remain static since discussion, modelling and conferencing alter individuals’ perspectives and continuously introduces them to further possibilities. Negotiation and co-construction of meanings are taken place. The melting pot period therefore covers a huge range of representations through an extended period of time. Some children in this period may choose different ways of representing their calculations as they refine their ways and ideas of representing addition and subtraction.
5b. Separating sets
Children use a range of strategies to show that the two amounts are distinctly separate. They do this in a variety of ways including:
· grouping the two sets of items to be added either on opposite sides (to the left and right) of their paper, or by leaving a space between them.
· separating the sets with words
· putting a vertical line between sets
· putting an arrow or a personal symbol between the sets.
5c. Exploring symbols
Children explore the role and appearance of symbols. Some children who have begun to make explicit use of symbols may move on to increasingly choose to use standard symbols. Other children show that they have an understanding of ‘+’ or ‘=’, but have not represented the symbols: the marks they make, or their arrangement of their calculation, show that the symbol is implied and that they understand the calculation in their head. For example Jax, 5 years 2 months (figure 12) implies symbols, using dots (iconic form) and numerals (symbolic). This can be read as ‘6 and 4 = ten’: Jax wrote the initial letter ‘t’ of the word ten. At this stage children may ‘read’ their calculation as though to include written features which are absent: speech is therefore used ‘as a means to make explicit the implicit dynamic aspects of the children’s intended meanings.’ (van Oers, 1997, p. 244). We believe this represents a highly significant point in children’s developing understanding.
Figure 12: Jax: 5 years 2 months
A noteworthy feature of dimension 5c is what is termed code-switching, a term that originated from second language learning (Cook, 2001). In terms of spoken language, code switching occurs when a speaker switches from one language to another in mid-sentence. This is similar to what we have found when children learn their second (abstract) mathematical language. For the speaker, a condition of code switching is that both speakers know the two languages used (Cook, 2001). In children’s developing understanding of abstract mathematical symbols, we repeatedly see examples of code-switching as children switch between different forms of mathematical graphics. They often use:
5d. Standard symbolic calculations with small numbers
This stage arises directly out of the preceding ones. All their previous knowledge combines to support simple calculations with small numbers. Calculating with larger numbers is challenging. There can be a problem here since children:
In the concluding chapter of his book, Hughes (1986) suggests that ‘work could be done with children’s own representations of addition and subtraction before introducing them to conventional plus and minus signs’. This could be misleading, like suggesting that in supporting children’s early writing teachers withhold standard letters, printed texts and punctuation. What is important is that we provide children with the whole picture - and for addition and subtraction this will include the standard symbols. Children can then choose what to use from a range of possibilities. They will put down what makes sense to them at the time and this might include conventional symbols.
5e. Calculations with larger numbers supported by jottings
In this dimension children will use a variety of ways of representing their mathematical thinking. They will be challenged by using larger numbers and may have to write down something to aid their memory. The mental methods that children use tend to cluster together and often reflect approaches used by their teacher. Often children can use these mental methods to help them visualise and work out the calculations, and at other times children put down some of the stages of these ‘mental methods’ on paper, to support their thinking.
Features for representing subtraction are very similar to addition on all points:
· using known number facts
· counting on from the larger of the two numbers
· using a number line with points marked on it
· using an ‘empty number line’
· partitioning numbers
· exploring alternative ways of working or checking a calculation
· the use of derived number facts
· some understanding of commutativity
In this overview of our findings we have outlined the development of children’s mathematical graphics (see figure.13). It is important to emphasise that the research was based on numerous examples of ‘written’ mathematics from children who were encouraged to really think about what they did and to build their understanding in ways that were personally meaningful to them. In this paper we have tried to show something of the level of challenge and thinking that children experience when working in more open ways and when selecting their own written methods.
Some significant findings of this study uncover ways that children move towards abstract mathematical symbolism when using their own mathematical graphics:
As we analysed the children’s mathematical graphics one overriding point became clear: that the essential reason for teachers to encourage children to represent their mathematical understanding on paper, in their own terms, is that children will come to understand the abstract symbolism of mathematics at a deep level. This allows children to translate from their informal, home mathematics to 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 symbols in appropriate and meaningful ways: developing their own written methods for calculation is an integral part of this. As children develop mathematically beyond what we have described here, continuing support for their own methods is vital, otherwise their confidence and cognitive integrity are sacrificed. There is no point where there is a definite separation of intuitive and standard methods. Children will adapt from the models they have been given and use what makes sense to them if they are encouraged to do so.
In England official documents support children’s own mathematical graphic representations (QCA, 1999, p.12 and QCA, 2000, pp. 71-2). Yet when we surveyed almost 300 Early Years teachers we found that a high percentage used worksheets for mathematics and most did not recognise and therefore did not support children’s own mathematical mark-making (Worthington and Carruthers, 2003b). Ofsted has reported that a weakness of current mathematics teaching is that teachers rely too heavily on published schemes and worksheets (OFSTED, 2002, p.554). This evidence could suggest that teachers may not have the knowledge to support children’s own graphics. We therefore return to the first question posed in this paper, which teachers have asked about children’s own mathematical graphics: ‘what do children’s own mathematical graphics look like?’. We and reflect that we have gone some of the way in answering this question and that this has considerable relevance for classroom practice. We argue that this original research provides striking evidence of the power of young children’s thinking in mathematics and their ability to make meaning. We believe that our findings have significant implications for mathematics education.
The field of children’s own mathematical graphical representations is under-researched and there needs to be many more classroom based studies in this area.
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