Young children’s multimodal mathematical explanations

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Young children’s multimodal mathematical explanations Maria Johansson#, Troels Lange¤, Tamsin Meaney¤, Eva Riesbeck¤ and Anna Wernberg¤ # Luleå Technical University; ¤ Malmö University

This paper investigates how three children provided mathematical explanations whilst playing with a set of glass jars in a Swedish preschool. Using the idea of semiotic bundles combined with the work on multimodal interactions, the different semiotic resources used individually and in combinations by the children are described. Given that the children were developing their verbal fluency, it was not surprising to find that they also included physical arrangements of the jars and actions to support their explanations. Hence, to produce their explanations of different attributes such as thin and sameness, the children drew on each other’s gestures and actions with the jars. This research has implications for how the relationship between verbal language and gestures can be viewed in regard to young children’s explanations.

1. Introduction

Mathematical explanations have been recognised in many curricula as integral to the learning and using of mathematics (Esmonde 2009). The Swedish preschool curriculum is no exception, with preschools being required to provide activities for children in which they “develop their ability to distinguish, express, examine and use mathematical concepts and their interrelationships; and develop their mathematical skill in putting forward and following reasoning” (Skolverket 2011, p. 10). It is difficult to imagine that children could engage in such activities without providing explanations, because an explanation with “a convincing argument makes a clear connection, using reasoning, between what is known about a problem and the suggested solution” (Meaney 2007, p. 683).

Nevertheless, preschool children are not school students in mathematics classrooms and they do not have the same degree of fluency, either verbally or in writing. Instead, their explanations are likely to reflect both their understanding of what counts as an adequate explanation as well as the resources that they have for providing them. For example, in research on young children’s explanations, Donaldson (1986) described three different kinds of explanations: the empirical; the intentional; and the deductive. It is the final one of these which uses logical reasoning. Donaldson did not consider children could use logical connectives, such as ‘because’ and ‘so’, in deductive explanations until about eight years old, much later than when they were used with empirical explanations. However, research in regard to science suggests that familiarity with phenomena is likely to contribute to preschool children using causal reasoning when providing explanations (Christidou and Hatzinikita 2006).

Body movements have been documented as a resource that children use to supplement their verbal language skills. Vygotsky described how babies begin to point as a response to not being able to grasp something (Seeger 2008). Seeger (2008) also described the work of Michael Tomasello who was interested in how very young children engaged with adults and others through producing and interpreting eye gazes as well as gestures. McNeill (2005) suggested that children from 4 years gesture like adults, children younger than 2 do not, and between two and four they learn/develop

Johansson, M. L., Lange, T., Meaney, T., Riesbeck, E., & Wernberg, A. (2014). Young children’s multimodal mathematical explanations. ZDM. 46(6), 895-909.

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gesturing – or their imagery develops and this finds expression in their gestures. In support of this, Göksun et al.’s (2010) study of two and a half to five year old children found that gestures seemed to supplement children’s verbal descriptions of causal events, with the use of gestures preceding the child’s ability to describe causal relationships in speech. Similarly, Whitebread and Coltman (2010) suggested that gestures in conjunction with speech support young children’s descriptions of their thinking. As well, the prominence given to verbal language in young children’s learning has been critiqued. For example, Flewitt’s (2006) study:

revealed how the children began to grasp unconsciously the rationale of series of patterned behaviours that corresponded to preschool structures and routines, activity types and interpersonal relationships, illustrating how they became apprenticed in particular communities of practice largely through observation and imitation rather than through discourse. (p. 46)

The results of this extensive study suggest that young children’s explanations are likely to make use of features gleaned from observing and imitating adults and possibly other children. This would be similar to how they incorporate the verbal utterances of others into their explanations as they test out different formulations (Peterson and French 2008).

Although it is acknowledged that young children are developing their verbal fluency and understandings about explanations, research has focused almost exclusively on their interactions with adults, teachers or parents and on verbal explanations (see for example, Peterson and French 2008). However, this focus on verbal explanations must be considered as a focus on what the children cannot do, rather than what they can do.

In contrast, in mathematics education, as with other education disciplines, older students’ explanations of their thinking are considered to involve extra-linguistic components, such as gestures (see Roth 2001; Radford 2003; 2009a; 2009b; Meaney 2007; Arzarello et al. 2009). Therefore it seems inconsistent that young children who are known to use gestures have not been researched in mathematics education in any systematic way. When young children’s gestures have been researched they are considered generally only as supplements to their verbal explanations, as noted in the next section. Consequently, young children’s gestures have not been theorised to the same degree as those of older children in mathematics classrooms have been. In our research, we investigate preschools children’s explanations and the different resources that they drew upon, including gestures. In this paper, we discuss two methodological approaches for analysing what young children did, as opposed to not being able to do, in regard to their mathematical explanations. However, before discussing issues of methodology, we summarise different perspectives on the relationships between gestures and language in regard to mathematical explanations. 2. Gestures, artefacts and verbal language

In recent times, gestures have received much attention in mathematics education research. Surprisingly, in much of this research a definition of a gesture is not provided, although generally it seems to be considered as movement of the hands or arms in a manner that can be considered to express something about the gesturing person’s thinking. Roth (2001) defined gestures as having a particular structure which

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begins and finishes at a place of rest. In the research, the relationship of gestures to verbal language and the kind of insights that they provide about children’s thinking and thus their contribution to mathematical explanations is contested (Radford 2009b). In the following section, we provide three different perspectives on this relationship.

2.1 Verbal language and gestures as an integrated process McNeill (1992), who is quoted often in research about gestures, considered that they were primarily movements of the hands and arms, which always accompanied speech and that “speech and gesture are elements of a single integrated process of utterance formation in which there is a synthesis of opposite modes of thought” (p. 35). McNeill (2005) considered that the motion of a gesture was a dimension of the meaning being conveyed and was adamant that it should not be seen simply as a representation. For him, gestures give a materiality to the meaning which cannot be provided by speech alone. Therefore, speech and gestures should be thought of as part of the same meaning making process. Acceptance of the premise that speech and gesture are part of an integrated whole has been the basis for research on mathematics activities, such as addition, location and explaining (Goldin-Meadow et al. 2009; Sekine 2009; Roth 2001). Broaders et al. (2007) found that Year 3-4 children in the US who explained how they had tried unsuccessfully to solve arithmetic problems often provided information about relevant problem solving strategies in their gestures that were not evident in their speech. When children were given more information about how to solve the problems, those who had gestured when giving their earlier explanations were more likely to learn than children who were told not to gesture. Thus, explanations that contain gestures are considered to provide more information about children’s thinking both to onlookers but also to the children, than explanations not accompanied by gestures.

From this perspective, gestures are considered to provide speakers with an alternative way to express their ideas, one that can reduce cognitive effort and serve as a tool for thinking. As well, they are thought to provide listeners with a second representational format, which gives access to the unspoken thoughts of the speaker and thus enriches communication (Goldin-Meadow 1999, p. 428). Summarising research based on the premise that gestures and speech are two components that provide information about students’ thinking processes, Roth (2001) stated:

As a whole, these studies make four core claims: First, gestures reveal knowledge that is not expressed in speech. Second, gestures reveal implicit or emergent knowledge that is expressed in speech only at some later point; gestures can be said to constitute the “leading edge” in children's cognitive development. Third, the mismatch between gestures and speech is an indication of the readiness to learn. Fourth, the changes in the gesture-speech relationship can be interpreted as reflecting a path of knowledge change. (p. 373)

2.2 Verbal language and gestures with two different functions Nevertheless, not all researchers accept that language and gestures have equally important roles in explanations of children’s thinking. Sfard (2009) presented a compelling case for language and gestures having different roles, with gestures and

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other visual mediators being a first step towards using verbal language to discuss abstract mathematical ideas.

For Sfard, it is the fact that (mathematical) language can be used to talk about (mathematical) language which allows a hierarchy of ideas to be compacted. She provided the example of abstract algebra being the compaction of a discussion about the complexities of algebra, which is itself the compaction of the meta-discourse of arithmetic. The compaction process occurs when the mathematical object is discussed and this can happen only through verbal language. As Sfard stated “mathematical objects are not any less material than any concrete, tangible thing, except that they are usually inaccessible to us in their entirety, that is, their numerous parts are never simultaneously visible” (p. 197). For Sfard, gestures have a role in making apparent what the words mean, as a process of realisation. She also noted that words can act to realise gestures. The manipulation of artefacts has a role in the realisation process, as they are “part and parcel of the act of communication and thus, in particular, of thinking processes” (Sfard 2007, p. 574). Nevertheless, although gestures, including actions on artefacts, and words can be used to provide more details about each, from Sfard’s perspective, it is only verbal language that has the capacity to describe abstract mathematical objects. Thus, mathematical explanations provided in verbal language describe abstract ideas in a way that gestures cannot, either alone or in conjunction with verbal language.

2.3 Verbal language, gestures as semiotic means of objectification Similar to Sfard’s (2009) discussion of realisation, Radford (2003) used the term objectification to discuss the process of making something, such as a complex mathematical object, available to our awareness. For Radford, gestures are one of many semiotic means of objectification:

The objectification of mathematical objects appears linked to the individuals’ mediated and reflexive efforts aimed at the attainment of the goal of their activity. To arrive at it, … individuals … may manipulate objects (such as plastic blocks or chronometers), make drawings, employ gestures, write marks, use linguistic classificatory categories, or make use of analogies, metaphors, metonymies, and so on. In other words, to arrive at the goal the individuals rely on the use and the linking together of several tools, signs, and linguistic devices through which they organize their actions across space and time.

These objects, tools, linguistic devices, and signs that individuals intentionally use in social meaning-making processes to achieve a stable form of awareness, to make apparent their intentions, and to carry out their actions to attain the goal of their activities, I call semiotic means of objectification. (Radford 2003, p. 41)

Radford’s view of the relationship between gestures and verbal language is different to the two discussed in the previous sections. Rather than considering verbal language and gesture to be two sides of the same coin, or for gestures to be a precursor to verbal language, Radford (2012) suggested that they can be used individually or together to restructure thinking. Drawing from Vygotsky, Radford (2012) suggested that thinking should not be considered as occurring solely within the head but as something that occurs as it is articulated. This is an important point of difference because it suggests that speech, gestures, or any other kind of semiotic means of

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objectification, do not merely reproduce, accurately or not, existing thoughts but the articulation process leads to the production of those thoughts.

Therefore, abstract mathematical objects cannot be considered to rely only on verbal language for their realisation. In Radford’s (2012) research which followed a young student from Year 2 to 4, verbal language did not replace gestures in the student’s description of algebraic patterns. Rather over time, the student was able to co-ordinate gestures, perceptions and speech more easily – “the coordination of these outer components of thinking is much more refined compared to what we observed in Grade 2” (p. 128). When considering mathematical explanations, the relationship between verbal language and gestures is different to the other perspectives, in that production of the explanation itself, using either one or more semiotic means of objectification is likely to reformulate the thinking. The explanation is not a reflection of the thinking but actually constitutes the thinking. Thus, the research of Broaders et al. (2007) discussed in section 2.1 could be seen as an example of how gestures combined with verbal language reformulate children’s thinking in a more productive manner than when verbal language is used alone to provide the explanation.

2.3 Gestures in mathematics education The role of gestures in mathematics education is contested with different, although related, perspectives being used in research. Gestures can be seen as having a role either as a representation of an image, or as a contributor to the realisation process, or finally as a component of thinking itself. Each of these perspectives affects perceptions about the role that gestures and verbal language have (or should have) in mathematical explanations, mainly because of the way they consider thinking to occur. Although young children are likely to include gestures either instead of or to accompany verbal utterances, there has been little research about how young children who are still gaining fluency in verbal language make use of bodily actions or gestures when providing explanations.

As well as developing verbal fluency, preschool children are likely to be developing their awareness and use of other semiotic means of objectification, including being able to integrate them into explanations. Radford’s (2012) research suggests that combining different semiotic means of objectification was easier for children to achieve as they became older and more acquainted with the cultural forms of reflection expected of them. Thus, it may be that preschool children’s use of gestures with or without verbal language is likely to be less smoothly integrated than on older children’s use of gestures. Therefore, we considered that a detailed analysis of preschool children’s mathematical explanations is required. In particular, we want to consider how verbal language is combined with children’s actions when providing mathematical explanations for themselves, other children and the teacher. As well, it seemed valuable to see how different gestures were adopted and adapted. In order to do this, we combined the theories of Arzarello et al. (2009) and Kress and van Leeuwen (2001/2010).

3. Developing a methodology In order to develop an appropriate methodology to investigate young children’s mathematical explanations, we decided to use the work of Arzarello et al. (2009) and that of Kress and van Leeuwen (2001/2010). Arzarello et al. (2009) working in

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mathematics education made use of the work of the linguist Peirce (1931/1958) whilst Kress and van Leeuwen’s (2001/2010) development of a theory of multimodalities drew on the work of the linguist Halliday (1985) in that it connects participants’ modal choices and representations with their functions in order to produce meaning (Jewitt 2008). The main difference between these theorists is that whereas Kress and van Leeuwen come at multimodality from a perspective of semantics, or the study of meaning (Jewitt 2009), Arzarello et al.’s (2009) approach is from semiotics, or the study of signs. Regardless of their theoretical inspirations, they share commonalities in regard to investigating gestures. By combining these theories, we anticipated that we would gain insights about the relationship between verbal language and gestures, in regard to the production of explanations. Arzarello, et al. (2009) developed their theories from research in mathematics classrooms. Semiotic bundling allows gestures to be analysed in relationship to inscriptions, children’s arrangement of physical artefacts and their spoken and written utterances. We considered that semiotic bundling would provide insights into how children who were still gaining fluency in verbal language made use of semiotic means of objectification, or signs in Arzarello et al.’s (2009) terms, individually and in combinations.

Arzarello et al. (2009) described a “semiotic bundle” as being composed of several signs, produced by different people over time.

The semiotic bundle dynamics can be analysed in two different and complementary ways. The first one is synchronic analysis, which considers the relationships among different semiotic resources simultaneously activated by the subjects at a certain moment. The second is a diachronic analysis, which focuses on the evolution of signs activated by the subjects in successive moments (in short or long periods of time). Together, synchronic and diachronic analysis allow us to foreground the roles that the different types of signs (gestures, speech, inscriptions) play in students’ cognitive processes. Considering semiotic bundles, we can fully grasp the evolution of learning processes and the role of gestures therein. (p. 100-101, italics in the original)

A synchronic analysis shows how one sign is used in relationship to another sign, or signs, at a particular moment in time. How different signs are integrated moment by moment and how they change over time – a diachronic analysis - provides insights into the learning process (Arzarello et al. 2009). The assumption is that if a sign, such as a gesture, changes then so does the student’s learning. This assumption seems to be built on an understanding, similar to that of McNeill (1992) that gestures and verbal language reflect inner thinking but the production of signs is not considered to affect that thinking. Artefacts, however, for Arzarello et al. (2009) are not signs in themselves because a sign has to be produced by a person so the artefacts can only become signs once they are acted upon. Yet, in our data, the attributes of different artefacts appeared to contribute to certain kinds of gestures and verbal descriptions being used in explanations and so the artefacts could not be considered merely as vehicles for the meaning making process. Such considerations led us to the work of Kress and van Leeuwen (2001/2010) who labelled a range of physical and ethereal things as modes. These included language,

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speech and writing, but also colour and sound. Meanings can be expressed through a number of different ways such as the choice of a theme colour for a magazine article which could appear clearly in the pictures but also be represented in the written language describing the pictures. In the digital age, they considered that there was a need to understand how meanings are expressed when there was an integration of resources.

The traditional linguistic account is one in which meaning is made once, so to speak. By contrast, we see the multimodal resources which are available in a culture used to make meanings in any and every sign, at every level, and in any mode. Where traditional linguists had defined language as a system that worked through double articulation, where a message was an articulation as a form and as a meaning, we see multimodal texts as making meaning in multiple articulations. (Kress and van Leeuwen 2001/2010, p. 4; italics in the original)

Kress and van Leeuwen’s (2001/2010) suggestion that meaning making occurs when different multimodal resources were used seems to be more in alignment with Radford’s conception that articulation produces thinking, rather than just describing it. However, their suggestion is more complex than this in that their focus is not on how one person provides an explanation, for example, but on how that explanation is interpreted by others. The different modes produce, singularly and together, different meanings both for the interpreter and for the producer.

4. Method In 2011, we recorded a series of videos at a preschool in a city in the southern part of Sweden. Our research was focussed on describing the mathematics that children engaged in, using Bishop’s (1988a; 1988b) six activities as the basis for our classification (Johansson et al. 2012). One of Bishop’s activities is explanation. Bishop (1988a) described this as an essential mathematical activity because:

It focuses attention on the actual abstractions and formalisms themselves that derive from the other activities and where they are related to answering the relatively simple questions of “How many?”, “Where?”, “How much?”, “What?” and “How to?”, explaining is concerned with answering the complex question of “Why?” (p. 48).

Our decision to focus on explanations in this paper was affirmed by the fact that the Swedish preschool curriculum (Skolverket 2011) draws on Bishop’s 6 activities (Johansson et al. 2012).

In the following section, we describe the first 2 minutes of an 11 minute video in which three young children play with some glass jars. The children were four and five years old at the time. This extract was chosen because it showed the children’s initial explorations and their explanations about these. In particular, we considered that much of the exploration contributed to the explanation about what was the same about two glass jars (“de är lika samma”) which becomes apparent at the end of the two minutes. The eight jars were similar in height and diameter of the opening. Figure 1 shows the children and the jars at the beginning of the session which was to be about making candle holders. The children are named here, Lena (the child on the left), Mira (the child in the middle) and Rita (the child on the right). The Swedish that the children use is in development and providing an English translation that reflects that development has proved difficult.

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Figure 1. The children began to play with the jars almost immediately.

In order to identify semiotic bundles, we accepted “the necessity of using an extremely fine grained analysis” (Arzarello et al. 2009, p. 98). Consequently, each member of the research group watched the video several times, noting what each child and teacher did moment by moment. These descriptions were incorporated into the table, reproduced in the following section. Finally, a comment was made about the purpose of the combined actions and speech. This purpose was added into the table only when all 5 members of the research team agreed that this seemed to be what the participant intended. In regard to determining if an explanation was given, we looked at whether the child’s speech and actions seemed to be responding to an explicit or implicit question, following Bishop’s (1988a) suggestion that explanations were a response to the question “why”. We then considered whether the speech and actions provided a link between a problem, expressed as the why question, and a solution (Meaney 2007). This link like the question often was incomplete, indicating to us that these children were still developing their knowledge of what constituted an adequate explanation. For example, at 30 seconds, we considered that Lena was providing an explanation to the teacher’s question, “Do you think all jars look the same?”. Lena explains that they are not the same because her jar has particular features. She does this by stating what the features are and touching them, rather than by answering the question directly. However, divining intention from young children’s actions does rely on the subjective interpretation of the observers. Although we have provided much information in the following table, it is not possible to provide all the rationales for our decision making here. As Sfard (2007) stated “Ever since audio and video recorders have become standard tools of the researcher’s trade, our ability to interpret human activities lags behind our ability to observe and to see” (p. 568). By looking across the rows in the table, we could do a synchronic analysis and identify which semiotic resources (Arzarello et al. 2009) or semiotic means of objectification (Radford 2003) were being used simultaneously by different participants. Looking down the columns, we could see how an individual made use of one resource, such as a specific gesture, at different times and possibly with different purposes. By looking across and down the rows and columns we could see how different resources were adopted and adapted by different participants over time.

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A multimodal analysis was done by looking at the adaptation and adoption of certain resources, verbal or gestural, and their relationship to the children’s purposes.

5. The data In this section, the data are presented in a table before the analysis using semiotic bundles and multimodality is provided. The final section of the paper discusses these analyses and the usefulness of the methodology in identifying the production of mathematical explanations for different audiences. Time Lena (L) Mira (M) Rita (R) Teacher (T) Speech Comments 00:00 Put left hand in

a jar in front of her.

Touched a jar on the outside

Held a jar in both hands.

Sat down next to the children.

T: The idea was that, you know

00:02 Put her hand around the top of a jar before putting it into the jar.

Picked up a jar. Runs her fingers around the ridges at the mouth of the jar.

Put her hands in her lap

T: When we are having Lucia then we must be out early

L & R: Exploring

00:06 Picked up the jar with her hand inside it holding it

T: In the morning.

L: Exploring

00:07 Pushed the jar in front of herself upside down

After gently tapping the jar with her finger, put her hand inside it.

Looked at Lena and the jar.

L: Exploring R: Imitating

00:09 Turned towards teacher.

Turned towards the teacher. Then put her hand inside the jar.

Held jar with two hands. Put one hand inside it and held it upside down.

L: We are going to make Lucia candles

L: Repeating M & R: Imitating

00:10 Looked at the other girls, whilst holding the jar with her hand inside it, in front of her.

Looked at jar with hand inside. Held jar upside down

Moved forward to face teacher

T: We are going to make Lucia candles

T: Repeating M: Imitating

00:13 Took hand out of jar. Then put it back in and turned the jar so the base is pointing upwards.

Took her hand out of the jar.

Sat back down facing the teacher. Picked up jar.

Looked at Rita. L: We took the almost longest

L & M: Exploring

00:19 Placed jar on floor but kept hand on the opening. Turned to look at Mira.

Picked up another jar.

Bent slightly outwards to see the children better.

T: But before we begin making these candles I thought we could play a little with these jars,

00:24 Picked up jar, put it in her lap.

Tapped the base of the jar with left hand.

T: Do you think all jars look the same?

M: Exploring

00:28 Put her right hand in and out of the jar.

Put her left hand into the jar.

Lowered the jar she was holding.

Children: Nooo L & M: Exploring

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Time Lena (L) Mira (M) Rita (R) Teacher (T) Speech Comments 00:30 Holding the jar

in front of her, left hand on the base and the other around the middle. Patted and tapped the jar with right hand.

Put down the jar and picked up another one and put her hand inside. Looked at Lena.

Took her hand out of the jar, held the jar with her other hand.

Lent forward to look at Lena.

L: This is a little oblong and thick

L: Explaining M: Exploring

00:34 Kept tapping the jar. Turned to face Mira.

Waved the jar around with her left hand inside it.

Held jar with both hands.

Turns to Mira. T: It is a little oblong and thick M: Mine is a little thin [same time]

T: Repeating L: Emphasising M: Explaining through adaptation of L’s earlier description.

00:36 Looked at Rita. Stroked side of jar with right hand. Looked at Rita.

Put her right hand into the jar and spread her fingers.

T: And yours is a little thin R: Mine is a little thick

R: Explaining by repeating L’s words and spreading her fingers.

00:39 Tapped the jar hard with her left hand.

With jar facing upwards on her hand, bends her arm backwards.

Points with her left hand to the place where the jar gets thicker.

T: And yours is a little thick. In what way is it thick, Rita, how is it thick?

L & M: Exploring R: Explaining

00:41 Banged the jar against the floor, first looking at it and then the teacher. Lifted up the jar.

Continued to hold jar with her left hand in it, up in the air.

Took her hand out of the jar.

L: Exploring

00:44 Started to say something while looking at her jar.

Tapped the base of the jar.

Held up the jar with her right hand round the middle.

Touched Lena on her arm.

R: It is thick at this width

M: Adapting L’s tapping R: Explaining

00:47 Looked at Rita. Left hand inside the jar, base upwards. Tapped the base with right hand.

Pushed the jar towards the teacher.

T: There at that width it is thick.

M: Repeating R: Emphasising T: Adapting R’s explanation

00:49 Tapped side of jar.

Tried to pull her hand out of the jar.

T: Does it have any other shape some other place?

L: Repeating M: Exploring

00:51 Held jar up in the air with both hands. Looked at Rita.

Held base of jar with right hand with left hand still inside. Looked at the teacher.

Moved her right hand to the top of the jar, and held it there.

R: Here it is thin.

R: Explaining

00:53 Held the jar on her lap with both hands.

Took her hand out of the jar with some effort.

Held jar up higher and then lowered it onto her lap.

T: There it is thin.

T: Repeating

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Time Lena (L) Mira (M) Rita (R) Teacher (T) Speech Comments 00:56 Turned to the

teacher and tapped the jar.

Put her hand inside the jar again, so the base faced upwards. Shook the jar and then pulled her hand out with some force again.

Placed both hands around the jar which was on the floor. Tapped the sides of the jar with both hands.

L: Mine is thick and in glass T: Is all your [jar] thick everywhere?

L: Explaining M: Repeating R: Adapting L’s tapping

1.01 Rotated her finger around the top of the jar

Turned jar on its side and held it with both hands.

L: Yes, except here. T: How do you see it there?

L: Explaining

1:04 Continued to circle top of the jar. Then put her right hand into the jar.

Looked at Lena or the teacher, while tapping the jar.

Ran her finger over the bumps on the bottom of jar in a circular motion.

Looked at Lena.

L: There are such small and thin stripes here for it is thick there inside also and then it is made of glass

L: Explaining M: Repeating R: Exploring by adapting L’s circular action

1.08 Took her hand out of the jar. Tapped the jar.

Looked at Rita. Then looked at Lena.

Still rotating her finger around the bumps on the bottom of the jar

1:12 Placed the jar on the floor.

Held the jar up. Flicked the side of the jar with her fingernail a couple of times, looking at Lena or the teacher.

T: It is, it is made of glass, so when we handle them here now, then we perhaps should be a little careful.

R: Exploring. T: Explaining

1:13 Placed her hands on her lap and turned to Mira.

Flicked the jar with her fingernails.

Looked at Mira L: Yes M: Adapting R’s hitting jar with finger nails to make noise.

1:19 Tapped the jar on the floor gently. Knocked over her jar. Turned to the teacher.

Hit the side of jar with finger tips. Picked up jar and touched the bottom first with one hand and then with the other.

Played with the label on the side of the jar.

T: How is yours shaped Mira? M: It is thin, you can put in,

L: Adapting tapping action M: Explaining

1:26 Looked at Mira.

Put her left hand in the jar and had the base facing upwards. Touched the base.

Bent behind Lena to look at Mira.

M: I can put in my hand, it (inaudible)

M: Explaining.

1:28 Put her hand on top of jar.

Waved jar around with hand inside.

Put her hand inside the jar.

M: Emphasising R: Adapting

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Time Lena (L) Mira (M) Rita (R) Teacher (T) Speech Comments 1:29 Looked at

Mira. Put her hand inside jar and lifted it so base was facing upwards.

Held jar with base facing upwards.

Held jar with base facing upwards.

Bent forward, supported her head in her hand and asked a question.

T: There is space for your hand in it. Is there any jar here, where there isn’t space for your hand?

L, M &R: Repeating

1:34 Started to put her hand into a jar.

Put her fingers into the top of a jar, picked it up and pushed it forward and upwards.

Continued to hold her jar with base facing upwards.

Looked at Mira.

M: Explaining

1:37 Put down the jar and put her hand in a jar close to Rita.

Continued holding jar in the air by her fingertips.

Put her hand into another jar, holding the first jar with her other hand.

T: That one. Any other? R: This has space. T: That has space

L: Exploring R: Explaining.

1:42 Took her hand out of the jar near Rita.

Put the jar down, picked up another one with her right hand in it. Her left hand still had a jar on it.

L: This, not that R: This has T: Yes

L & R: Explaining

1:44 Put her left hand in another jar.

Put the bases of the two jars together. Face expressed surprise

Put her right hand into another jar.

M: Ha, ha. L, M & R: Exploring

1:45 Picked up a jar with her hand in it so base is facing upwards. Looked at Mira.

Taps the bases against each other. Takes them away from facing each other.

Picked up the jar with her hand inside it. Turned it so base began to face upwards. Begins to say something.

Looks at Mira, then Rita.

L & R: Exploring

1:49 Picked up a second jar, held it around the middle

Pushed the jars in front of her.

Places jar on the ground. Puts her hand on another jar.

Moves her body upright while looking at Mira

M: They are equal same. T: These are equal same, you think?

M: Explaining T: Adapting M’s description.

1:50 Placed the base on top of the other jar with her hand in it. Looked at Mira.

With the jars still on her hands, she pointed to parts of the jars.

Looked at Mira.

Moved so she sat almost between Lena and Mira.

M: Yes because they have (inaudible) and they have the same there and they have the same there (inaudible) T: They have bumps and they have glass

L: Adapting M’s actions M: Explaining T: Repeating

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Time Lena (L) Mira (M) Rita (R) Teacher (T) Speech Comments 1:57 Placed the

bases of the jars close together. Put her head on the side.

Turned a jar upside down and ran her finger along the bumps on the base in a circular manner.

M: (inaudible) T: Are they halfway big M: They have same bumps [on the bases] T: They have same bumps.

M: Explaining T: Repeating M: Repeating T: Repeating

5.1 Semiotic bundles

A synchronic analysis shows that the children accompanied a spoken utterance with a bodily action and/or an action immediately preceded or followed it. For example, at 1:37, Lena and Rita responded to the teacher’s question about which jars they could not get their hands in (Är det någon burk här som, som man inte får plats med sin hand i?) by putting their hands in the jars and by saying that their hands fitted.

Figure 2. Jars in the air. Teacher lent forward to better see.

In contrast, the teacher never handled the jars. Her actions entailed moving her body so she could see more easily what the children were doing (see Figure 2). Thus, her semiotic bundles were different to those of the children, as was her role in the exchanges. Apart from explaining at 1:12 that the children must be careful when they played with the jars because they were made of glass (den är gjord av glas, så när vi håller på med dem här nu så får vi kanske vara lite försiktiga), she did not provide any explanations. Thus it could not be said that the children learnt about the structure of an explanation from the teacher in this exchange. Instead, she prompted for explanations by asking questions. For example, at 1:29 she seemed to notice the children’s exploratory touches and by asking a question, she gave words to the children putting their hands in several jars. However, it was the children who used mathematical terms like thick, thin, and same and used specific actions to indicate a circular shape.

A diachronic analysis shows that some contributions to the semiotic bundles reappeared, either as repetitions or as adaptations. Parrill and Kimbara (2006) suggested that the more people see others using mimicry, the more likely they are to mimic the actions and speech of others. They also suggested that it could be one way that participants develop a rapport with each other. Although Parrill and Kimbara’s research was with adults, this video suggests that mimicry may play a similar role in children’s interactions. As the time progressed, more repetitions and adaptations were seen. For example, at 0:07 Lena pushed the jar forwards and upwards with her hand in it, perhaps to emphasise the point she was making as this action positioned the jar in

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front of her and the teacher’s face. Six seconds later, she did a similar action with another jar, although this time its purpose seemed to be exploratory. At 0:39, Mira held a jar in the same manner, perhaps because Mira was aware that in looking at Lena, who was talking, the teacher also could see Mira and her actions (Figure 2).

Figure 3. Mira using Lena’s jar holding action

At 0:47, when she placed her hand in and out of the narrow jar, Mira also held the jar upwards in a similar motion. At 1:29, while she was talking with the teacher she put her hand in a jar and pushed it forward and upwards in a similar manner to Lena’s original action (see Figure 2). A second or so later, Lena and Rita copied her. At 1:34, when she found the jar in which she could only fit her fingers, Mira showed it to the others by holding it in the same manner (upside down and in front of her body, see Figure 5). At 1:49, she pushed the two jars, that she nominated as being the same, forward and slightly upwards (see Figure 4), seemingly to emphasise the importance of her point. This action is not an institution sign as it has not been constituted by an institution (Arzarello et al. 2009) but it seems to have become more than one individual’s personal sign as it appears to have a specific local meaning, that of emphasising. It appeared to be transformed from a semiotic bundle about exploring to be included in a semiotic bundle for emphasising.

Figure 4. Showing the two jars that were the same

The children’s putting their hands into and out of jars, we did not consider to be gestures. Rather we saw these as exploratory actions and did not seem to contribute to their explanations. However, emphasising actions were considered to be gestures and, as will be discussed later, they seemed to provide a visual indication that the children

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were explaining something, such as that their hands did not fit into specific jars to the teacher.

Like the synchronic analysis, a diachronic analysis shows differences between the children and the teacher’s semiotic bundles. In contrast to the children’s predominant use of actions, the teacher transformed the children’s spoken utterances and actions, into just speech by repeating word-for-word what they said. The children did repeat some terms, but generally they produced new descriptions of the jars. They did not repeat whole sentences as the teacher did. Thus, the teacher’s contributions to the interactions were different to those of the children and this can be seen in the changes to the semiotic bundles in use.

5.2 Multimodal interactions For the children, the explanations that they provided were multimodal as they involved forming and transforming semiotic bundles using different resources. The jars provided an opportunity to develop and use some descriptions of material qualities in their explanations. When the children used terms such as oblong, thick, thin, and same, they also provided actions involving the jars. Kress and van Leeuwen (2001/2010) stated “material qualities can also acquire meaning, not on the basis of ‘where they come from’, but on the basis of our physical, bodily experience of them” (p. 74). Thus, the experiences of exploring the jars provided substance to their understanding of what these qualities signified, in McNeill’s (2008) words, the image of the qualities was in its most material form. At the same time, because the jars had the potential to illustrate the qualities, they can be said to have what Kress and van Leeuwen (2001/2010) labelled experimental meaning potential from which the children could draw on when they gave an explanation. For example, by 1:19, all three children had used the term “smal”, meaning “thin”. Following Mira’s description of how she could fill the jar with her hand, the teacher asked if there were any glass jars that the children were not able to get their hands into. Mira immediately picked up a jar by placing her finger tips in it and pushing it forward (see Figure 5).

Figure 5. The thin jar

Although she did not say anything, her previous spoken utterances provided a history to the jar which added extra meaning to the action. Not only could it be interpreted as

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“here is a jar in which I cannot fit my hand”, a direct response to the teacher’s question, but also as an explanation, “here is a very thin jar, because I cannot get all of my fingers into it, let alone my whole hand”. Interpretation of this non-verbal explanation relied on the teacher’s question and the other children’s joint exploration of these jars, as well as Mira’s previous experiences with jars. Therefore, the glass jars were a medium, which because of their features and the children’s previous interactions with them, could be considered an integral component of the mathematical explanations. They were not just a means for supporting the communication of the children’s explanations, but the meaning embedded in them through previous actions added dimensions to the mathematical explanations in a similar manner to that of speech and gestures. 6. Mathematical explanations

This paper began with questions about the mathematical explanations of young children who do not have the verbal resources of older children or adults. We were interested in how the children came to understand what constituted a mathematical explanation and the situations in which one was needed.

In this video, as was the case with many of the videos that we collected from this preschool, the teacher did not formally teach the children and so mastery of mathematical explanations relied on what the children gained in interactions with the teacher and the other children. It seemed that the children constantly gave explanations, in the sense of Bishop’s mathematical activity, such as why a jar was thin (because it had a narrow neck, which was difficult to get their hands in). Sometimes the explanations were to questions put by the teacher and included words and actions. At other times the explanations seemed to be to themselves, such as when Mira put her hand into a jar twice (00:51-00:56), with both times having to apply some force to remove it. In repeating the action, it seemed Mira wanted to understand more about why her hand became stuck. Consequently, we considered this to be an explanation to herself which answered her unspoken question about why it was hard to pull out. The discussion of sameness provides an example of how Mira’s explanation drew on and modified gestures and actions on the jars that had previously been used by the other children. It also gives insights into how the teacher’s repetition of the children’s verbal expressions during the two minutes contributed to the recognition of what a mathematical explanation was.

Our interpretation of the video suggested that Mira’s placing of her hands in two jars simultaneously, at 1:42, and feeling how much space there was inside led to her recognising an example of the mathematical object, sameness. She then put the bases of the jars together and found they covered the same area, which prompted her to state at 1:49 that they were “lika samma” (equal same). Her actions may have contributed to her expanding her understanding of sameness, even if the term that she chose to label this was a non-standard Swedish expression. From Sfard’s (2009) perspective, it could be said that “lika samma” realises the actions. Later, when she provided actions and words to illustrate what she meant to the teacher, these realised “lika samma”. The glass jars was implicit in both sets of realisation processes. They prompted the recognition of sameness and also allow sameness to be exemplified, first to Mira herself and then to the others, through different actions on them.

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Radford (2008) stated that to recognise what is equal about two things is “to select some features of a and b and to dismiss some others. This cognitive act entails the formation of a concept—a generalized entity—that does not fully coincide with any of its instances” (p. 83). Mira’s actions with the glass jars supported her to notice the “sameness” of certain features. Initially, the sensation that both hands occupied a similar space in the two jars seemed to make Mira aware that there was something different about the relationship between these two jars than between the other jars she had put her hands into. By this point in time, the children had tested many jars by spreading out their fingers in them to give a bodily sensation of their volume. Although the teacher (00:24) had originally asked if the jars looked the same, by this point the teacher seemed to be focused on the space that the children’s hands took up in the jars by asking them to find jars in which their hands could not fit. Thus, although sameness had not been the focus of this exploration, the previous interactions had led Mira to notice the volume of the jars. This bodily sense of sameness seemed to contribute to her putting the bases together (see Figure 6). At this point, exploration of the jars becomes an explanation, beginning with the verbal expression “Ha, ha” and a little later “De är lika samma” (They are equal same). Now sameness refers not to the sense of space, but to the two bases being congruent or having the same area.

Figure 6. Mira noticing that the bases are the same

Nevertheless, her verbal description of what was the same seemed to be restricted by her lack of fluency which may have contributed to her using gestures to show the teacher what she considered to be the same. These gestures resembled Rita’s earlier actions (1:07) when she ran her finger around the bumps at the base of the jar, which may have supported Mira to identify another attribute that was the same, that of the bumps at the bases of the jars which she then described to the teacher. Mira’s actions with the jars both immediately before and after her utterance of “De är lika samma” (“they are equal same") were adoptions and adaptations of actions that she had watched the other children use. Thus, Mira’s developing understanding about sameness seems different to that proposed for older students. For example, Sfard (2007) stated “it is rather implausible that learners would initiate a metalevel change by themselves. The metalevel learning is most likely to originate in the learner’s direct encounter with the new discourse” (p. 576). However, spoken language did not seem to be the stimulus for developing Mira’s understanding of sameness, rather it seemed to be grounded in her actions. As Flewitt (2006) indicated, to focus exclusively on children’s verbal utterances leads to a restricted understanding of their meaning making.

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As observers, we considered Mira’s purpose in responding to the teacher’s implied question was to convince the others that the two jars were the same, according to how her actions matched her developing knowledge about sameness. She was not just talking about what she experienced, but was trying to explain why those experiences could be constituted as examples of sameness. Thus, her recognition that she was dealing with examples of sameness supported her to see that there was a need for a mathematical explanation, provide one and have it assessed as adequate, even if this did not eventuate in a formal evaluation. In this way, she was learning culturally appropriate understandings of sameness and mathematical explanations. In regard to explanations, Mira’s use of the forward and upward thrust of the jars (see Figure 4) to emphasise that she had an important point to make suggests that she understood that there was a difference between exploring the jars and explaining a point to others.

7. Conclusion Our analysis of two minutes of video recording can only be seen as a beginning of coming to terms with how young children provide explanations. However, it seems that conceptions of what older children do may not be appropriate in considering what preschool children do. The teacher’s focus on verbal language, although well intended and justifiable, meant that she missed opportunities to develop children’s actions and thus broaden the resources that they could use for their mathematical explanations. As adults ourselves, it took time, discussion and thinking to see that these children were doing something valuable in the way that they adopted and adapted each other’s actions on the glass jars. Therefore it is likely to be hard for teachers “to start from where learners are” (Mercer 1995, p. 84) if they as adults cannot recognise what that start actually is.

In future explorations of this issue, we anticipate the use of semiotic bundles and multimodality analyses will be valuable. Combined, they provide insights into how the children’s mathematical explanations not only utilised verbal language, gestures and exploration of the jars but were developed from adoptions and adaptations of each other’s earlier contributions. The synchronic and diachronic analyses of semiotic bundling provided insights into the potential verbal and non-verbal resources that the children drew upon to form their mathematical explanations. For example, it showed how the thrusting forward and upwards of the glass jars came to move from being an exploratory action initially to being a gesture that marked an explanation by indicating that they had an important point to make.

The multimodality interaction analysis enabled us to understand how the glass jars had an active role in determining what could and could not be discussed. The shared knowledge that glass could be broken led to the teacher providing an explanation about why the children had to play with the jars carefully. The composition of the jars also allowed the children to make sounds by flicking their fingers against the sides. Holes at the top of the jars allowed the children to wear them like mittens. The roundness of the jars contributed to the children running their fingers around the neck, the middle and the base, which was supported further by the bumps on the base of the jars that Rita, in particular, found fascinating. The range of jars meant that an exploration of sameness was not a trivial activity for the children. Thus, the artefacts were not mere vehicles for the transmission of children’s explanations, whether pre-existing or being formed in the moment. They brought with them an experiential meaning potential (Kress and van Leeuwen

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2001/2010) that contributed to the mathematical objects that could be explored, discussed and utilised in the children’s mathematical explanations.

8. Acknowledgement We would like to thank Götz Krummheuer who was with us as we did our initial fine-grained analysis of this video and who challenged us to think more deeply about what it was that we were seeing.

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