text:
Sue Gifford
email: s.gifford@roehampton.ac.uk

Rose Griffiths

Jenni Back

Making numbers: Issues in using manipulatives with young children

Keywords: Early childhood mathematics, finger counting, patterns, representations, manipulatives

The Making Numbers research Project, funded by

Background to the project

Making Numbers is a Nuffield Foundation funded project to develop research-informed guidance for teachers of three to nine year olds on the use of manipulatives to teach arithmetic. The project has run for two years and includes a literature review, a survey of current practice and the development of exemplars of good practice through observation and small-scale teaching investigations. The resulting guidance has been published as a fully illustrated book for teachers with accompanying animations for use with children [START_REF] Griffiths | Making numbers: Using manipulatives to teach arithmetic[END_REF]. The findings of the project are reported in full elsewhere (Griffiths, Back, & Gifford, in press): this paper gives a brief overview, highlights some issues concerning the use of manipulatives with young children and exemplifies key pedagogical principles.

Our definition of manipulatives is "objects that can be handled and moved and are used to develop learners' understanding of a mathematical situation" (Gifford, Back, & Griffiths, 2015, p.1). This includes the use of everyday and structured materials, characterized by a pedagogical intention, which concurs with recent studies [START_REF] Swan | Revisiting mathematical manipulatives[END_REF][START_REF] Carbonneau | A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives[END_REF]. The project focus was on number sense, prioritizing for this age range counting, cardinality, comparison and composition of numbers. Hence the project title, 'Making numbers', emphasises flexibly decomposing and recomposing numbers, which was identified by [START_REF] Boaler | The elephant in the classroom[END_REF] as key to mathematical achievement.

From the survey and interviews with teacher groups, we found that manipulatives were mainly used in the early years and with older low achievers [START_REF] Gifford | Making numbers: where we are now[END_REF]. The most common were counters, interlocking cubes and Numicon 10-frame based number plates (Wing, 2001), followed by place value apparatus. Teachers' choice was influenced by commercial availability and past government policy, but for some it was serendipitous. Most expressed a lack of confidence about how to use manipulatives to teach different aspects of number.

The methodological approach of the literature review was to consider studies and theories from a range of perspectives, in order to gain insights into factors affecting children's learning. Sources included the history of pedagogy, cognitive and social constructivist theories of learning and quantitative and qualitative empirical studies. We found that experimental studies were contradictory and inconclusive, identifying only crude factors, such as length of the study or the amount of instructional guidance, as shown by Carbonneau et al.'s (2013) meta-study.

The conclusion was that the effective use of manipulatives depended on some key pedagogical principles [START_REF] Gifford | Making numbers: where we are now[END_REF]. These included:

 the careful matching of both manipulatives and activities to the mathematical focus  the identification and assessment of children's prerequisite understanding  activities involving comparison, equivalence, analysis and generalisation  discussion, requiring children to use manipulatives to justify reasoning  linking manipulatives to abstract symbols  creating an inclusive mathematics learning community.

Here we identify some issues from the literature review about teachers' use of manipulatives with young children. These concern fingers as manipulatives, discrete and continuous models for number and the learning potential of pattern activities. The pedagogical principles above are exemplified in a small case study from a teaching investigation.

Fingers as manipulatives

The benefits of drawing from a range of perspectives are evident in the consideration of fingers as manipulatives, which are of particular relevance to teachers of young children. Their significance has been underlined by neurological research: brain areas representing fingers and numbers are closely related, according to [START_REF] Wood | Numbers, space and action-from finger counting to the mental number line and beyond[END_REF]. Young children use mental finger representations for numbers more than adults, suggesting that finger use is significant for the development of number understanding. Gracia-Bafalluy and Noel (2008) found that young children's finger awareness was predictive of their mathematical competence and that training in distinguishing fingers resulted in improvements in subitising, counting and comparing numbers. However, the way that fingers are used for counting varies in different cultures, including counting three to a finger, or counting hands as fives. [START_REF] Bender | Nature and culture of finger counting: Diversity and effects of an embodied cognitive tool[END_REF] argued that the resulting number concepts also vary according to different languages, some of which support 10s structures more transparently. [START_REF] Jordan | Arithmetic fact mastery in young children: a longitudinal investigation[END_REF] also found that children from low-income families tended not to use fingers to solve problems. This suggests that home practices differentially support children's number understanding and need to be taken into account by teachers.

How should children be taught to use fingers? [START_REF] Sarama | Early childhood mathematics education research: Learning trajectories for young children[END_REF] advised teachers not to discourage children from using fingers until they were confident with mental strategies, in order to prevent reliance on finger counting. [START_REF] Marton | Constructivism, phenomenology and the origin of arithmetical skills[END_REF], using a phenomenological approach, found that older higher attainers, who used recall and derived facts, showed numbers of fingers 'allat-once', whereas low attainers continued to rely on using fingers to count on and back. 'Finger numbers' encouraged children to analyse numbers, developing subitised images and part-whole number understanding. This model therefore represents key number concepts more effectively. [START_REF] Sinclair | Mathematics using multiple senses[END_REF] reported that using the app 'Touchcounts' rapidly resulted in three year olds showing 'all-at-once' finger numbers. This seems an important skill which young children might also learn in other ways, for instance, when singing number rhymes.

Discrete and continuous models of number

One current issue relating to young children is about the relative merits of discrete and continuous models of number, for instance using either counters or colour rods which represent numbers as lengths. Usually in England numbers are introduced by counting separate items. However, when children are later introduced to number lines, numbers are represented by intervals on a continuous line and children often count the numerals or marks, rather than the intervals between them. Sarama & Clements (2009, p. 119) recommended caution in using the number line "as a representation for beginning arithmetic", raising the issue of when and how to introduce it. However, neuroscientific evidence suggests that people intuitively see some kind of mental number line (Wood & Fischer, 2015). This supports renewed interest in teaching young children about number based on measuring lengths, as proposed by [START_REF] Bass | Quantities, numbers, names and the real number line[END_REF] following the approach of Davydov (1975). Bass pointed out that number "arises from measuring one quantity by another, taken to be the 'unit'" (2015, p. 100). He argued that introducing numbers as chosen units for measuring quantities provides a more coherent model of numbers which can also include fractions. It supports multiplicative and proportional reasoning and early understanding of algebraic principles such as inverse and commutativity. This argument also supports the use of colour rods. Fuson (2009) argued that number paths or tracks are more comprehensible for young children, as they present adjacent squares which are more obviously countable than intervals on a line. [START_REF] Laski | Learning from number board games: You learn what you encode[END_REF] found that children who regularly played number track games, reading the numbers aloud as they counted moves, improved their awareness of number magnitude and arithmetical achievement. It has subsequently been argued that children's engagement with ordinal number has been overlooked as a means of developing understanding. Sarama & Clements (2009, p. 93) suggested that children were linking the distance model with counting moves and numbers spoken: connections between the numerical magnitudes and all the visual-spatial, kinaesthetic, auditory and temporal cues in the games (i.e., all the magnitudes increase together: numerals, distance moved, number of moves, number of counting words etc.) may provide a rich mental model for a mental number line.

It therefore seems that we may underestimate children's capabilities to combine the various models to construct complex networks of numerical understanding. According to [START_REF] Clements | Learning and teaching mathematics: The learning trajectories approach[END_REF], four year olds may be in the process of developing a mental number line connecting different 'quantification schemes', including discrete number ideas based on subitising and counting with continuous ideas about duration and length. Some manipulatives have attempted to integrate discrete and continuous models of number. For instance, bead strings present countable beads, usually coloured in groups of five or 10, along a line: [START_REF] Beishuizen | The empty number line[END_REF] reported that teaching children to find a number on a bead string and then on a number line helped them to develop mental calculation. Some older apparatus, such as Montessori's numeric rods and the Stern counting board, present rods arranged as 'staircases' of discrete cardinal numbers, with numerals alongside. These afford a clear image of the value of numerals in sequence increasing by one, but are not common in schools, having been replaced by rods made of interlocking cubes. An approach we have developed is to use centimetre cubes alongside a ruler: this has the advantage of providing countable objects, while demonstrating that the individual items counted are in the intervals between the numerals. This not only shows the numbers increasing by one and but also that each successive number includes all previous ones, the idea of hierarchical inclusion or 'nested numbers' (Clements & Sarama, 2009, p. 20). Combining both models also provides opportunities for children to discuss what is the same and what is different, as advocated by [START_REF] Harries | What's in a picture? Understanding and representation in early mathematics[END_REF]. This is an area where teachers need to be aware of the complexities involved in the differences between continuous and discrete models and further investigation is needed of how children reconcile these.

The potential of pattern

There is a range of possible factors affecting young children's understanding of manipulatives used to represent number relations. From Piagetian theory young children are seen as needing sensory experiences in order to learn, and from Vygotskian theory as being able to use symbols, such as fingers, from an early age. Recent theories suggest differences in young children's spontaneous and intuitive appreciation of number and pattern. [START_REF] Hannula | Spontaneous focusing on numerosity and mathematical skills of young children[END_REF] found that some very young children displayed a tendency to spontaneously focus on numerical features of a situation (SFON), while others did not, and this predicted later achievement. [START_REF] Mulligan | Awareness of pattern and structure in early mathematical development[END_REF] found that children varied considerably in their awareness of mathematical pattern and structure (AMPS) and that this was also linked to mathematical achievement. They identified different stages of AMPS: some children focused on non-mathematical surface features of patterns, some children noticed one or two mathematical elements in a pattern and others could reproduce and continue patterns, by identifying the components and relationships. Mulligan and Mitchelmore also found that pattern awareness could be taught to young children, suggesting that this facility is learned from experience: Papic, Mulligan and Mitchelmore (2011) reported a successful intervention with four year olds, which improved awareness of pattern and mathematical structures. This presents a potentially powerful approach, both in building on young children's strengths with visuo-spatial memory and in developing pre-algebraic understanding.

Staircase patterns

One of our teaching investigations, into the learning of numbers to 20 with a class of 30 six year old children, exemplified the importance of pattern awareness as well as some of the key principles identified in the literature review. In a series of small group teaching sessions, we aimed to teach children about the numbers from 10 to 20, sometimes referred to as the 'teen numbers' [START_REF] Gifford | Using pattern to inspire rich mathematical discourse in mixed attainment groups[END_REF]. We used a 'staircase' pattern of 'teen numbers' made of rods of interlocking cubes arranged in number order. The rods for one to 10 were of different colours, with the rod for ten consisting of a stick of green cubes; the rods for 11 to 19 then had 10 green cubes with the rods for numbers 1 to 9 attached, repeating the same colour sequence. Numerals were presented alongside the rods 1 to 9. We invited the children to find a single rod, such as 18. We found that children used a variety of strategies to do this: most picked a rod at random and counted the cubes from one. Magda counted in ones along the 20 rod and then along the 18 rod and explained "I knew one less would be 19 and one less would be 18". Only one child identified rods by counting on from ten. Some children made errors in counting, either in the word sequence, or in matching words to cubes when pointing. This activity demonstrated the pedagogical principle of carefully selecting manipulatives to match mathematical concepts and showed how this affected the accessibility of those concepts. We wanted the children to recognize the sticks as representing the teen numbers as composed of ten and another number, but it was not easy to see there were ten green cubes and recognize the 'ten', unlike with Numicon plates which children had previously identified as 'ten'. However, the staircase arrangement enabled some children to recognize the 'one more than' pattern, which would not have been so evident with Numicon. When trying to find the 'thirteen' rod, six year Lucasz, who was not yet fluent in English, said, 'because after 2, 3. Because it's like ten, 1, 2, 3.' In doing this he made a sweeping gesture across the 11, 12 and 13 rods, then across to rods 1, 2 and 3. Later, Magda, whose home language was Czech and who was more fluent in English, said, 'It's a bit like counting up stairs. Like counting 1, 2, 3 but 11, 12, 13.' The children thereby implicitly identified two types of pattern, a repeating units pattern and the stair pattern of equal intervals of one, which is the most basic arithmetic sequence. This was evidence of early algebraic thinking, in that they noticed mathematical features, identified the relationship between elements and observed regularities (Blanton et al., 2015, cited by Kieran, Pang, Schifter, & Ng, 2016). This activity thereby also demonstrated the principle of prompting analysis, by simply asking children to say how they identified a number. In our teaching investigations we have been impressed by the way patterns can fascinate children. On one occasion, when children were rushing by our table outside a classroom on their way to lunch, two six year old boys halted in their tracks in order to gaze at the staircase pattern and one child sat down to join us. We found that patterns like these engaged a range of children, who had differing expertise with numbers, in solving problems and noticing relationships. While some children may have had high levels of AMPS [START_REF] Mulligan | Awareness of pattern and structure in early mathematical development[END_REF], the striking pattern made with manipulatives helped all children to focus on number relationships. Presenting number patterns in this way contrasts with other approaches which advocate that early years mathematics pedagogy should be based on realistic "context situations" (van den Heuvel-Panhuizen, 2008, p. 20).

The activity also prompted discussion, stimulating both Lucasz and Magda to express themselves creatively using language and gesture, as shown above, making connections and elaborating ideas. Lucasz may have been an example of a child using gesture to express emergent mathematics learning, as described by [START_REF] Garber | Knowledge conveyed in gestures is not tied to the hands[END_REF] or he may have already been familiar with this pattern and been able to articulate it in Polish: we do not know. However, this showed that a lack of fluency did not prevent a child in the early stages of learning a language from trying to express a mathematical relationship, and also the importance of gesture in supporting mathematical discussion between children who are not using their home languages. The manipulatives pattern thereby supported the linking of different modes of representation, not only verbal and visual, but also kinaesthetic and, through the use of numeral cards, abstract symbols. This activity also exemplified the need to identify and assess prerequisite understanding. Some children could not reliably count objects to 20, despite having been assessed as doing this the previous year. Most of the children counted from one instead of counting on from 10, even when it was pointed out to them that there were always 10 green cubes. This may have been because they lacked skills of counting on or were not yet able to unitise ten as an item. As [START_REF] Cobb | Cultural tools and mathematical learning: A case study[END_REF] pointed out, if children do not have a concept of 'a ten' they will not be able to 'see' it even when demonstrated. Similarly, some children could add 10 and 3 and instantly say "13", but seemed not to understand the inverse relationship sufficiently to apply this to decomposing 13 into 10 and 3 and did not use this knowledge to identify 13 as 10 and 3. Therefore some children may have lacked the prerequisite understanding and skills to access what the manipulatives were intended to represent.

The final principle shown by this example, is that of an inclusive learning community. The children were not in 'ability' groups, as is common in English early years classrooms, in which case Lucasz and Magda would not have been in the same group. We do not know if Magda would have articulated the pattern in this way, if she had not first listened to Lucasz, who was able to hear his idea expressed differently. Latoya, in her second session, identified the 15 rod by counting on from 10, saying, '10, 11, 12, 13, 14, 15'; this was a skill she had not used previously but had observed another child using in the previous session. Both the lack of grouping and the open activity, allowing for a range of solution strategies, facilitated children in learning from others.

In our study we also observed successful practice, particularly with colour rods, which similarly showed children engaged by patterns and stimulated to discuss mathematical relationships. These experiences imply that developing children's focus on patterns is a promising avenue for mathematical pedagogy in the early years.

Implications

The issues discussed here suggest some potentially profitable avenues for future research with mathematical manipulatives and young children and some implications for practice. Firstly, we might build on home practices in finger counting and develop young children's use of 'all at once' finger numbers. We also might investigate how children reconcile differences between discrete and continuous models of number, through comparing and discussing representations. These issues highlight the need to consider carefully exactly how manipulatives might foster learning of particular aspects of mathematics. There are promising avenues for early years mathematics pedagogy in using manipulatives to build on children's interest in patterns and to develop children's own expressions of mathematical relationships. However, an important prerequisite for all of these is a focus on children's understanding of mathematical relationships rather than on performing calculations, as suggested by [START_REF] Ma | The theoretical core of number relationships[END_REF], which also has implications for early years mathematics curricula.

Proceedings of CERME10

Acknowledgements

Helen Thouless was our co-researcher in the classroom-based research described here. The Making Numbers Project is based on 'Using manipulatives in the foundations of arithmetic', a Nuffield Foundation funded research and development project.

the Nuffield Foundation, has developed guidance for teachers of 3 to teacher use and small-scale teaching investigations, identified key principles and issues for the use of manipulatives.


domain: math shs

text:
Svanhild Breive
email: svanhild.breive@uia.no

Kindergarten children's argumentation in reflection symmetry: The role of semiotic means

Keywords: Argumentation, kindergarten, gestures, semiotic means

 model for substantial argumentation, I illuminate structural aspects of the ongoing argumentation. In addition, I analyse the children's argumentation with respect to their use of semiotic means. Results show that children are able to argue for a claim in a quite complex manner. The study also illustrates the extensive use of semiotic means in children's argumentation. In every element in the argumentative structure, children use gestures and other semiotic means to mediate their ideas. It is actually impossible to make sense of the ongoing argumentation without considering the use of semiotic means.

Introduction

How children communicate their mathematical ideas is an important aspect in the attempt to understand children's reasoning in mathematics. In kindergarten children experience mathematical concepts through play and interaction with others. In their communication they justify and explain their mathematical ideas and in return they need to consider other's ideas and arguments. Thus argumentation can be seen as important for fostering children's mathematical learning.

This study is situated within a research and development project called the Agder project i (AP). One of the aims in the project is to investigate how researcher designed mathematical activities, developed in the project, stimulate mathematical competences. In this case study I observed one kindergarten teacher (KT) and a group of six 5-year-old children engaged in mathematical activities about reflection symmetry. The aim of this paper is to examine the characteristics of children's argumentation when they work with reflection symmetry. Furthermore, I examine what role 'semiotic means' (e.g. objects, linguistic devices and signs) play in the ongoing argumentation.

Following [START_REF] Toulmin | The uses of argument[END_REF] and [START_REF] Krummheuer | The ethnography of argumentation[END_REF], I regard argumentation as the practical business of choosing statements that serve the purpose of making an initial assertion reasonable and accountable for others. Argumentation is the production of an argument. An argument is then "the final sequence of statements accepted by all participants" (Krummheuer, 1995, p. 247). In addition, several arguments can serve as units in an expanded argumentation which again constitutes a new and extended argument. Toulmin also recognises that single statements can contain argumentative features. Just by making a statement you put yourself in a position of potentially being questioned.

Argumentation, acknowledged as an important means for enabling young children's mathematical reasoning, can be promoted through a dialogic approach to teaching [START_REF] Mercer | Teaching children how to use language to solve maths problems[END_REF][START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF]. Despite the increased focus on the role of mathematical argumentation for enabling young children's mathematical reasoning, little research has focused explicit on the role and characteristics of mathematical argumentation at kindergarten level. [START_REF] Pontecorvo | Learning to argue and reason through discourse in educational settings[END_REF] found that children's reasoning in preschool activity unfolded "through complex argumentative Thematic Working Group 13 Proceedings of CERME10 patterns" (p. 133). They emphasised that teachers should pay attention to the different ways children argue in order to facilitate children's "possibilities to practice, enrich and refine argumentative resources they have already acquired" (p. 139). [START_REF] Tsamir | Exploring the relationship between justification and monitoring among kindergarten children[END_REF] investigated different types of justification given by children between five and six years old, working with number and geometry tasks. Their study shows that young children are able to justify their statements by using appropriate mathematical ideas. Some children, in contrast, used their 'visual reasoning' as a way to justify their statements. When the researcher asked how they could know which bunch of bottle caps had more they answered "because we see", and they felt no further need to justify their answer or did not know how to do it. [START_REF] Dovigo | Argumentation in preschool: A common ground for collaborative learning in early childhood[END_REF] investigated how argumentation promoted collaboration and problem solving in preschool (age 3-5). By comparing different ways of how argumentation took place in teacher-talk and peer-talk they found that peer-talk contributed very positively for promoting collaboration and problem solving. But at the same time they emphasised that if the teachers were able to guide the debate in a careful and exploratory way the teacher guidance could be a positive contribution to the development of the argumentation.

Theoretical framework

My theoretical stance is rooted in a sociocultural paradigm where interaction is regarded as the very engine of learning and development, [START_REF] Vygotsky | Mind in society. The development of higher psychological processes[END_REF]. As a consequence of adopting this theoretical stance, I regard argumentation as a cultural and historical activity. Children are not naturally born with the ability to argue. Argumentation is a communicative pattern which they learn through interaction with more knowledgeable others.

Interaction, specific for human beings, is characterised by the use of tools and especially by the use of language [START_REF] Vygotsky | Mind in society. The development of higher psychological processes[END_REF]. In recent years there has been a growing interest to study the interplay between gestures, language and thought both in mathematics education and in other domains. [START_REF] Mcneill | Gesture and thought[END_REF] developed a theory where he regarded gestures as an integral part of language, not merely as a support for language. He regarded gestures as having an active and inseparable role in language and thought. Not only gestures have been recognised as important for human reasoning. [START_REF] Radford | Introduction: Beyond words[END_REF] talk about the importance of the multimodal nature of cognition; how different sensorial modalitiestactile, perceptual, kinesthetic become integral parts of our cognitive learning processes. [START_REF] Radford | The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge[END_REF]2003) theory of knowledge objectification emphasises how gestures, bodily actions, artifacts, (mathematical) signs and speech in cooperation affect mathematical reasoning. A special category of semiotic means of objectification that [START_REF] Radford | The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge[END_REF] considers is deixis. Deictic terms are words that have the function "to point at something in the visual field of the speakers" (p. 17), and cannot be fully understood without additional contextual information (e.g. "here", "there" "that", "this" etc.). All semiotic means play a significant role in mathematical mediation and reasoning. "Each semiotic means of objectification puts forward a particular dimension of meaning (signification); the coordination of all these dimensions results in a complex composite meaning that is central in the process of objectification" (Roth & Radford, 2011, p. 78).

The concept of argumentation used in mathematics and mathematics education is often related to the production of proofs. It is nevertheless important not merely to connect the concept of argumentation to formal logic as found in mathematical proofs. [START_REF] Toulmin | The uses of argument[END_REF] distinguishes between analytic argumentation, which is used in production of mathematical proofs, and substantial argumentation which is informal argumentation used in everyday practices. Substantial argumentation does not necessarily have a strict logical structure. Substantial argumentation gradually supports a statement by presenting relationships, explanations, background information, etc. [START_REF] Krummheuer | The ethnography of argumentation[END_REF]. [START_REF] Toulmin | The uses of argument[END_REF] strongly emphasises that substantial argumentation should not be regarded weaker as or less important than analytic argumentation. [START_REF] Toulmin | The uses of argument[END_REF] developed a model for analysing structural and functional aspects of substantial argumentation with the aim to illuminate how statements are organised for the purpose of constituting an argument, and how a conclusion is established through the production of an argument. In Toulmin's model the core of an argument is based on three elements: claim (C), data (D) and warrant (W). The claim is an initial statement, for example an assertion or an opinion about something. To support the claim, the arguer needs to produce data. Data are facts or statements on which the claim can be grounded. A warrant is a justification of the data with regard to the claim. The warrant holds the argument together. It points to the relation between the data and the claim.

In addition, [START_REF] Toulmin | The uses of argument[END_REF] model contains three other elements, backing (B), qualifier (Q) and rebuttal (R). A backing is a statement that supports the warrant. It is like a special case of data that is provided as evidence for the warrant. The purpose of a backing is to answer "why in general this warrant should be accepted as having authority" (Toulmin, 2003, p. 95). A qualifier says something about the extent to which the data confirm the claim. Words like 'probably', 'presumably' etc. are often used as qualifiers. Rebuttals refer to exceptions or conditions under which the claim is true, often used subsequent to a qualifier, exemplified as "The claim is true except/unless/only if …".

Method for data collection and data analysis

In this case study I observed one KT in the focus group of AP and his group of six 5-year-old children engaged in mathematical activities about reflection symmetry. The activities had been developed in the AP, and as part of an in-service program for the focus group the KTs were asked to implement a number of activities with their children. I visited the kindergarten on two occasions with a one week interval. It was the KT himself that decided to work with reflection symmetry activities on both occasions. The method for data collection was observations and the sessions were video recorded and field notes were made.

I regard argumentation as a sequence of statements (both verbal and non-verbal) that serve the purpose of supporting an initial claim. Thus one criterion for selecting episodes from the transcript was that they should contain verbal communication and have more than one utterance from the children. Another criterion was that the episodes I selected should contain mathematics, and they should be linked to the lesson aim (reflection symmetry). In total I found 11 episodes from the transcript using these criteria. Ten of these episodes had more or less an argumentative structure. Six of the episodes had more than two turns and more than two argumentative utterances from the children. These episodes were analysed in depth according to [START_REF] Toulmin | The uses of argument[END_REF] model to identify the argumentative structures. In addition, I analysed each of the six episodes from a multimodal perspective. In fact, I had to look at multimodal aspects in order to be able to differentiate between the elements in the argumentative structure. I did not focus on any specific semiotic means and their significance for children's reasoning. Rather I focused on what kind of semiotic means children used with respect to the different elements in the argumentation, and what role they played in constituting the argument.

Results

In this section I will present the analysis of one of the six episodes to illustrate the structure of children's argumentation, and what role semiotic means play in the ongoing argumentation.

In advance of this episode the children have been asked, by the KT, to find things in the room which they think are symmetric or as the KT says "equal on both sides". Each child is then asked to explain why they think the toy they have chosen is equal on both sides. In this particular episode one of the boys (John), who has chosen a trolley, is being asked to explain why he thinks the trolley is equal on both sides (or more precise; he is being asked if he think the trolley is equal on both sides).

KT:

Maybe we should start with John, since he has a very large thing. John, is this equal on two sides?

John: Mmm (2) ii There (2) ((He lifts his trolley up from the table, and holds it in a straight forward position. Then he says "there" and nods his head)).

John: and there… ((He rotates the trolley 90 degrees, showing the side of the trolley and then nods his head while saying "there")).

KT: Aha! John: and (2) there. ((He rotates the trolley 180 degrees, showing the other side of the trolley and nods his head again while saying "there")).

Elias:

And there and there. ((Elias has already paid attention to the situation)).

KT:

Can you see if this is equal Elias?

Elias: Look…

Elias: There, there (.) there, there (.) there, there and there, there. (.) And there and there, and (1) everywhere. ((He is pointing with his index finger to show where he thinks the trolley is equal. When he says "everywhere" he is letting his whole hand swipe over the trolley)).

KT: ((The KT lifts up the trolley and tries to show the symmetry line and explaining how the trolley is equal on both sides of that line)).

Elias:

Everything is equal on both sides, even the wheels.

The structure of children's argumentation

Before this episode each child was asked by the KT to choose a thing that they thought was equal on both (two) sides. In this episode John has picked a trolley and just by doing so he has implicitly made the claim that the trolley is equal on both (two) sides. When the KT asks John; "John, is this equal on two sides?" John's claim is being challenged. The KT actually asks a yes-no question, but the question is still a quest for explanation or justification.

To argue for his claim, John lifts the trolley up in the air, like he wants to make it visible to the others. The first "there" and the first nod is also a part of this visualization which together constitute the data he presents. It is important to notice that at this moment he holds the trolley in a straight forward position from his own point of view. He is not referring to any particular equal points on the trolley. The trolley itself is being presented as the data that supports the claim.

Then he is trying to present a warrant for his data by rotating the trolley 90 degrees while saying "there" and then back again 180 degrees while saying "there" again. Each time he is saying "there" he is nodding his head. John is presenting the warrant as two particular sides that are equal, and exemplifies the equality. The warrant (the example) relates the data (the presentation of the whole trolley) to the claim (the trolley is equal on two sides). Considering the time John is using while presenting the warrant and the way he utters the second and third "there", John does not seem very confident in his presentation of the warrant. Nevertheless, when John was walking around in the kindergarten trying to find a thing that had two equal sides, he considered the trolley for some seconds before he took it back to the table. This indicates that his choice was not completely random. John seems certain that the trolley is equal, but he is not quite certain how to justify it.

After John has presented the second "there", the KT utters "aha" (with a rising intonation at the end). By this utterance the KT gently appreciates John's contribution, even if the two sides that John presented thus far were not equal. While presenting his data and his warrant, John waits several seconds, and it seems that the KT thinks that John has finished his explanation after the second "there". From the children's point of view, the KTs "aha" gives Johns contribution authority and can be regarded as a backing of John's warrant-attempt. But from the KTs point of view, the "aha" was not meant as a backing, only as a gentle appreciation of his contribution.

Elias then contributes to the argumentation. By the utterance "and there and there" and a pointing gesture he is presenting another warrant for John's data. Elias is talking faster and more concisely than John. Because he is using his index finger rather than nodding he is also more precise in his communication and is able to point on specific points on the trolley, like the joints and the handles.

The way Elias communicates indicates that he is more confident and has more knowledge about reflection symmetry. Elias is actually presenting several warrants for the data. Every time he says "there and there" and points at different corresponding points, he gives a new warrant. By presenting particular corresponding points, each warrant is exemplifying exactly where the trolley is equal. By repeating several almost identical warrants (presenting several examples) it seems that he is trying to communicate that every point on one side has a corresponding point on the other side.

After presenting several warrants Elias says "everywhere" while he is swiping his whole hand over the trolley. I interpret this as a generalisation of his previous statements (warrants), and thus a backing for the warrants because it answers "why in general this warrant should be accepted as having authority" (Toulmin, 2003, p. 95). The warrants are not independent examples of equality rather examples of a more general property of reflection symmetry.

When the KT shifts his attention to Elias in the middle of this episode, Elias answers by saying "look". The intonation of the utterance indicates that he is only introducing his coming explanation. I interpret his utterance as synonymous to "let me explain".

At the end of this episode, Elias says "everything is equal on both sides, even the wheels". The use of the word "even" in this sentence is very interesting. The word "even" I interpret as a qualifier for the claim. It says something about to what degree the data confirm the claim. Usually words like "probably" or "presumably" are used as qualifiers, but in this case Elias is indicating that he is very certain that everything is equal on both sides, by saying "even the wheels". It seems that the probability for everything being equal increases since 'the critical points', the wheels, are equal. Why Elias regards the wheels as important points is hard to tell. This example illustrates the complexity of children's argumentation. They are able to present more than only the core of an argument. In this episode I found that some children are able to present both data, warrant, backing and even qualifier for a claim. In another episode that is not provided in this paper (because of the limited space) Elias was also able to present a rebuttal. He was able to modify his claim by giving examples of exceptions.

Discussion

This study shows that young children are able to argue for a claim in a quite complex manner. Using [START_REF] Toulmin | The uses of argument[END_REF] model to illuminate structural aspects of the children's argumentation, the results show that some children are able to use several of the elements in the model in their argumentation.

This study also illustrates the extensive use of semiotic means in children's argumentation. In every element in the argumentative structure, children use gestures and other semiotic means to mediate their ideas. This illuminates the significant role that gestures and other semiotic means play in children's communication and especially in their argumentation. (cf. McNeill, 2005;[START_REF] Radford | The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge[END_REF]2003;[START_REF] Roth | A cultural-historical perspective on mathematics teaching and learning[END_REF]. Deixis, in particular, are extensively used in the argumentation above. Both the data that John presents and the warrants that John and Elias present are based on the deixis "there" and the related pointing and nodding gestures. namely to give contextual information to the deixis "there". It is actually not possible to get the whole meaning of the words "there and there" without including the pointing and nodding gestures.

The deixis and the related pointing and nodding gestures are not the only significant semiotic means in this argumentation. To be able to distinguish between the data and the warrant that John provides I had to interpret his related actions. When he presents the data he holds the trolley in a straight forward position, he is not referring to any particular equal points, only presenting the trolley as a whole, as if he wants to show the equality. The way he presents his claim corresponds with one of the findings in Tsamir, Tirosh and Levensons (2009), that some children based their justification on 'visual reasoning'. The trolley itself is being presented as the fact that supports the claim. In the warrant he is presenting two corresponding sides, as if he wants to give an example of the equality. Without interpreting these actions, it is impossible to distinguish between the data and the warrant, and thus fully understand the structural aspects of the ongoing argumentation.

The repetitive presentation of Elias' warrants and the swiping hand that generalises the repetitive warrants are other important semiotic means in the argumentation. By repeating "there and there" with corresponding pointing gestures Elias indicates that every point has a corresponding point on the other side of the symmetry line. When Elias says "everywhere", he swipes his hand over the trolley. This swiping gesture plays a significant role in the generalisation process of the points.

The results from this study point to significant features of children's argumentation and give important insights into how children argue. I think teachers could benefit from paying attention to the different ways children argue and being aware of the structural aspects in children's argumentation in order to provide opportunities for improving children's mathematical communication and reasoning (cf. [START_REF] Dovigo | Argumentation in preschool: A common ground for collaborative learning in early childhood[END_REF][START_REF] Pontecorvo | Learning to argue and reason through discourse in educational settings[END_REF]. But to be able to do so, the KTs also need to pay attention to how children make use of semiotic means in their argumentation. The Toulmin model revealed structural aspects of children's argumentation, but these structural aspects would not have emerged without considering the use of semiotic means. In line with [START_REF] Roth | A cultural-historical perspective on mathematics teaching and learning[END_REF] I would argue that all the different semiotic means play a significant role in the constitution of meaning.

In the example above we saw that Elias was able to use several elements in the model and demonstrated more confidence in his argumentation than John. A possible explanation could be that Elias is further in his appropriation of the properties of reflection symmetry than John. Maybe there is a correspondence between how far children have appropriated a certain subject and their ability to use several elements in the Toulmin model. This is thus a suggestion for further research on this interesting topic.

Figure 1 :

 1 Figure 1: The structure of children's argumentation

  Even if John and Elias use different signs for mediating their ideas, both the nodding and the pointing gestures serve the same purpose, There, there. There, there. There, there and there, there. And there and there [pointing with his index finger at equal points on two sides of the trolley]

	Data (John): There [Lifts the trolley in a straight forward position and nods his head]	Claim (John): The trolley is equal on two sides	Qualifier (Elias): even the wheels
	Warrant (John): [pause] and there [pause] and there [He is first showing one
	side, then the other side]
	Warrant (Elias): And there and there [pointing with his index finger at equal
	points on two sides of the trolley]
	Warrant (Elias):

Backing (Elias): Everywhere [swiping his hand over the trolley]

Proceedings of CERME10


text:
Ingvald Erfjord
email: ingvald.erfjord@uia.no

Christiane Benz
email: benz@ph-karlsruhe.de

Esther Levenson
email: levenso@post.tau.ac.il

Bozena Maj-Tatsis

Introduction to the papers of TWG13: Early years mathematics Introduction

The working group on Early Years Mathematics was established at CERME 6. The aim of this working group has always been to share scholarly research concerning mathematics for children aged 3-8. In this age group, the transitions within preschool, and from preschool to the early grades of primary school are important areas of attention. The working group on early years mathematics has also considered that in different countries preschool education has different objectives and children in different countries begin primary school at different ages (e.g., in Sweden there is one preschool curriculum for children aged 1-6 followed by primary school, and in UK there are different curricula for with nursery, preschool and primary school: 0-2, 3-4 and 5-8 years). These differences have always stimulated fruitful discussions and CERME 10 was no exception.

The 15 papers were each allocated 30 minutes of attention during the TWG 13-sessions; 12 minutes for presentation, 8 minutes where a different participant prepared a response or query, and 10 minutes for open discussion. The five posters were allocated 15 minutes each for a brief presentation and open discussion. One 30-minute slot was reserved for group work where participants from different countries were grouped and discussed quite openly topics of common interest (such as, e.g., the role of aesthetics in mathematics and the use of manipulatives in mathematics), and another 30-minute slot was reserved for discussing the plenary by Lieven Verschaffel on children's number sense. It was very fortunate to have him as a participant in TWG13..The final 90-minute session on Saturday was used to collaboratively prepare the report for Sunday, which together with what was presented and discussed in the other sessions, formed the basis for this introduction.

One of the Friday sessions included a 60-minute presentation and discussion of the Early Years Mathematics chapter draft for the soon to be published ERME book, where one of the editors of the book (Kenneth Ruthven) was present. Participants had been asked to read and send their questions and comments to the chapter in advance of the conference, and these inputs, as well as comments during the session, contributed important input to the ongoing discussion in the group.

The number of participants in TWG13 has been quite stable over the past five conferences, and each time the group has benefitted from a quite broad attendance when it comes to the area of Europe. Twenty-nine participants from 11 different countries attended the working group in Dublin (see the table below).

Country

General summary of presentations

Seventeen of the 20 papers and posters concerned preschool children aged 3-6 years, preschool/kindergarten teachers' professional knowledge and competence development or teaching materials for activities in preschool. The remaining three contributions focused on children aged 6-7 years, which in most countries means Grade 1 in school. Compared to previous years, there was an increased attention given to preschool compared to early schooling.

Broad attention was given to the roles of preschool teachers' competencies in promoting early years mathematics learning. Vanegas, Gimenez and Samuel studied school mathematics narratives in early childhood teacher education, focusing on how the future preschool teachers recognized the potential of two professionally designed geometrical tasks to promote mathematical processes. Palmér and Björklund considered how preschool teachers characterize their own mathematics teaching in terms of design and content, while Hundeland, Erfjord and Carlsen analysed preschool teachers' orchestration of researcher-designed mathematical activities. They used this analysis to discuss what can be learned about teachers' knowledge by adopting the Knowledge Quartet by Rowland and colleagues for kindergarten level. Tirosh, Tsamir, Levenson and Barkai studied preschool teachers' variations when implementing a patterning task, where they investigated the impact of the various implementations on children's success in extending repeating patterns. Gifford, Griffiths and Back presented a theoretical paper concerning the use of manipulatives with young children, while Skoumpourdi presented a framework for designing inquiry-based activities for early childhood mathematics. Silva, Costa and Domingos offered an nterdisciplinary approach to linking science and mathematics in early mathematics. A new issue presented and discussed during our WG was mathematics for special needs children during the early years. Gasca, Clemente and Colella outlined and discussed the design of mathematical instructional activities to foster achievements in mathematics for children with Trisomy 21 (Down's Syndrome). Peter-Koop and Lüken considered the role of inclusive compared to exclusive settings for the learning of mathematics for children with special needs.

In total, eight papers and posters focused on the learning and development of mathematics for preschool children. The paper by Tzekaki and Papadopoulou was based on a teaching intervention for developing generalisation in early childhood. Björklund's paper considered the importance of adults challenging two year old toddlers' evolving concepts of numbers in their play. Her paper stimulated a discussion as to whether 3 years of age should be the lower limit of attention for our group as it was in the past. Sundström and Levenson reported on an exploratory study of young children's aesthetic development in the context of mathematical explanation, while Pettersen and Volden presented a poster considering the use of maps in kindergartens and children's development of spatial orientation and navigations. Bjørnebye reported from a multi-case embodied design study on early learning of numbers, while the paper by Schöner and Benz studied preschoolers' perception and use of structures in sets, adopting eye-tracking as a tool. Breive studied preschool children's argumentation as part of an inquiry approach in reflection symmetry and Rinvold investigated children's learning of numerocity. Both Breive and Rinvold adopted Radford's theory of knowledge objectification to kindergarten settings.

Regarding the first year of primary school, Maj-Tatsis and Swoboda reported on an intervention study on first grade school children's ability in noticing and using regularities in three-dimensional geometrical objects while using playing blocks. Van Bommel and Palmer focused on 6 years old children's representation of the semi-concrete and semi-abstract, as connections between concrete (objects) and abstract (signs) representations. Finally, Thoules presented a self-study action research study on the role of gestures in supporting mathematical communication for first grade students with language delays.

Main characteristics and issues in focus

The group had ongoing discussions about the importance of fostering children's early development of mathematics for later success. As such, the group experienced presentations and discussions of several intervention and exploratory studies, teacher development and ways of fostering children's mathematics learning. The role of play and the term "playful learning" was given great attention. This proposed term takes into account that distinguishing between play and learning at this age-level makes little sense. The argument offered was that an activity designed, planned and guided by the teacher may indeed promote learning among children, but can be considered as play by the children.

As usual, the group presented a mixture of small scale studies (case study design), large scale studies and several pilot studies for larger studies. These different studies contributed valuable insights. We also experienced a broad range of theory in discussions of papers and posters, including the theory of objectification, variation theory, socio-cultural theories, the use of Clements and Sarama's framework, the core knowledge system, as well as development of conceptual frameworks. The theoretical discussions of the group were supported by Lieven Verschaffel's perspective and contributions, generating fruitful discussions of theory use and development.

The most prominent "news" for our group came from presentations discussing the following issues: ways in which special needs children can engage with mathematics, toddlers and mathematics, and young children's appreciation for the aesthetical dimensions of mathematics. The group was also introduced for the first time to the possibility of using eye tracking as a tool in early years mathematics research. We encourage researchers to continue with their important work regarding early years mathematics and hope to see new insights in the next CERME meeting.

Proceedings of CERME10