Sandy Bakos
email: sbakos@sfu.ca

Nathalie Sinclair

Simon Fraser

Pips (times) Pods: Dancing towards multiplicative thinking

Keywords: Touchscreen technology, multiplicative reasoning, primary, tangible

This paper examines the design of an iPad touchscreen application (TouchTimes), that provides children with the opportunity for direct mathematical/multiplicative mediation through fingers and gestures. We describe the explorations of a pair of third-grade students with this technology, in which engagement with number in a multiplicative sense draws on a singular interaction between the eyes and both hands. Using a theoretical perspective informed by tool use and by embodiment in mathematical thinking and learning, we seek to gain insight into the affordances of the TouchTimes app in the development of multiplicative awareness in young children, with a specific focus on the multimodal nature of their mathematical interactions.

Introduction

Multiplicative reasoning is the ability to work flexibly and efficiently with "the concepts, strategies and representations of multiplication (and division) as they occur in a wide range of contexts" (Siemon, Breed & Virgona, 2005, p. 2), concepts that include direct and indirect proportion. Among other things, such reasoning involves learners viewing situations of comparison in a multiplicative rather than an additive sense. As students progress to larger whole numbersand then to decimals, fractions, percentages, ratios and proportionsmultiplicative reasoning becomes key to a large number of mathematical situations found in upper elementary and middle school [START_REF] Brown | The struggle to achieve multiplicative reasoning 11-14[END_REF].

In the primary grades (K-3), however, the action of repeated addition is commonly used as the initial (and, for some, the sole) means for introducing and working with multiplication, and it tends to become firmly entrenched as the dominant perception of multiplicative situations, both for students and for primary teachers [START_REF] Askew | Multiplicative reasoning: Teaching primary pupils in ways that focus on functional relationships[END_REF]. This becomes problematic when students begin to engage with mathematics that requires a direct capacity to think multiplicatively (e.g. [START_REF] Siemon | From additive to multiplicative thinking[END_REF]. Consequently, multiplication may prove a crucial turning point in student learning, possibly a turnstile for mathematical competence. Rather than rely exclusively on repeated addition, approaches need to be developed and implemented in the early grades that highlight the function aspect of multiplicative reasoning that can be so critical to future success with mathematics.

Since [START_REF] Vergnaud | Multiplicative structures[END_REF] wrote about the conceptual field of multiplicative structures, considerable attention has been given to the comparison of quantities using multiplicative thinking. Extensive research has documented difficulties students have with employing it in middle school and beyond, which [START_REF] Brown | The struggle to achieve multiplicative reasoning 11-14[END_REF] claim has not improved since the 1970s. Limited student experience with different multiplicative situations is proposed as a contributing factor to this significant challenge (e.g. [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF]. Furthermore, [START_REF] Askew | Multiplicative reasoning: Teaching primary pupils in ways that focus on functional relationships[END_REF] contends that the lack of development of multiplicative reasoning in the primary grades is "a consequence of predominant approaches to teaching multiplication limiting access to opportunities through which thinking functionally can emerge" (p. 1).

Multiplicative reasoning and digital technology

Digital technology is providing new resources and means that show promise in supporting the mathematical learning of young children (e.g. [START_REF] Sedaghatjou | Exploring cardinality in the era of touchscreen-based technology[END_REF]. In particular, the multi-touch affordances of TouchCounts (hereafter TC) allow children to produce and transform objects directly on an iPad screen using fingers and contact gestures, providing new counting and early arithmetic opportunities that are enabled by the app's visible, audible and tangible design. [START_REF] Sinclair | The haptic nature of gesture: Rethinking gesture with new multitouch digital technologies[END_REF] show how the interaction in TC between fingers and eyes-between the tangible and the visual-enables new ways of thinking about number.

TouchTimes (TT), a novel extension of TC, is a gesture-based, multi-touch environment for multiplication designed to improve children's flexible and relational understanding of multiplication. Multiplicative rather than additive, TT allows children to create the "visual images of composite unit structures in multiplicative situations" (p. 306) that [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF] argue are fundamental to developing multiplicative reasoning directly. Children can produce and transform objects directly on an iPad screen, using new gestural experiences of multiplication that provide direct feedback through both symbolic and visual representations. Our research focus involves exploring the influence of TT as a gesture-based modality for conceptualizing, visualizing, experimenting with and communicating about multiplicative relationships with students in primary grades (2-4).

Brief description of TouchTimes

TouchTimes is an iPad application, which when initially opened displays a blank screen that is split in half by a vertical bar (Figure 1a). A user can place and hold her fingers on one side of the screen to create coloured discs, which we call "pips". (In what follows, we presume that the user has chosen the left side of the screen, but the description is also symmetric for the right side of the screen.) Each finger that maintains continuous contact with the left side (LS) produces a different coloured pip (Figure 1b). When the user taps her finger(s) on the right side (RS) of the screen, a unit of coloured discs appears. These units, which we call "pods", are comprised of the coloured pips that correspond to those pips being created by the user's fingers in contact on the LS (Figure 1c). As each tap creates a new pod, TouchTimes displays the number sentence that corresponds with the pips and pods created by the user. When a finger is taken off a pod, it remains on the screen, but becomes slightly smaller, so that users can create numerous pods. As long as at least one finger remains in contact with the LS, the pips are maintained within the pods, but when the user removes all fingers, the pods disappear ("multiplying by 0"). Contact with the screen can be made either one finger at a time or several fingers simultaneously. Pods can be dragged into the trash, at the bottom of the screen. [START_REF] Pimm | How do you make numbers?[END_REF] note that TouchCounts 'takes care of the counting' through both symbolic and auditory means in response to the user's fingered requests. Although done in a different manner, TT 'takes care of the multiplying', both in terms of making sure that the pods on the RS are reflective of the number of pips on the left, and in terms of ensuring that the equation on the screen corresponds to the pips and pods that are displayed in response to the fingered requests of the user. [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF] argue "that the co-ordination of composite units is the core of multiplication, and that young children's (8 year-olds) concept of multiplication is based on the meaning they give to the composite units they construct" (p. 306). Thinking multiplicatively involves the ability to simultaneously think about units of one and units of more than one. TT embodies a multiplicative model that involves the co-ordination of two quantities similar to Figure 1d below (Boulet, 1998, p. 13). One way to conceptualise this action is to see the LS touches as the number of pips which are then unitised into pods, with the pods then unitised into the product, as in the Davydovian approach (see Boulet, ibid.). In this view, the sentence 3 x 4 = 12 is read as the multiplicand times the multiplier equals the product, which reverses the typical North American approach (3)(4x). Of course, it is also possible to see the pods as being groups of pips, which can be understood in terms of repeated addition. However, the simultaneity of the two-handed touching retains less of the temporal, sequential sense of repeated addition. Part of the research goal is to develop tasks that promote a more multiplicative sense of multiplication.

Theoretical Framing

The theoretical orientation of this study draws principally on theories of embodiment and the relation between bodily movement and mathematical meaning-making (see [START_REF] Nemirovsky | Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit[END_REF][START_REF] Radford | Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings[END_REF]. We take the monist, ontological position found in inclusive materialism (de Freitas & Sinclair, 2014) on the nature of body and mind, which does not subordinate sensorimotor actions to thinking, but instead recognizes the way in which new ways of moving one's body are new ways of thinking. For this reason, we are less interested in studying TT as a manipulative involving acts of moving structures and are more interested in the structured acts of moving that TT modulates. Given this orientation, and the gesture-rich design of TT, our focus will include not only verbal explanations but also gesture-based actions, as we are particularly interested in the structured acts of gesturing that arise through the use of the app. The epistemic and communicative nature of gestures has been well documented in the literature (see Sinclair & de Freitas, 2014) and warrants our attention to gestures as particularly relevant structured acts of moving. Since the multitouch environment also enables children to work together, we will also focus on jointly structured acts of moving. Therefore, we will be investigating how a pair of students' interactions with TT prompts new gestures and how these new structured acts of moving are related to multiplicative thinking.

Methods

The data for this paper comes from an exploratory conversation conducted by one of the authors as part of an iterative design experiment aimed at refining the TT prototype and developing appropriate tasks for use with grade two and three children. After the researcher requested a pair of students who had not yet used TT (three pairs of children had already participated), the two girls, whom we refer to as Jacy and Kyra, were selected by their classroom teacher to explore TT with the researcher. This interaction occurred in an elementary school in a culturally diverse and affluent neighbourhood in British Columbia, where the interviewer worked for approximately 30 minutes with the pair. A video-recording of this interaction was created, and the drawings produced by the girls were kept. The children were initially given time to become familiar with TT through independent exploration prior to any specific requests or questions from the interviewer. Given that this was the pair's first encounter with TT, it provided an opportunity to observe how they made sense of the app and if their interactions with TT would lead to an ability to identify certain multiplicative aspects with distinct handedness. After a period of free exploration that lasted about seven minutes, the interviewer began to use prompts to focus the attention of the children on certain features of their creations on the iPad screen. Using TT, in conjunction with the significant presence of an adult, the initial aim of this research was to understand better the impact of the affordances of TT on young children's multiplicative thinking, with a particular focus on the multimodal and joint nature of the mathematical activity of two girls working together on a single iPad.

Data analysis

In order to account for the multimodal, distributed nature of the phenomenon seen in the videorecordings, following our theoretical tenets, we produced an orchestral transcription of ten-second increments involving three separate, but interacting modalities: voices, hands and the iPad screen itself (Figure 2). The top three rows were designated for the voices of each child and the interviewer, thus providing a way to sequence the speaking visually in a manner that would effectively display overlapping voices. Descriptions of what the children's hands were doing on both sides of the iPad screen, were sequenced in time with the voices transcribed in the section above.

As the importance of the iPad screen itself became apparent, additional rows were created to reflect what could be seen on the top, left and right sides of the iPad screen, supplemented by screen shots to illustrate these ideas. The orchestral transcription enabled us easily to identify patterns of structured acts of moving throughout the course of the interaction, by comparing the images.

Results

From the orchestral transcript we chose two intervals that illuminate some of the gestures made by the girls during their exploration of TT that we see as being relevant to multiplicative thinking. Prior to presenting these two intervals, however, we describe the pair's initial interactions with TT, which began with the researcher giving the pair permission to "play a little bit". Each girl placed and removed her pointer finger on and off the screen in what appeared to be random motions. After approximately fifteen seconds, Jacy placed her thumb on the screen in addition to her pointer finger, creating two pips. Kyra next touched the screen, creating a pod composed of two pips (Figure 3a). Jacy then began to use both hands. It is shortly thereafter that Jacy said, "How do you get those mini ones?", thus marking a transition from random tapping to more intentional actions on the iPad screen.

Disappearing pips and 'dancing' pods

Approximately five minutes into the exploration, Jacy instructed Kyra to "Wait, just press a lot. Press your whole hand." Kyra responded by placing all five RH fingers simultaneously on the LS and waited while Jacy created pods on the RS one at a time with her index finger. The top of the iPad screen read 5x1, 5x2, 5x3, 5x4, 5x5, 5x6 until Kyra abruptly removed her hand from the screen, causing both girls to laugh. At this point, Jacy began directing the creation of pips through the placement and removal of Kyra's fingers on the LS: when Kyra returned her five fingers to the iPad, Jacy physically removed her partner's pinky and thumb from the screen, causing some confusion. Jacy tried again, telling Kyra to "Wait. Put those [indicating Kyra's RH fingers] and then take up your pinky and thumb." Kyra placed three fingers on the LS, while Jacy created one pod on the RS. This was clearly not what Jacy was after, and she further instructed Kyra to "Press your pinky and thumb away. No wait. Put your pinky and thumb down [Jacy physically presses Kyra's pinky and thumb down (Figure 3b)] and now take them away." Each time Kyra's pinky and thumb touched the screen, the resulting pips changed colour, and the composition of the pod that Jacy was 'holding' changed from three pips to five pips and back to three pips as Kyra alternated rhythmically, making the pod appear to be swinging back and forth (Figure 3c). Jacy created a second pod and at this point Kyra pointed to the pods with her free hand and declared with a laugh that "They're dancing!". Throughout this brief episode, Jacy became interested in exploring how the creation and deletion of pips on the LS affected the shape and colour of the pods on the RS, as demonstrated by both her comments and her actions. This complex interplay between the girls and TT involved a coordination of two pairs of hands, which resulted in a joint holding and repetitive-tapping gesture.

The appeal of the 'dancing' pod seemed to draw the girls' attention to the relation between the RS and LS finger touches, which we see as significant in terms of multiplicative thinking. It involved the intentional co-ordination of two quantities: the multiplicand (three or five pips) and the multiplier (the one pod). The girls' joint gestures expressed their attention to the way in which the pods on the RS are directly affected by changes related to the number of pips controlled by the finger actions on the LS.

Counting by fours and fives

After seven minutes of exploration, the interviewer asked Kyra to put four fingers down, which she did with a simultaneous four-finger gesture, creating four pips on the LS. Without prompting, Jacy created a single pod on the RS with her index finger (Figure 4a). When questioned about what she noticed, Jacy replied, "Four times one equals four". She then created additional pods with her thumb while maintaining her index finger on the original pod (Figure 4b). When asked how Jacy could make five with one finger on the RS, Kyra's thumb touched the screen to create five pips, while Jacy continued to add pods on the RS (Figure 4c). After a pause, the interviewer asked, "Are you making a five now?" At this point Jacy, who had been creating additional pods with her thumb, glanced towards the top of the screen which displayed 5x10=50, then 5x11=55. She said, "Fiftyfive, sixty…" and with excitement, "Wait! It's counting up by fives! Sixty-five, seventy, seventyfive, eighty, eighty-five, ninety, ninety-five…". The pair were then asked what they could do to count by fours. Kyra removed one of her five fingers from the screen. Jacy said, "Count up by fours?" and physically moved closer to the left side of the screen, causing Kyra to remove her hands. While placing her thumb and all fingers except her pinky on the screen, Jacy explained that you would "Put four fingers down". Meanwhile, Kyra used her index finger on the RS (reaching over Jacy's arm) to create pods (Figure 4d). Jacy then said, "Two, four, six, eight, twelve" then "wait, what?" after she noticed the product displayed on the iPad screen go from eight to twelve. She silently observed the next product and then began to count aloud again: "Twenty, twenty-four, twenty-eight…We're counting by fours." The girls had transitioned away from single touches and were now making a simultaneous touch gesture, as in the four-finger gesture described above. The interviewer's first question was meant to draw attention to the number/colour of pips in the 4-pod they had created, but Jacy described the multiplication statement instead. The interviewer then hoped to elicit the making of a 5-pod in order to prompt the unitising of five pips into one pod. With Kyra's additional finger on the LS, Jacy did indeed begin creating 5-pods, but she did not seem to be aware of the shift from 4-pods until she noticed the multiplication statements, and that the product was increasing by five with each RS touch. This induced a particular kind of joint skip-counting gesture in which one girl was responsible for holding pips while the other girl's sequential touches created pods. The pair then switched roles to skip-count by four, with Jacy holding pips and Kyra sequentially making pods. Unlike the skip-counting that occurs in many classrooms, where children count intransitively by the unit ("four, eight, twelve", etc.), the skip-counting in TT explicitly involves gesturally expressing the unit (the pips) and the number of units (the pods).

Jacy's counting self-correction seems to indicate a shift from rote counting to co-ordinating the joint skip-counting gesture (tangible expression) with the products (symbolic expression) that she could see on the iPad screen. When asked how Jacy could create a 5-pod, Kyra who was in control of creating the pips, immediately added a finger to the screen and later removed a finger when prompted to count by fours. Kyra was controlling the effects of the multiplier (the pips) on the multiplicand (the pods) with her fingers. Jacy, however, did not attend to the effects of each added pod on the product until after she noticed the product changing in the equation at the top of the iPad screen. Skip-counting in TT using the joint skip-counting gesture involves co-ordinating two quantities (instead of one) and is therefore more multiplicative than additive.

Discussion

The intent of the TT design was for learners to notice the relation between the number and colour of pips, and the shape and content of the pods, as this is the basis for the multiplicative operation.

Although the girls created numerous pips and pods, it was not until the joint holding and repetitivetapping gesture that they seemed to co-ordinate this relation. Indeed, it appeared to be the shifting pod shape, and then the changing pip colour, that initially drew their attention. The joint holding and repetitive-tapping gesture, which arose from manipulating the screen in exploratory ways, became a gesture for expressing the relation between pips and pods. The girls were able to make visible the effect of changing the unit through the tapping on and releasing of pips, which resulted in the changing size, shape and colour of the pods. When there was more than one pod on the screen (as in Figure 4c), the joint holding and repetitive-tapping gesture produced the multiplicative effect in which one new tap produced a new pip in every pod simultaneously.

In the previous section, we outlined how the joint skip-counting gesture can be viewed as more multiplicative than additive. However, we think that a gestural shift from tapping one finger at a time to create new pods, to tapping several fingers simultaneously would provide an even stronger multiplicative effect. We know from prior TouchCounts research that sequential tapping was more frequent than simultaneous tapping, during children's initial interactions. Since the girls rarely created several pods simultaneously, we hypothesise that such a gesture involves a more difficult co-ordination and may need to be prompted by a particular task.

Another significant aspect of the actions described throughout this paper involves the joint nature of the girls' interactions. In both of the gestures identified, the pair must co-ordinate their hand movements and there were a few instances during the 30-minute exploration in which one of the girls "took over" TT. We wonder how a situation with two children co-ordinating multiple hands would differ from one in which a single child coordinates only her own hands.

Conclusion

Designed to support the development of multiplicative thinking, TT provides young learners with ways of thinking about multiplication that are not solely dependent upon the use of repeated addition. Two intervals were described in the 30-minute episode reported above, in which the girls created and sustained particular structured ways of moving their hands that we hypothesise as being relevant to the development of multiplicative thinking. The two primary gestures discussed were the joint holding and repetitive-tapping and the joint skip-counting gestures. The former prompted and enabled the girls to attend to the relation between the number/colour of pips and the pods, and thus the co-ordination of the two quantities. The latter enabled the girls to produce multiples of a number determined by the number of pips. Attention to the symbolic expression on the screen was particularly salient in the second interval. We highlight that the girls primarily engaged in free exploration throughout this experiment, with few prompts from the interviewer. In future explorations, we intend to build upon the insights gathered here to design tasks that can effectively prompt and support similar types of gestures, and to link these gestures to other off-line and TTbased actions.

Figure 1 :

 1 Figure 1: (a) Initial screen of TT; (b) Creating pips; (c) Creating pods; (d) Multiplicative model

Figure 2 :

 2 Figure 2: Orchestral transcription

Figure 3 :

 3 Figure 3: (a) Two fingers; (b) Pinky and thumb down; (c) Pinky and thumb up

Figure

  Figure 4: (a) First pod of 4; (b) Second pod of 4; (c) Many pods of 5; (d) Crossover pod making


Svanhild Breive
email: svanhild.breive@uia.no

Kindergarten teachers' orchestration of mathematical learning activities: the balance between freedom and structure

Keywords: Kindergarten, mathematics, orchestration, zone of free movement, zone of promoted action

 zone theory to investigate the relationship between zone of free movement (ZFM) and zone of promoted action (ZPA) which the kindergarten teachers set up to canalise children's actions and thinking and thus development. The results show that in the kindergarten where the ZFM is gently set up and limited to mathematics, and where the kindergarten teacher sensitively sets up the ZPA and promotes children to share, argue for and explain their mathematical ideas, and explicitly promotes the children to collaborate, is where most problemsolving interaction occur and thus facilitate children's learning possibilities the most.

Introduction

This paper reports on a multiple-case study which aims to investigate four kindergarten teachers' (KTs') orchestration of pre-designed mathematical learning activities and what impact their orchestration has for children's learning possibilities. The current debate about mathematics in kindergarten is seldom about whether or not mathematics should be part of the curricula in kindergarten, rather on how mathematical activities in kindergarten should be orchestrated1 [START_REF] Gasteiger | Fostering early mathematical competencies in natural learning situationsfoundation and challenges of a competence-oriented concept of mathematics education in kindergarten[END_REF][START_REF] Hirsh-Pasek | A mandate for playful learning in preschool: Presenting the evidence[END_REF][START_REF] Van Oers | Emergent mathematical thinking in the context of play[END_REF]). Children's opportunities to take part in mathematical discourses are important in learning mathematics, but in activities where the kindergarten teacher has a pedagogical aim, it may be difficult to balance teacher-talk and child-talk [START_REF] Dovigo | Argumentation in preschool: a common ground for collaborative learning in early childhood[END_REF]. The study reported here focuses on four KTs' orchestration of mathematical learning activities with respect to the degree of freedom2 and aims to:

 Investigate the characteristics of four kindergarten teachers' orchestration of pre-designed mathematical activities with respect to the degree of freedom, and;  Investigate what impact the four kindergarten teachers' orchestration of the mathematical activities has for children's mathematical learning possibilities.

Theoretical framework

The study reported here draws on [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] zone theory to investigate the relationship between freedom and structure in four KTs' orchestration of mathematical learning activities. The balance between freedom and structure is the main focus when [START_REF] Hirsh-Pasek | A mandate for playful learning in preschool: Presenting the evidence[END_REF] and [START_REF] Weisberg | Making play work for education[END_REF] discuss 'playful learning' as an educational approach in kindergarten. Playful learning captures both 'free play' where children play without interference from adults, and 'guided play' where the KT organises the learning environment and guides the play in desired directions with respect to a learning aim. [START_REF] Weisberg | Making play work for education[END_REF] argues that, although the KTs initiate and guide the activity in guided play, the KTs must make room for children's self-directed exploration, and it is this balance between freedom and structure that makes guided play such an effective teaching tool. Similar van Oers (2014) argues that playful learning activities should contain some elements of instruction. "The nature of the actions embedded in play can vary with respect to their degree of freedom allowed, as long as the activity as a whole remains a playful activity" (van Oers, 2014, p. 121). The learning activity must be engaging and give possibilities for the players freedom to explore the (mathematical) objects in their own manner.

In his study on preschool children's argumentation, Dovigo (2016) investigates children's learning opportunities in different types of conversations (peer-talk and child-teacher talk). Dovigo found that children had richer opportunities to participate and asked more questions in peer-talk than in child-teacher talk. It was a clear tendency that in child-teacher talk, the KT talked more than the children. However, the children's abilities to build arguments were limited in peer-talk and were facilitated in child-teacher talk. The KTs facilitated children's explanations and helped them to elaborate their argumentations, which again improved children's critical thinking and abilities to collaborate.

Through his zone-theory, [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] explores the development of children's actions and thinking through organisation of person-environment relationships. The physical environment of the child is the cultural frame which the child is acting within and thus develop its thinking. Valsiner's theory emphasises that both the developing child and the environment are structurally organised, however the structuring nature of the child and the environment is continuously and dynamically transformed. [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] uses three zone concepts to conceptualise the dynamic environmental structures that organise the child's development: Zone of Free Movement (ZFM); Zone of Promoted Action (ZPA); and Zone of Proximal Development (ZPD). The ZFM is co-constructed by the child and the adult and organises the "child's (1) access to different areas in the environment, (2) availability of different objects within an accessible area, and (3) ways of acting with available objects in the accessible area" (p. 97). Within the ZPA there may be activities or objects which the child is promoted to engage with. An important characteristic of the ZPA is its non-binding nature:

The child does not need to follow the ZPA and can act with other objects (in other ways) within the ZFM. The child cannot be 'forced' to accept the ZPA unless the ZPA is turned into ZFM. The ZFM and ZPA must be considered as a unit and [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] labels it the 'ZFM/ZPA complex'. The ZFM/ZPA-complex work as a mechanism to canalise the child's actions and thinking and thus development in culturally accepted ways. In addition, [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] discusses how ZPD relates to the ZFM/ZPA-complex, but due to space limitations, this study focuses primarily on the relationship between ZFM and ZPA in four kindergartens with respect to mathematics.

Methodology

The study reported here is a multiple-case study [START_REF] Yin | Case study research: Design and methods[END_REF], which aims to characterise four KT's orchestration of pre-designed mathematical activities and its impact for children's mathematical learning possibilities. It is part of a larger study on mathematical teaching and learning in kindergarten and situated within a Norwegian research and development project called the Agder Project3 (AP). The intervention of the AP was based on mathematical activities which were predesigned in collaboration between researchers (including myself) and the KTs in the focus groups of AP. The mathematical activities suggest how to organise learning sessions, what materials to use and suitable questions to ask etc. in order to promote a playful and inquiry-based approach to the teaching and learning of mathematics. The activities were meant as suggestions not as scripts which the KTs needed to follow to the letter and this was communicated both verbally and in written form to the KTs before the intervention. The activities are described in [START_REF] Størksen | Lekbasert laeringet forskningsbasert førskoleopplegg fra Agderprosjektet[END_REF], a book containing one-page outlines of the activities. This study takes a qualitative approach to data collection and data analysis and the empirical material was collected over two observation periods (autumn 2016 and spring 2017) during the intervention of the AP. Observations were conducted of approximately 40 minutes sessions where the four KTs, who were part of the focus groups of the project, implemented the pre-designed activities. Interviews were conducted in each kindergarten after each observation period. Overview over data sets (observations and interviews) in each kindergarten is illustrated in Table 1 below. All four KTs have at least 15-years working experience.

KT1 KT2 KT3 KT4

Autumn 2016 5 obs. + 1 interv. 4 obs. + 1 interv. 5 obs. + 1 interv. 3 obs. + 1 interv.

Spring 2017 6 obs. + 1 interv. 7 obs. + 1 interv. 0 obs. + 0 interv. 4 obs. + 1 interv. The empirical material was collected through ethnographic field notes [START_REF] Emerson | Writing ethnographic fieldnotes[END_REF] from observations and interviews. Field notes were written during and/or straight after each observed session and were supplemented when having conversations with the KTs after each session. Field notes from the interviews were made straight after the interviews and were extended when the video-recordings of the interviews later were watched. Field notes should not, ideally, include interpretations of the observed interaction [START_REF] Emerson | Writing ethnographic fieldnotes[END_REF]. However, as [START_REF] Emerson | Writing ethnographic fieldnotes[END_REF] argues, when the fieldworker starts to work with the field notes, he/she orders patterns of interactions and decides what to include and leave out from the field notes. In this research study the research questions were made before conducting the field work, which guided what I was especially looking for and thus what I included and left out from the field notes. The field notes must therefore, to a certain extent, be regarded as interpretative or analytical.

Processing and analysing data went through an iterative process. Instead of using the field notes as 'raw material' from which I started a coding process, I worked directly with the field notes and carefully contracted them into profiles of the KTs orchestration and their interaction with the children. Since the field notes were already to some extent analytical, they served as a useful starting point for this purpose. The profiles concern four key points: (1) Children's access to different areas in the environment;

(2) children's freedom to act (physically) within the accessible area;

(3) children's freedom to contribute with questions, mathematical ideas and argumentations etc.; and (4) the degree to which problem-solving interaction was promoted (that is the degree to which the children were promoted to ask questions and explain and argue for their ideas to solve mathematical problems). Key points 1-3 concern the KTs' orchestration of the mathematical activities with respect to the degree of freedom and intends the ZFM. Since the ZFM is coconstructed by the child and the adult, children's eagerness to participate and take initiative are also used to identify ZFM. Key point 4 concerns how the KTs promote children to ask questions, explain and argue for mathematical ideas to solve mathematical problems, which intends the ZPA.

As [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF] argues, ZFM and ZPA are related and work as a unit to canalise children's development, and in this case, learning and development related to mathematics. In this study I identify learning possibilities whenever the KTs and the children together solve mathematical problems through questions, explanations and argumentations.

The profiles, which will be presented below, are of course tendencies not absolute characteristics. The KTs' orchestration with respect to the degree of freedom changes dynamically during each session and from session to session. In addition, the profiles are relative, which means that the degree of freedom in one kindergarten is relative to the three other kindergartens and cannot provide an indication for how it relates to other KTs' orchestration in other contexts.

Profiles of four kindergarten teachers' orchestrations

Kindergarten teacher 1 KT1 orchestrated the mathematical activities with a relatively high degree of freedom, which is based on the way that the children were allowed to move around in the room (and even walk out of the room) and to talk about almost whatever they wanted, like birthday parties or their parents' occupation etc. The KT never told the children to sit down and pay attention, instead she promoted the children to do so by the way she enthusiastically presented the activities, which captured the children's attention. For example, in an activity about reflection symmetry, the KT introduced the activity having diverse reflection symmetrical objects in a plastic bag without telling what was inside. She shook the bag and whispered, "Listen!", which made the children curious and created joint attention. Another characteristic of the KTs orchestration was that the KT listened to almost every child's contribution (not only related to mathematics). In one of the conversations with the KT, she expressed that her desire to listen to and appreciate every child's contribution could be a hinderance for her, because her attention became very shifty. She rapidly turned her attention from one child to another. The children eagerly participated in the activities, however, as mentioned above, they often contributed with other ideas than mathematics. The KT expressed that she had a challenging group of children but their ability to pay attention to mathematics grew during the intervention. There were a lot of 'golden moments' for problem-solving interaction. The KT and the children initiated a lot of interesting 'topics' for investigation, but very few ideas were thoroughly discussed. Mathematical questions were often (not always) considered briefly, and the children seldom had to ponder about problems and to express mathematical ideas, argue for and explain their ideas in order to solve the problem. Because the KT gave the children a lot of freedom to talk and payed attention to almost every contribution, the conversations moved very quickly from one topic to another.

Kindergarten teacher 2

KT2 gave the children a relatively high degree of freedom to talk, however she often restricted children's talk to mathematics by ignoring some of the contributions that were about the children's everyday experiences. The KT also restricted the children's freedom to act (physically) to areas or with objects relevant to the mathematical activity. Although the KT for the most part focused attention to mathematics, she gave the children freedom to suggest other mathematical issues than what she initially introduced. Similar as KT1, the KT2 presented the activities in an exciting way, by use of for example excited face expressions and whispering, which captured the children's attention and promoted the children to contribute. But sometimes she also asked questions directly to children to capture their attention. For example, when Carl was distracted by something else, she said: "Carl, do you know how many building blocks there are in the red tower?" When Carl said that he didn't know, the KT further asked "Would you like to help me count?" This helped Carl, who often had difficulties paying attention, to focus his attention on mathematics. The conversations between the KT and the children were almost always mathematical, and sometimes the KT and the children had longer conversations about mathematical problems. The children had to argue for and explain their ideas in order to solve the problems, and the children eagerly participated with mathematical ideas and explanations. In addition, the KT seemed to focus on collaboration. For example, the KT had a conversation with the children about the meaning of collaboration, and the KT promoted the children to help each other if needed. She also promoted the children to listen to each other by for example asking the group of children: "Did you hear what Ada suggested?".

Kindergarten teacher 3

KT3 was a football trainer in his spare time, which was somehow recognisable from his orchestration of the activities. He gave the children a relatively high degree of freedom to act (physically) and focused on 'doing' mathematics, which for him was when the children got opportunities to use their hands, their body and various artifacts to solve mathematical tasks. In one of the conversations with the KT he expressed that 'doing' mathematics was for him an important feature of mathematics in kindergarten and therefore he especially liked physical outdoor activities.

In addition, he was giving short 'missions' for the children to perform. For example, in the 'Sorting Shoes' activity, when the children had to figure out how many shoes there were in each category, the KT gave each child a 'mission' to draw equally many lines in the bottom of the diagram as there were shoes in each category. The KT expressed several times that it was important to give the children challenging but manageable tasks, so they felt they succeeded. He often encouraged the children, in an enthusiastic manner, with comments like "good" or "great" etc. It seemed that the children enjoyed the activities and the way that the KT encouraged them. The children eagerly participated and seemed to have fun. There was relatively little problem-solving interaction and the children often solved tasks without having to explain or argue for their ideas. For example, in the activity called 'Tripp, Trapp', where the children should count stairs in a staircase and find out what number each stair had, the KT made A4 papers with numbers from 1-24 on and the children, one by one, had to pick an A4 sheet and place it on the correct stair. (Stair number 15 should have the A4 sheet with the number 15 on). The children just performed the tasks, without having to explain what they did, and why they did what they did. Sometimes the KT promoted the children to reflect on their solution strategies in retrospect, however the children's explanations were seldom helping them to solve problems in the first place.

Kindergarten teacher 4

KT4 gave the children relatively little freedom to act (physically) or talk which is based on the way that she, to a large degree, controlled who was going to talk (or 'do' something) and when. For example, in an activity called 'The Farm' the children were, at one point in the activity, supposed to find how many animals there were on the farm. First the KT asked a girl, "Helene, can you figure out how many animals there are all together?". After Helene had counted and answered the KT asked a boy, "John, can you find how many different animals there are?". The KT continued to give similar 'missions' to each child. The KT made sure that each child got the opportunity to answer or 'do' something mathematically, and she appreciated children's contributions by comments like 'that's correct', 'very good' etc. The KT expressed in one of the conversations that it was important that the children learnt to respect the other children and to wait for their turn in an activity. The KT also expressed that some activities were difficult to implement as outdoor activities, because the children often got disturbed by other things. These characteristics are of course tendencies, and sometimes the activities were a lot more open where the children had a lot more freedom to act. But, as she also expressed in one of the interviews, she thought it was difficult to 'hold back' and give room for the children to figure out the problems themselves without too much interference. It is difficult to state how 'eager' the children were to participate, because they seldom answered or did something without being asked. They accepted the KTs request to sit and wait for their turn. In some activities, like when they measured how much water there was room for in a tank, the children laughed and were having fun and showed eagerness to participate. Still they were asked to wait for their turn and respect that each child got the same opportunity to fill water. There were few incidents where the children together solved problems by expressing ideas and arguing for solutions. The children were waiting for their turn to answer or to perform 'missions'. The KT sometimes asked the children to explain what they did when they solved a task, but this explanation did not help the children to solve the problem, but to reflect on their strategy in retrospect.

Discussion

From the results above, it seems that KT1 is very concerned about freedom, and the ZFM is relatively wide compared with the ZFM the other Considering the ZPA, the results illustrate how the KTs promote children to ask questions, explain and argue for their ideas in order to solve mathematical problems. The children do not need to accept the ZPA set up by the KT, but instead of turning the ZPA into ZFM the KT may, I hold, 'advertise' for the ZPA to promote the child to act in a desired manner. In K1 the KT promotes children to act mathematically by acting in an exciting way, and by introducing the activities in a manner which makes the children curious. The children sometimes accept the ZPA, but sometimes they do not. The KT1 is carefully promoting the children to think mathematically, but the ZPA (related to mathematics) is never turned into ZFM. The KT2 also promotes children to think mathematically by acting in an exciting way. The ZPA is related to specific mathematical areas, however the ZFM is related to mathematics in general. Sometimes, especially related to some children, the KT2 carefully turns the ZPA into ZFM, that is the KT limits the ZFM to specific mathematical tasks whenever the children do not pay attention. In addition, the KT explicitly promotes the children to help each other and to collaborate which, I hold, is important for the way that the children solve problems together. KT3 promotes children to think mathematically or 'do' mathematics by a quite tight ZFM related to mathematics. The ZPA is often turned into ZFM by asking the children to perform 'missions'. However, the ZFM is relatively wide related to physical actions. KT4 almost always turns the ZPA into ZFM. What the KT promotes the children to do is also what the KT allows the children to do.

Considering the characteristics of ZFM/ZPA complex in each kindergarten which according to [START_REF] Valsiner | Culture and the development of children's action: A cultural-historical theory of developmental psychology[END_REF]1997) canalise children's actions and thinking and thus development, it seems that the KTs who limit children's actions to mathematics, but where the children's freedom is relatively wide related to what the children may talk about within mathematics and who is allowed to speak, promotes most problem-solving situations (KT2), and thus children's opportunities for learning. How the ZFM/ZPA complex canalise children's development is not only related to the boundaries of the zones itself, but how the ZFM and ZPA are set up. The KT2 is relatively mild in her way of setting up the ZFM, and instead of turning the ZPA into ZFM she acts in an exciting way which promotes the children to accept the ZPA. The KT2 'advertise' for the ZPA by the way she presents the mathematical problems and makes the children want to pay attention and accept the ZPA. The KT4 is not that enthusiastic, and perhaps that is why she must turn the ZPA into ZFM to make the children pay attention and to accept the ZPA. The KT1 is also enthusiastic in setting up the ZPA, but since the ZFM is relatively wide, the children often choose to act in other ways than what the KT promotes. The results support Hirsh-Pasek et al. ( 2009), [START_REF] Weisberg | Making play work for education[END_REF] and van Oers (2014) who emphasise that playful learning activities should have some structure as long as the activity as a whole remains a playful activity and as long as the KTs gives freedom for children's self-directed play, so they may explore the content in their own manner. The results also supports [START_REF] Dovigo | Argumentation in preschool: a common ground for collaborative learning in early childhood[END_REF] results in the sense that whenever the KT is structuring the environment around mathematics, but opens up for children's own exploration around mathematical ideas, the children are canalised into more problem-solving activity and thus create possibilities for children's mathematical learning.

The results indicate that instruction which structures children's actions around mathematics but introduces the mathematics in an 'exciting' way and allows and promotes children to contribute with various mathematical ideas not necessarily related to the aimed subject area, captures children's attention and promotes their voluntarily participation in the problem-solving activity and thus facilitates children's possibilities for mathematical learning.

Limitations of the study

This study focuses on the KTs' orchestration and its consequences for children's learning possibilities and does not consider how the children, in light of for example their background, influence the KTs' orchestrations and the nature of interaction in each kindergarten. Moreover, in this study the ZFM/ZPA complex is considered on a group level. It would be interesting to investigate the ZFM/ZPA related to each child in the groups, to see which children benefitted the most from the KTs' different orchestrations.

Table 1 : Overview over data sets (observations and interviews) in the four kindergartens

 1

  KTs set up, both with respect to what the children are allowed to talk about and what they are allowed to do (physically). The children are even allowed to walk out of the room if they want to, and they can talk about whatever they want. KT2 restricts the ZFM to mathematics, both what the children can physically do and what the children are allowed to talk about. However, KT2 gives the children freedom to talk about and work with other mathematical objects than what she initially promotes. KT3 restricts the mathematical talk to specific mathematical areas, however the ZFM is relatively wide when it comes to what the children are allowed to do (physically). The KT3 gives the children freedom to move physically and to make loud voices when they solve the mathematical tasks. KT4 is the most controlling of the four KTs, and the ZFM is relatively narrow both with respect to what the children are allowed to do (physically) and what the children are allowed to talk about. KT4 decides, to a large degree, who is allowed to talk (or 'do' something), when the children are allowed to talk (or 'do' something) and what the children are allowed to talk about.

The term 'orchestration' is used in accordance withKennewell (2001), as a broad metaphor for how the KTs structure or organise the activity through use of questions, cues, prompts, information, demonstrations etc.

The term 'degree of freedom' is used in accordance with van Oers (2014) as a characteristic of the way an activity (in cultural-historical activity theory) is carried out and refers to the "degrees of freedom allowed to the actor in the choice of goals, tools, or rules" (p. 113, emphasis in origin), which in turn initiates the actor's choices of actions.

The Agder project is funded by the Research Council of Norway (NFR no. 237973), The Sørlandet Knowledge Foundation, The Development and Competence Fund of Aust Agder, Vest Agder County, Aust Agder County, University of Agder and University of Stavanger.


Ingvald Erfjord
email: ingvald.erfjord@uia.no

Martin Carlsen
email: martin.carlsen@uia.no

Sigurd Hundeland
email: per.s.hundeland@uia.no

Characterising the mathematical discourse in a kindergarten

Keywords: Discourse, inquiry, kindergarten, mathematics, playful learning

In this study we investigate the mathematical discourse in a kindergarten. The mathematical learning activity engaged with was initially designed by researchers for 5-year-olds, and the kindergarten teacher orchestrated the mathematical activity. Observational data was quantitatively analysed by measuring how time and talk were distributed between the kindergarten teacher and the children. We also analysed whether the talk was focused on mathematics or not. Our analysis shows that the time elapsed during the activity was distributed unequally, the nature of the participants' utterances shared both similarities and differences, while the engagement nurtured upon the kindergarten teacher's request varied from each of the requests. Based on these results we characterise the mathematical discourse and hypothesise about the children's potentials for mathematics learning.

Introduction

Researching teaching and learning of mathematics in a kindergarten setting has been gaining improved focus over the last decade. Since the establishment of the early years mathematics research group at the 6th Congress of the European Society for Research in Mathematics Education in Lyon in 2009, a multitude of research foci have been launched and 50-100 research studies have been conducted (see CERME proceedings, www.mathematik.uni-dortmund.de/~erme). This is not to neglect journal articles published in the same period as well as before 2009. Early years mathematics has gained its place as a research field with mathematics education research also due to the four POEM conferences we have seen so far (POEM -A Mathematics Education Perspective on early Mathematics Learning). A huge portion of the conducted studies have been qualitative in nature, as case studies have been conducted, observations and interviews have been made [START_REF] Levenson | Early years mathematics[END_REF]. This study is an attempt to broaden the qualitative research scope and include quantitative aspects of mathematics discourse in a kindergarten context.

The aim of the analyses made in the current study is thus to quantitatively characterise the mathematical discourse evolving as a kindergarten teacher (KT) orchestrates a mathematical activity for her five-year-olds. In the analysis we seek to develop an analytical approach which quantifies the collected qualitative data. We use the term mathematical discourse in line with [START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF], as a type of communication featuring mathematical words that "bring some people together while excluding some others" (p. 573). According to Sfard, a mathematical discourse has the features called visual mediators, narratives, and routines as well. However, in the current study we focus at the verbalisations made by the KTs and the participating children. The results of our analysis will be discussed in light of the framework of mathematics discourse in instruction (MDI), developed by [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF], which heavily build on [START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF][START_REF] Sfard | hin in as communicatin uman de elopment de elopment of discourses, and mathematizing[END_REF] work. For the present study we have formulated the following research question:

What characterises the mathematical discourse in a kindergarten in which a kindergarten teacher and five-years-old children engage with a planned mathematical learning activity?

In our analysis we have particularly focused at three dimensions: (1) the distribution of time in the activity;

(2) the nature of the participating persons' utterances; and (3) the engagement nurtured through the KT's initiatives, requests, and prompts. Our focus at adult-child interactions has been studied elsewhere (e.g. [START_REF] Carlsen | Orchestration of mathematical activities in the kindergarten: The role of questions[END_REF][START_REF] Dovigo | Argumentation in preschool: A common ground for collaborative learning in early childhood[END_REF][START_REF] Vogel | Videocoding -A methodological research approach to mathematical activities of kindergarten children[END_REF]. [START_REF] Dovigo | Argumentation in preschool: A common ground for collaborative learning in early childhood[END_REF] studied argumentation as a basis for collaborative learning and problem solving in a preschool setting. Dovigo adopted a sociocultural approach in his analysis, compared teacherchildren talk and peer-talk, and investigated the role of argumentation in empowering children's collaboration and problem solving through discursive practices. Results from quantitative and qualitative analyses show that argumentation is effective in cultivating shared and critical thinking amongst the children.

The mathematical activity that was orchestrated by the KT (one of many) was designed by researchers in mathematics education (the authors of this paper among others) in collaboration with the KT (among others), as part of the research and development project called "The Agder project"1 (AP). This is why we use the term planned mathematical learning activity in the research question.

These planned mathematical activities were designed based on the principles of playful learning and inquiry approach to the learning of mathematics. Playful learning as a construct comprises free play (child-initiated and child-directed play) and guided play (adult-initiated and child-directed play). In AP we drew on the principle of guided play as the KTs were empowered to orchestrate and guide the children's play in the mathematical activities in purposeful directions in order to plausibly reach aims for the activities. Furthermore, The KTs functioned as catalyst for the interest, curiosity, engagement, and mathematical sense-making of the children (cf. [START_REF] Weisberg | Making play work for education[END_REF]. Adopting inquiry as an approach to the learning of mathematics is in accordance with [START_REF] Jaworski | Learning communities in mathematics: Creating an inquiry community between teachers and didacticians[END_REF] inquiry as "a way of being in practice" (p. 103). Inquiry as a way of being is empowering when five-year-olds and their KT interact to achieve insights in mathematics. By seeking answers to mathematical prompts and questions, the children's curiosity and excitement are nurtured and met. [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF] developed the MDI framework to analyse the mathematics made available for students to learn in the classroom. The framework was developed for a South African classroom setting, deviating from a Norwegian kindergarten context. We have thus slightly adapted the framework in order for us to purposefully employ this framework in our analyses of orchestrations of mathematics activities. However, the main components are adopted in our use of the framework. But as will be seen, we have adapted the definitions of these components to also encompass a Norwegian kindergarten context.

Mathematical Discourse in Instruction

According to [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF], it is important to focus on what mathematics students are supposed to learn, when analysing teaching. Thus, the object of learning, e.g. a concept, procedure, algorithm, is the starting point for the MDI framework. In our context and analysis, acknowledging and accommodating the social pedagogy tradition of Norwegian kindergartens' enterprise, we use the concept of intention to address what the children are supposed to meet and make experiences of in the mathematical activities. Further, [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF] argue that the mediational means used in teaching to reach the object of learning, are exemplification (examples and tasks), explanatory talk (naming and legitimations), and student participation. We believe learning takes place in social, communicative settings, in which children and adult(s) actively participate, engage, and argue in interaction. Hence, it is of importance to consider children's possibilities to talk in the orchestrations, their opportunities to "speak mathematically and to verbally display mathematical reasoning" (Adler & Ronda, 2015, p. 245).

The mediational means of exemplification manifests itself as examples and tasks. Examples used in mathematics teaching/orchestration serve the purpose of being a representative of a larger class, one particular case of that class from which it is possible to generalise [START_REF] Zodiak | Characteristics of teachers' choice of examples in and for the mathematics classroom[END_REF]. Deliberate use of examples is important to reach the intention/object of learning. Tasks are also commonplace constituents of mathematics teaching/orchestration. Tasks are given by the teacher/KT to bring to the fore certain mathematical properties, actions etc. Tasks are thus what the children are supposed to do with the examples given [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF].

The mediational means of explanatory talk comprise what is to be done and known, that is to name and legitimate the focus and content of the talk between the teacher/KT and the children. To address this, [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF] coin the terms naming and legitimation. Naming is "the use of words to refer to other words, symbols, images, procedures or relationships" (p. 244). Naming may be either colloquial, non-mathematical everyday language, or mathematical, either mathematical words used or reading strings of symbols (Ms) or formal mathematical language used (Ma). Legitimation comprises four domains: mathematical, non-mathematical, curriculum and teacher. Within the mathematical domain there are local (L) criteria (specific case, convention) and general criteria (partial (GP) and full (GF) that give authority. Within the non-mathematical domain there are also criteria that give authority, called visual (V), positional (P), assigning the authority to the speaker, and everyday (E) knowledge and experience. The significance of addressing all these criteria is, according to [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF], "the opportunities they open and close for learning" (p. 244). The KT's utterances will be analysed according to the levels in the MDI framework (cf. [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF], where level 1 is a type of mathematical discourse in which the children contribute with answers to yes/no questions or single words to the KT's unfinished utterances. Level 2 is a type of mathematical discourse in which the children contribute with answers to what/how questions in phrases/sentences. Level 3 is a type of mathematical discourse in which children contribute with answers to why questions and present ideas and where the KT revoices/confirms/asks questions (cf. [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF].

In our discussion we adapt the MDI framework in order to analyse the KTs' orchestration of mathematical activities. Thus, not all facets of the MDI framework will be used in the discussion. The reason for our adaptation is that the MDI framework is developed to analyse mathematics teaching in (South African) schools. The Norwegian kindergarten context differs significantly from that school context. As will be seen in the analysis, to approach the intention/object of learning of the mathematical activity, the KT exemplifies by the use of examples, use of tasks made explicit through questioning, and explanatory talk. The children participate in the interaction with the KT, question and argue for their opinion(s).

Our analytical approach

In this study we collected video data from the orchestration of a mathematics activity by one kindergarten teacher. The activity involved five five-years-old children in this kindergarten who participated in AP, selected by the KT. The studied KT and children were chosen out of convenience, based on the consideration that the only criterion for selection was that the observed KT and children had to participate in AP and voluntarily participate in this specific study.

In order to analyse the mathematical discourse emerging from the KT's orchestration of a mathematical activity, we developed an analytical tool that fitted our purpose of quantifying qualitative data. Quantification of qualitative video data has also been studied by [START_REF] Vogel | Videocoding -A methodological research approach to mathematical activities of kindergarten children[END_REF], however from a slightly different perspective. The first step of our analytical process was that we agreed upon making 'quantification of qualitative' data the main focus of our analysis. With this idea in mind, the second step of our analysis continued as two of the researchers met and discussed ways of making this quantification. These two researchers agreed that use of time was a critical element in the orchestration, who had the floor? Were any questions asked, and by whom? Were the questions mathematical ones or not? What other comments were made, and by whom? Were these comments mathematical or not? etc.

The result of that meeting was submitted to the third researcher, who had not been taking part in the discussions so far, comprising a third step in our analysis. This researcher critically scrutinised the preliminary analytical tool, asked critical questions and made modifications to the tool.

All three researchers met for the fourth step of our analysis as we refined the analytical approach, agreed upon the various dimensions necessary to scrutinise in order to analyse the mathematical discourse in the kindergarten, as well as pinpointed sub-categories within the three dimensions. The idea behind our analytical approach is to steadily funnel down to the core of our analysis. The distribution of time is seen as a first attempt to separate the various elements of the activity. Secondly, we discriminate between what we call verbal mathematical contributions and nonmathematical or colloquial contributions. Thirdly, we have chosen to make a deeper analysis of those verbal contributions in which mathematics is focus, what [START_REF] Sfard | hin in as communicatin uman de elopment de elopment of discourses, and mathematizing[END_REF] would label a mathematical discourse. Our analytical approach is summarised in the following table (

Analysis of the mathematical discourse associated with 'The secret bag'

The mathematical activity called 'The secret bag' was orchestrated by the KT Tone (pseudonym). The activity addresses geometrical shapes of both two and three dimensions and an in-transparent fabric bag. The KT and the children talked about and discussed the various geometrical shapes, and they talked about the connections between two-dimensional shapes and three-dimensional shapes, as some two-dimensional shapes may be turned into three-dimensional ones and vice versa. Continuing from this, the KT put all the shapes in the in-transparent bag, and let one child at a time, hold his/her hand into the bag, pick one of them and tactilely reason what shape the picked one has. As seen in Table 2, the KT holds the floor for approximately 1/3 of the total time used, the children are active discussing more or less on their own for about 15 % of the time. About 50 % of the time was used for dialogic communication between the KT and the children.

Category

Distribution of time KT active 00:07:16 Dialogic communication 00:10:54 Children active (KT interference tolerated) 00:04:11 Concerning the verbal communication going on in the activity, we found that the KT made 125 mathematical utterances and 99 non-mathematical utterances. The children contributed with 103 mathematical utterances and 86 non-mathematical utterances. An utterance is here viewed as an instance of speech. However, the non-mathematical utterances enable the mathematical utterances to emerge. Thus, all utterances contribute to the ongoing mathematical discourse (cf. [START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF][START_REF] Sfard | hin in as communicatin uman de elopment de elopment of discourses, and mathematizing[END_REF]. In this study we are occupied with the KT's utterances that we characterise as mathematical and the verbal engagement nurtured amongst the 5 children involved. Upon the KT's verbal utterances, we counted the number of contributions of the various children, see Table 4. We further registered the number of children responding to the KT's questions and prompts, thus we see this number as a measure of the mathematical engagement nurtured, see Table 5. Three questions from the KT were not responded to at all, and none of questions/prompts engaged all five children. At 46 occasions it was only one child responding to the KT's question/prompt. However, each child might contribute with several responses to the initial question/prompt. Thus, the total number of responses in Adopting an inquiry approach to the orchestration of mathematical activities in kindergarten, will plausibly create mathematical engagement amongst the children. We thus further scrutinised each of the 77 contributions of the KT that engaged the children mathematically, and we analysed the mathematical discourse that these contributions initiated according to the levels of discourse in the MDI framework (cf. [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF]. From this analysis we found that 49 mathematical discourses were at level 1, 24 of the discourses were at level 2 and four discourses were at level 3, see the right-hand column of Table 6. Amongst the 49 mathematical discourses at level 1, 31 engaged only one child, 14 of those 49 discourses at level 1 engaged two children, 2 discourses engaged 3 children and 2 discourses engaged four children. From Table 6 we also observe that approximately 64 % of the mathematical discourses were at level 1, and 36 % were at level 2 & 3. We also observe that rarely more than 2 children were engaged by the KT's questions and prompts, 69 (46 + 23) out of the 77 mathematical discourses, and in those occasions 45 were at level 1 and 22 at level 2 and only two discourses were at level 3.

Verbal utterances

# of children

Examples of the KT's questions and prompts that we analysed as initiating a mathematical discourse at level 1 were: "Are all the edges (of the rectangle) of equal length?" and "Is it (the rectangle) two dimensional?" Examples of the KT's questions and prompts that we analysed as initiating a mathematical discourse at level 2 were: "What is this (a sphere)?" and "What is your thinking with respect to that one (a cone)?". Examples of the KT's questions and prompts that we analysed as initiating a mathematical discourse at level 3 were: "Why is that a rectangle?" and "Why does this quadrilateral (a square) have another name?".

Discussion

We have addressed the question: What characterises the mathematical discourse in a kindergarten in which a kindergarten teacher and five-years-old children engage with a planned mathematical activity? Firstly, the mathematical discourse analysed is dominated by the KT. She holds the floor on individual basis for about 1/3 of the elapsed time for the whole activity. She is also dominating the dialogic communication going on, a category comprising approximately 50 % of the elapsed time. This result is also documented by [START_REF] Dovigo | Argumentation in preschool: A common ground for collaborative learning in early childhood[END_REF]. Secondly, we observe that the distribution of verbal utterances amongst the KT and the children is fairly equal, both regarding utterances categorised as mathematical and utterances categorised as non-mathematical. However, we also observe that the children do not contribute equally to the mathematical discourse. Erik is dominating and the other four children are contributing approximately at the same level. Thirdly, the mathematical discourse is characterised by engaging only one or two children. Additionally, 64 % of the mathematical discourses were at level 1 and 36 % at level 2 and 3 according to [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF].

The sparsity of occasions where three or four children were engaged, and at the same time engaged in a mathematical discourse at level 2 and 3, we hypothesise has consequences for the mathematical learning potential in this orchestrated mathematical activity. According to [START_REF] Sfard | When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint[END_REF], "Learning mathematics may now be defined as individualizing mathematical discourse, that is, as the process of becoming able to have mathematical communication not only with others, but also with oneself" (p. 573). We thus hypothesise that the children in this activity have limited opportunities to individualise the mathematical discourse occurring.

We took the KT's questions and prompts as point of departure in our analysis, and scrutinised the mathematical discourse associated with these various questions and prompts. The mathematical discourse was then analysed according to the levels of Adler and Ronda (2015) regarding children's participation. From this analysis, we came to the result communicated in Table 6 above. Our initial hypothesis was that it seems reasonable to argue that the levels of children participation comprise an increasing opportunity for learning. Thus, the potential of learning associated with a level 3 mathematical discourse is more promising than mathematical discourses at level 2 and level 1 respectively. Our analysis shows that most of the mathematical discourse was at level 1 (64 %).

From the analysis we argue that the KT often initiated a mathematical discourse at level 1 through asking a yes/no question. However, these mathematical discourses evolved into mathematical discourses at level 2 and level 3 in several occasions. Thus, mathematical discourses that from the outset may seem to have a limited learning potential evidently may improve into mathematical discourses at levels comprising a higher mathematical learning potential.

Table 1 ): Distribution of time between the KT and the children Who Number of minutes Description of category

 1

	KT active

Table 1 : The analytical approach to quantification of qualitative data

 1

Table 2 : Distribution of time between the KT and the children

 2

Table 3 : Categorisation of utterances

 3

Table 4 : Number of verbal contributions distributed over the children

 4

	Erik	Ivar	Ida	Mari	Nina	>1	Sum
	Number 62	29	23	24	22	29	189

Table 5

 5

	(120 responses) is less than the 189 verbal contributions mentioned
	above.
	# of children engaged	0	1	2	3	4	5	Sum
	Frequency	3	46	23	4	4	0	77
	# of children responding in total	0	46	46	12	16	0	120

Table 5 : Children responding to the KT's questions/prompts

 5

Table 6 : Number of children involved versus the levels of mathematical discourse

 6

The Agder project is funded by the Research Council of Norway (NFR no.

237973), The Sørlandet Knowledge Foundation, The Development and Competence Fund of Aust Agder, Vest Agder County, Aust Agder County, University of Agder and University of Stavanger.

We also find that the KT adopts an inquiry approach to mathematics teaching and learning (cf. Jaworski, 2005), due to her mathematical questions and prompts. However, how to engage the children in mathematical inquiries is a different matter. An implication from our study is thus how to empower the children's inquiry by discussing more between themselves. More research regarding what characterises mathematical activities that empower children's own inquiry into mathematics is needed.


Maria Johansson
email: maria.l.johansson@ltu.se

Timo Tossavainen
email: timo.tossavainen@ltu.se

Ewa-Charlotte Faarinen
email: ewa-charlotte.faarinen@ltu.se

Anne Tossavainen
email: anne.tossavainen@ltu.se

Student teachers' definitions of the concept "teaching mathematics in preschool"

Keywords: Young children, Teaching, Learning, Mathematics, student teachers, preschool

This article reports on how a group of preschool and primary student teachers define the concept "teaching mathematics in preschool" in the beginning of their studies. The background for this pilot-study is the recent change in the Swedish curriculum, which means a shift from play-based

Introduction.

In 2011, a new national curriculum for Swedish preschool was implemented. In this text, and also in the previous curricula for preschool, the concept of teaching is not discussed. Instead of that, the guidelines for preschool teachers highlight the learning opportunities that should be provided to children, and these indicate indirectly what kind of mathematics preschools should provide to children (Swedish National Agency for Education, 2011). The concept of teaching is however used in the Swedish educational act (SFS, 2010:800). According to [START_REF] Öqvist | Teaching in preschool: heads of preschools governance throughout the systematic quality work[END_REF], this was the first time the concept of teaching was used in a steering document for preschools in Sweden. On the other hand, it seems to be important for preschool teachers not to work in the school-like way and, hence, they tend to avoid using the word "teaching" to describe their professional work [START_REF] Hedefalk | Teaching in preschool[END_REF].

In Sweden, there is an ongoing discussion about what teaching in preschool means [START_REF] Rosenqvist | Undervisning i förskolan? En studie av förskollärarstuderandes förestallningar[END_REF][START_REF] Doverborg | Att undervisa barn i förskolan[END_REF]Hedenfalk, Almqvist, & Lundqvist, 2015;[START_REF] Jonsson | Behöver de yngsta barnen undervisas i förskolan? Undervisningsbegreppet och dess innebörder uttryckta av förskolans lärare[END_REF]. [START_REF] Palmér | Different perspectives on possible-desirable-plausible mathematics learning in preschool[END_REF] provide an overview of eight Nordic articles which show a large diversity of aims and goals within preschool mathematics. There is not a common understanding or view of what teaching in preschool is, could or should be. Quite recently, this question was raised when the Swedish National Agency for Inspection (2017) reported that there is a lack of teaching in the Swedish preschools and that there is uncertainty about what the concept teaching in preschools is. This debate is our primary motivation for the present study.

Secondly, according to Clarke, [START_REF] Clarke | The mathematical knowledge and understanding young children bring to school[END_REF], the research into mathematics in preschools has often been driven by the school curriculum with its emphasis on number knowledge. Researchers have investigated various aspects of children's learning of mathematics and numerous theories concerning children's mathematical learning exist. Yet the everyday of the teaching of early mathematics has received much less attention in Nordic countries [START_REF] Saebbe | Initiating a conceptualization of the professional work of teaching mathematics in kindergarten in terms of discourse[END_REF][START_REF] Vallberg Roth | What may characterise teaching in preschool? The written descriptions of Swedish preschool teachers and managers in 2016[END_REF].

A third motivation for this study is the fact that the views and beliefs of student teachers (and teachers) affect the way they plan their teaching, what kind of materials they use, and their performance in mathematics during their teacher studies [START_REF] Wilson | Mathematics teacher change and developments[END_REF][START_REF] Tossavainen | A survey on the permanence of Finnish students' arithmetical skills and the role of motivation[END_REF]. [START_REF] Hannula | Attitude towards mathematics: emotions, expectations and values[END_REF] emphasises, in his widely applied framework for analysing individuals' attitude towards mathematics, the role of expected consequences and relating the attitudes to personal values. So, by surveying prospective preschool and primary teachers' views of the teaching of mathematics for small children we hope to get knowledge about their personal values and what they expect the outcome of their own mathematics education during teacher education and mathematics education in preschool will be.

Theoretical perspective

Benz (2016) synthesised the professional competences needed for supporting children's early mathematical thinking. These competences are also important for teacher education and hence can contribute to the understanding and the impact of beliefs on the teaching of mathematics. She specified three categories of competences found in previous research: (a) content knowledge, pedagogical content knowledge, and knowledge of children's development, (b) action competencies, and (c) attitudes and beliefs. In this article, we will focus on the third point, the attitudes and beliefs. These play an important role when it comes to providing learning opportunities for children in mathematics in preschools. Indeed, teachers' view of what counts as mathematics and what mathematics teaching for young children is have a remarkable effect on their everyday practice. Hannula (2002, p. 30) defines attitude as a category of behaviour that is produced by different evaluative processes which concern emotions, values and expectations. These processes are influenced by the social setting and former experiences. In the present paper, we are interested especially in students' values and expectations.

Concerning the beliefs, we use the following definition: "…teachers' pedagogical beliefs refer to pedagogical attitudes and values such as educational goals and norms, the definition of their own pedagogical role, beliefs about developmentally appropriate practices, as well as the educational goals of preschools" (Anders and Rossbach, 2015, p.308). More specifically "mathematics-related beliefs, which include the implicitly and explicitly held subjective conceptions about mathematics education, the self as a mathematician, and the social context, i.e., the class-context" (Op't Eynde et.al. 2002, p.14). In this paper, we focus on the explicitly held subjective conceptions about teaching in mathematics education.

Our research questions are:

(i) How do student teachers describe mathematics teaching in preschool in the beginning of their university studies? (ii) Are there differences between the different groups of students, and if there is, how can they be described?

The study

This study aims to investigate student teachers' beliefs about the concept of teaching in mathematics education for young children, specifically in preschool. In order to do that, we collected data from three different groups of students by using a printed questionnaire. The participants of the study (N=94) represent Swedish university students from three different teacher education programs. The first group studies (N=27) to become preschool teachers, the second (N=42) to lower primary teachers (Swedish classes F-3) and the third group (N=25) to upper primary (Swedish classes 4-6) teachers. All of the respondents were in the beginning of their university studies and had therefore not yet taken any mathematics courses contained in their university program.

The motivation for including these three different groups is the fact that the primary teachers continue the work of the preschool teachers, so, it is important to see if they share the same ideas and goals related to mathematical education. Another reason for the three groups is that we want to see whether students' views on teaching arise during the teaching education programme or are they based on their earlier beliefs. Therefore, we need to examine whether or how these groups' views differ from one another already in the beginning of university studies.

The first part of the questionnaire surveyed the participants' educational background concerning mathematics. In the second part, the participants were asked to make a concept map of their definition of mathematics education in the Swedish preschool. The third part contained a set of statements related to teaching of mathematics for young children with the seven-point Likert-type scales. In this article, we focus on the participants' concept maps i.e. the second part of the survey. The task was presented in the questionnaire as follows: "The definition of mathematics teaching in preschool: Spend a few minutes on reflecting teaching mathematics to young children and what it brings to your mind. Then, using a concept map, define what mathematical teaching in preschool education in Sweden should in your opinion be. Focus on the essential features, more detailed questions follow on the next page."

The students then provided concept maps where they had a centre of the map and then added essential features of the question around this centre. Using a concept map may provide other information about teachers' views and beliefs than open questions and Likert-type scale questions. This method is mostly used for investigation concept understanding [START_REF] Rosas | Quality and rigor of the concept mapping methodology: a pooled study analysis[END_REF] and, hence, we found that it might be a way also to investigate beliefs and views on the concept of teaching mathematics in preschool since this is in line with the definition and focus of the paper, the conceptions about teaching of mathematics. When analysing the students' concept maps we realised that they often had more features of a mind map than a concept map. According to [START_REF] Davies | Concept mapping, mind mapping and argument mapping: what are the differences and do they matter[END_REF], one important difference between a mind map and a concept map is that a concept map has a hierarchical structure with several levels. This was not the case with these students, most of their maps had only one level.

We used content analysis in order to find the essential categories of sub-concepts that occur in the students' concept maps. The analysis focused on qualitative differences in students' values and expectations. It did not however take into account at what levels in the concept maps these subconcepts were mentioned. The categories were derived purely from the data, i.e., the words found in the maps are grouped and after the grouping the categories are defined. When it comes to the written answers (a few students gave a written response instead of drawing a concept map), they were added to the categories after the grouping of the words.

Result

In this section, we first present the categories found in the concept maps and then give examples of different categories and their content. Secondly, we discuss the differences or similarities that can be seen between the three different groups.

The following categories were found in the concept maps: Motivation, Methods for teaching, Children-centred, Everyday, and Content.

Motivation is the first one of these categories. Here we have words like fun, enjoyable, meaningful, play based, play, inspiring, interesting and creative. We interpret these words as describing how mathematics teaching should be for young children. They focus on feelings as a tool for motivation. We also attached play and play-based to this category even though they could be placed also in the category methods for teaching.

The category Methods for teaching also answers the question how mathematics teaching should be for young children, but with a focus on the organization of the teaching or learning rather than the motivational aspects. Here we find words like practical, concrete, concrete materials, games, digital tools, mathematics in all activities, variation, exploration and group work.

The next category can also be seen to answer the question how, but it focuses on children and their world. Therefore, we call it Children-centred. Here we find words like draw on children experiences, on the child's level, and suitable level.

Everyday is another category putting a child in the centre. In this category, we find words like everyday situations, connect to the everyday, connected to reality and use the everyday.

Content is the last of the categories and here the students have given examples of mathematical content that the children should meet in the preschool. Here are words like numbers, counting, shapes, develop concepts, space, symmetry, geometry, arithmetic, problem-solving and sorting. This category is not analysed in detail here due to limited space but it is of interest for further investigations. In Motivation, we have few ( 7) response from upper primary student teachers, and of these seven, six mentions play. On the other hand, play is not mentioned at all by the preschool student teachers, this is surprising given that play as a base for learning is a foundation for the Swedish curriculum for preschool. For the preschool student teachers, we have 17 responses and for the lower primary student teachers we have 26 responses in this category. To conclude this category seems to be important for preschool student teachers and for lower primary student teachers.

The category Children-centred seems to be important for the preschool student teachers but not so important for the other groups, only one of the upper primary student teachers has a word in this category, and there are only few words from the lower primary student teachers and then even less for the upper primary student teachers.

Surprisingly, Everyday does not have that pattern. Here the lower primary student teachers have the mode while preschool student teachers and upper primary student teachers only have 4 and 3 responses, respectively.

In Methods for teaching, the preschool student teachers have few words in this category and they all say variation. This is in contrast to lower primary student teachers group where they have 16 and upper primary student teachers have 11 and gives a variety of methods.

Content is the largest category overall and seems to be important for all the groups, and it will be a subject to a separated analysis later on. It is the only category that is almost the same in all three groups and almost all student teachers have words in this category.

One important thing that was noticed in the analysis was the fact that the students had different words in the centre of their concept maps. The formulation of the task mentioned explicitly "The definition of mathematics teaching in preschool" and from this expression the students had chosen different centre words for their maps. This may have had an impact on the concept maps and, hence, it is important to present the different centerwords as well. They are: mathematics, mathematics learning, mathematics teaching and mathematics in preschool. As we can see in Table 2, only one of the student teachers for preschool has written about mathematics teaching. This is not surprising since we know from previous research that this is typical for in-service preschool teachers [START_REF] Hedefalk | Teaching in preschool[END_REF]. So, the fact that the same pattern appears for upper primary student teachers is more interesting. On the other hand, none of the primary student teachers has used the mathematics learning in their centre. The lower primary student teachers have focused on mathematics in preschool or mathematics teaching and the upper primary student teachers on mathematics and mathematics in preschool. The difference between the groups is noticeable and can only partly be explained by the fact that they have started their studies. They are, as mentioned before, in the beginning of their studies and have not yet taken the mathematics course.

Discussion

As mentioned above, this is just a pilot study with a relatively small group of students, and the results discussed in this article are only due to an analysis of one of the parts of the survey. The next step would be to investigate how these results are connected to the students' responses to the statements measures on a Likert-type scale; some preliminary results have been reported by [START_REF] Tossavainen | Swedish primary and preprimary student teachers' views of using digital tools in preprimary mathematics education[END_REF]. Nevertheless, the above results give us an overview of what prospective preschool and primary teachers expect mathematics education in preschool to be and what they value in the teaching of mathematics for small children.

One point that needs to be addressed here is that the students had chosen different words in the middle of their concept maps. The formulation of the task mentioned explicitly "The definition of the teaching of mathematics in preschool" and it seems that, from this expression, the students have chosen different centre words for their maps. This may have impacted the concept maps and, hence, it is important to present the different subcategories as well. They are: mathematics, mathematics learning, mathematics teaching and mathematics in preschool. The fact that most students have mathematics learning as the concept centre is compatible with previous research [START_REF] Hedefalk | Teaching in preschool[END_REF] but a new finding is that this view is already present when they start their studies at university. Similarly, it is surprising that their concept centres differ already at this stage.

The lack of the word play in the preschool student teachers' concept maps is another point is worth more investigations. One way to interpret this is that, for this group of students, play is so obvious that it does not have to be mentioned, yet this is not self-evidently supported by our data.

The fact that the lower primary student teachers seem to have a broader view than the other groups would be even more evident if we studied the students' concept maps at individual level.

To answer our research questions, the three groups of students give different descriptions about what teaching in mathematics in preschool is, and they focus on different parts of the teaching of mathematics. We can say that they in fact have different pedagogical beliefs and expectations regarding at least what mathematics teaching should be in preschool, how it should be provided, and also which focus the teachers should have. The category Content gives us some indications of attitudes and values regarding their educational goals. Since this category is shared by all groups and is the largest one, we can conclude that this is an important base for all the teachers' pedagogical beliefs. The categories Everyday and Children-centred give information regarding expectations and values conserning the teachers' pedagogical role. Here the focus for the preschool student teachers is on the child and the focus for the lower primary student teachers is on the everyday. These are not so important for the upper primary student teachers, and this could possibly be explained by the fact that they have not thought about the pedagogical role of the teacher since they will not teach themselves. Methods for teaching and Motivation give information about attitudes and values about appreciated practices. These can be found in all the groups but with different focus. For the preschool student teachers, the focus is on motivational issues and, the for the primary student teachers, on the methods. This is one of our main results since it gives us information about the preschool student teachers' beliefs that teaching in preschool is not their main interest but they prefer to focus on other parts of the educational practice.

Table 1 : Distribution of the categories for each group

 1

		Motivation	Children-	Everyday	Methods	Content
			centred		for teaching
	Preschool	17	12	4	5	17
	N=27	64%	44%	15%	19%	63%
	Lower Primary	26	7	9	16	18

  The remaining 19 students either does not have a centre word, have written a short text describing, or it is not obvious what the centre word is.

	Lower Primary N=42	8 18%	17 39%	0	14 32%
	Upper Primary N=25	9 39%	6 26%	0	1 4%
		Mathematics	Mathematics	Mathematics	Mathematics
			in preschool	learning	teaching
	Preschool N=27	7 26%	3 11%	9 33%	1 4%

Table 2 : Concept map centres across the student groups

 2

Acknowledgment

We thank Anna Öqvist for the assistance with collecting the data.