Hanna Palmér Camilla Björklund Elin Reikerås Jessica Elofsson  *Editors*

Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives

Results from the POEM5 Conference, 2022

Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives Hanna Palmér • Camilla Björklund Elin Reikerås • Jessica Elofsson Editors

# Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives

Results from the POEM5 Conference, 2022

*Editors* Hanna Palmér Department of Mathematics Linnaeus University Växjö, Sweden

Elin Reikerås Department of Early Childhood Education University of Stavanger Stavanger, Norway

Camilla Björklund Department of Education Communication and Learning University of Gothenburg Gothenburg, Sweden

Jessica Elofsson Department of Education Communication and Learning University of Gothenburg Gothenburg, Sweden

#### ISBN 978-3-031-37662-7 ISBN 978-3-031-37663-4 (eBook) https://doi.org/10.1007/978-3-031-37663-4

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### **Editorial**

The POEM conference – A Mathematics Education **P**erspective **o**n **E**arly **M**athematics Learning Between the Poles of Instruction and Construction – has become a strong tradition since the frst one was held in 2012. Every other year it brings together researchers within the feld of early childhood mathematics education to exchange current research fndings and to collaborate and discuss new ideas. After a break due to the Covid-19 pandemic, the ffth edition of the conference was held in Gothenburg, Sweden, in May 2022. The conference gathered more than 60 researchers and included 24 presentations. The theme of the conference was "Teaching mathematics as to be meaningful – foregrounding children's play and perspectives". To give inspiration and depth to this theme, professors Bert van Oers and Ingrid Pramling Samuelsson were invited as keynote speakers.

Bert van Oers (Vrije Universiteit Amsterdam) gave a lecture on the development of mathematical thinking in young children's play, focusing particularly on the role of communicative tools. He has been researching this topic over the past two decades and developing theory for the early childhood mathematics education community. In his paper, he summarizes some of the main fndings of this research programme on emergent mathematical thinking in young children. The research programme is based on a cultural-historical activity theory perspective (CHAT) and focuses on identifying the productive conditions that may promote mathematical thinking in young children (4–8 years old). He lists several conditions (play format, productive dialogue, schematizing, narrative competence and intertextuality) that have been empirically shown to promote meaningful mathematical thinking in young children, helping them communicate about mathematical objects.

Ingrid Pramling Samuelsson (University of Gothenburg) shared her view on the notions of "Child Perspective" and "Children's Perspectives" in mathematics learning in the early years. In her lecture, she asked if the two views of child perspective and children's perspectives can be of help in teaching young children mathematics. With more than 40 years of research in the feld of early childhood education, Pramling Samuelsson promotes children's perspectives as key in all education activities. In early childhood education, there are several other recurring notions that are central, such as "listening to children" and, not least, the notion of play, but

Pramling Samuelsson raises our awareness as to whether these become merely rhetoric or actual parts of praxis. In teaching according to Developmental Pedagogy, this comes through as a joint venture between children and teachers, in which the teachers know the direction for learning to point out to children, why the children's experiences and ways of talking about a content are key objectives for the teacher to challenge for the children's further development of understanding.

The presentations of both keynote speakers address issues that have been put at the centre of their research agenda for a long time, providing the early childhood mathematics education research community valuable insights for the development of theory and practice. The many scholars they have collaborated with and the research initiatives originating from their work are amply refected in the research presented at the POEM conference and in these proceedings.

The overarching theme of this year's POEM conference is also well refected in the papers presented at the conference and in these proceedings. We recognize that all parts of the theme "Teaching mathematics as to be meaningful – foregrounding children's play and perspectives" can be found in the relation between teachers, children and the mathematical content, and this is further framed within the context of play. Four sub-themes appear in these proceedings: *Play and learning, Children's perspectives on mathematics, Teachers' competencies* and *Theorizing aspects of early mathematics education*.

### **Play and Learning**

Globally, most guidelines and curricula for early childhood education mention play as one of the key features for young children's learning. Still, there are quite different views on the defnitions of play (and on whether it is even possible to defne play) and in what ways play should become part of children's learning. We chose to emphasize play in the theme for this year's POEM conference, and three of the papers presented specifcally focus on play. The broad spectrum of how to understand play and learning is refected in these papers.

Amrar, Clerc-Georgy and Dorier study pretend play (playing hospital in the classroom) and point to the close link found between pretend play and mathematics, as both are cultural and semiotic activities. They particularly discuss the need to dialectically scaffold play and mathematical thinking where the teacher guides children through the refective process. The other two papers within this sub-theme focus on digital play and learning of mathematics. During the last two years of the pandemic and lockdown in many countries, digitalization has become a familiar part of education, including in the early years, as a means to uphold education standards. But even before covid restrictions, the notion of digitalization appeared in curricula and guidelines for early childhood education and mathematics teaching. Christiansen raises the question of whether digital tools' entrance into kindergarten might entail a risk of changing children's pedagogical environment to one with less focus on children's play. She addresses this question in a study of 1-year-olds' interaction with three applications that provide the children with a playful environment. The kindergarten teacher in the study is shown to seize the opportunity that arises when the child faces digital challenges and uses it as an opportunity to discuss specifc mathematical objects. A similar interest in research is found in the paper by Birklein and Steinweg. They take play as a framework for learning a step further in a study comparing differences in free versus supervised use of an application developed to support children's mathematical competencies; more specifcally, they examine how the supervision in the implementation infuences the children's behaviour and their progress through the game. The common assumption that a lack of limitations and instructions will inevitably lead to excessive media consumption is shown not to be true in their study. This highlights the need for more research taking different approaches, including both explorative and designed studies, in order to deepen our understanding of play and learning as intertwined phenomena.

In these three papers where play is foregrounded, whether in a digital or physical environment, the authors conclude that the teacher has a key role as a mediator of mathematical objects because the play environments do not give suffcient support in themselves for children exploring new mathematical meaning.

### **Children's Perspectives on Mathematics**

One way to foreground children's perspectives is to focus on how children understand and deal with different mathematical ideas. This is in focus in four of the papers.

Eriksson, Hedefalk and Sumpter investigate the tension between division and fair share when 5-year-olds are faced with different sharing tasks, varying the sizes of dividend and divisor and sharing equal parts (groups) and unequal parts. Results of their observations show that children use different strategies when performing division. The study also shows how children use ethical reasoning claims to convince their peers in the sharing tasks rather than mathematical reasoning. The other three papers within this sub-theme are focused more specifcally on problem solving as a mathematical activity. In the frst, Vogler, Henschen and Teschner focus on the different heuristics emerging in peer interactions in block-play situations and on the extent to which the interactions and collective problem solving create conditions for mathematical learning. They fnd that children use various heuristic procedures in their interactions during block play such as decomposing into sub-problems, systematic guessing and testing and working backwards when working on problems in block-play situations. The paper by Meaney, Severina, Gustavsen, Hoven and Larsen focuses on children's mathematical and computational thinking. Problems to be solved are identifed through children's actions, or rather observed signs of uncertainty, which guide the authors in their study of what mathematical understanding children use when engaging in problem solving with robots in naturalistic settings. By focusing on the children's uncertainty, the authors identify problems from the children's perspective rather than against predetermined outcomes. The authors show how children's different understandings about counting contribute to their abilities to solve these problems. Finally, Palmér and van Bommel focus on how young children explain their choice of representation when working on a problem-solving task on combinatorics and how the representations used are compatible with children's understanding of abstraction. The results indicate that diffculties in representing the context of the problem-solving task may force some of the children to work with a representation on a level of abstraction not suitable for them. This in turn infuences how they manage to solve the problem-solving task.

Altogether, these four papers contribute to the feld of early mathematics education research from the perspective of the learner, which in this research domain includes young children to whom mathematical concepts and skills are novel. The children's perspective should thereby be signifcant to further the understanding of children learning mathematics.

### **Teachers' Competencies**

Five of the papers focused on the competencies of early childhood education teachers. Three of the papers direct attention to teaching skills and what informs educational choices. In their paper, Carlsen, Erfjord and Hundeland present a modifcation to a framework "Knowledge Quartet" developed in a school context, and they suggest that this framework could be used as an analytical tool to investigate the mathematical competence of kindergarten teachers. The framework is trialled on observations from a case study where a mathematical activity includes children's guided inquiries into features of two-dimensional geometrical shapes. A broader approach to teacher competence and, in particular, factors that may infuence educational choices, is found in the large-scale interview study by Torbeyns, Verbruggen and Depaepe*,* who investigate early childhood education teachers' use of educational technology in mathematics education and its association with school and teacher characteristics. They fnd a broad variety of factors that infuence the use of technology and that teachers need to have well-developed ICT competencies, both digital skills and pedagogical-technological competencies, to enable constructive mathematical learning processes. Beck and Vogler empirically investigate whether teachers being trained in "design patterns of mathematical situations" have a favorable infuence on the responsiveness in the interactions between teachers and children. The results indicate that the implementation is a promising form of preparation for mathematically rich and responsive interactions.

The other two papers within this sub-theme concern teachers' assessment competence regarding children's knowledge and skills. In their paper, Benz, Reuter, Maier and Zöllner study in-service kindergarten teachers and their selection of and refection on suitable diagnostic situations and diagnostic tools. Diagnosing is considered one main facet of teachers' professional competence, generating information about children's understanding and eliciting students' cognitive skills. The participating teachers express positive effects of implementing adaptive learning support, that is, showing advanced professional competence. In *Rinvold and Skorpen's* paper*,* the focus is on kindergarten teachers' play-responsive assessment of children's mathematical profciencies. They found that situations refecting reallife activities where children are given a large amount of freedom are important assets in the assessment planning process. This is discussed both in play-responsive participation in children's self-initiated play and when the kindergarten teachers organize activities for the sake of observation.

Similar to the papers in the frst sub-theme, "Play and learning", these four papers emphasize the key role of the teacher as the mediator for children's opportunity to learn mathematics.

### **Theorizing Aspects of Early Mathematics Education**

To move the feld of early childhood mathematics education forward, and to legitimate it as a part of the broader mathematics education research area, there are also attempts to theorize aspects of mathematics learning and teaching as well as to develop the methodologies within this research area. The fve studies within this sub-theme are based both in different theoretical frameworks and in empirical observations. For example, Reikerås investigates the relation between language and mathematics skills observed in 2-year-olds' play and everyday activities, a relation that is signifcant to gain a better understanding of in the thematic and play-oriented early childhood education. The study is large scale and thus may provide statistical correlations, something that qualitative studies (which make up most of the contributions) cannot do.

Some of the papers take a specifc interest in unpacking theoretical concepts such as multimodality and mathematizing in relation to empirical observations. In the paper by Billion and Huth, the starting point is empirical fndings from a case study of a 4-year-old boy's mathematical play with wooden fgures while he interacts with three peers and a teacher. The situation was an open offer for the learners, who were free to address several different mathematical ideas. The study gives rise to potentially rethinking the notion of multimodality as not always an interwoven but sometimes a phased-wise parallel proceeding construction in mathematical interaction.

Two papers take up the notion of mathematizing, an expression introduced by the Dutch mathematician Hans Freudenthal. Palmér and Björklund focus on mathematizing and, in particular, on how to understand the "real world" of very young children, which signifcantly contributes to mathematizing taking place. The results show that "real world" can involve both imagination and play, but mathematization is only noticed when there is a problem that, from the perspective of the child, needs to be solved. Björklund and Elofsson bring in the notion of playfulness as part of mathematics teaching as a motor in the mathematizing process. Their paper discusses the signifcance of playfulness as a feature of mathematics teaching in early education, and particularly brings to the fore nuances in different ways of making playfulness an asset in teaching.

Finally, Lembrér focuses on methodological choices in research on early mathematics education, discussing two different data collection methods used when investigating parents' views on mathematics education for young children. She found that the different narratives produced were affected by the ways the data were collected and gave different insights into parents' views on early mathematics education. Such inquiries of methodological outcomes are of course valuable to all future studies involving people of all ages.

The fve papers in this sub-theme are of importance to moving the feld of early childhood mathematics education forward, not only empirically but also theoretically and methodologically.

### **Closing the Fifth POEM Conference**

As mentioned in the introduction and as is apparent in the presentation of papers described above, the theme of the conference is recognized in all studies because they all concern, in one way or another, the relation between teachers, children and the mathematical content. A special focus of this year's conference was the framing of these relations within the context of play. We found this theme important in relation to the debate on whether teaching should be integrated with or separated from children's play, a debate we fnd to be contradictory as both teaching and play are highlighted as central in the curricula of preschool education in most countries. Thus, within the context of play, we fnd some papers putting the teaching of mathematics to young children as the essential issue of discussion. This means research in early childhood mathematics education has a strong ambition to facilitate mathematical learning in ways that adhere to the children's experiences and lived world. Other papers focus on meaningfulness in the learning process, particularly from the child's perspective. Meaningfulness indeed appears in the presented research as an essential feature for learning, but by emphasizing play, as in some of the papers, this becomes even more evident. One thing we noticed in the papers of this year's conference is that more papers than before include toddlers. As early mathematics in some countries are limited to children over the age of 3, we wish to stress the importance of POEM also making visible the signifcance of mathematics education with toddlers.

Even though the papers described above are presented under the four sub-themes of *Play and learning, Children's perspectives on mathematics, Teachers' competencies* and *Theorizing aspects of early mathematics education*, these themes are visible in all of the papers. The division into sub-themes is based on what is being foregrounded in the papers, but a focus on, for example, teachers is not possible without also including content and children and vice versa. The papers, together with the keynote lectures, move the feld of early childhood mathematics education forward – empirically, theoretically and methodologically. We are eagerly looking forward to forthcoming POEM conferences to see where this young but strongly up-and-coming area in mathematics education research is heading.

### **Contents**

### **The Development of Mathematical Thinking in Young Children's Play: The Role of Communicative Tools**

**Bert van Oers**

### **On the History of Developmental Education: The Case of Mathematical Thinking**

In the Netherlands we have been working since the 1980s at the implementation of a curriculum for primary education, called *Developmental Education*. This curriculum aims at a broad identity development of pupils and teachers and requires that all learning should be culturally meaningful and make personal sense for the learners. In this way, we (i.e., the teachers, teacher trainers, researchers) aim at enabling pupils to become agentive critical participants in all kinds of cultural practices. *Developmental Education* is not a mandatory curriculum in the Netherlands: every school is free to opt for this approach or not, and can get funding from the government for its implementation and professionalization of its teachers.

*Developmental Education* is based on Vygotskij's cultural-historical theory of human development, elaborated with Leont'ev's Activity Theory and El'konin's theory of child development (El'konin, 1978; Leont'ev, 1978; van Oers, 2012a; Vygotsky, 1978). On the basis of this Cultural-Historical Activity Theory (CHAT), it is assumed that higher psychological functions emerge from meaningful interactions with adults or more knowledgeable peers. Rather than being naturally emerging and waiting for maturation stimulated by experiences with the outside world, psychological functions develop in a process of a person's enculturation that starts out from participation and communication with more knowledgeable others in cultural practices that make sense for them, such as household, hospital, supermarket, museum, post-offce, library, artist studio, gardening, etc. Due to this meaningfulness, children are mostly interested to be engaged in these well-known activities,

B. van Oers (\*)

Faculty of Behavioural and Movement Sciences, Educational Studies, Vrije Universiteit Amsterdam (VU University Amsterdam), Amsterdam, The Netherlands e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2024

H. Palmér et al. (eds.), *Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives*, https://doi.org/10.1007/978-3-031-37663-4\_1

and are willing to acquire new actions with new cultural means (like instrumental tools or concepts, etcetera) to make it look 'like the real world' for them. Participation in real life cultural practices always requires the adoption of a specifc role within that practice, and eventually it calls for the appropriation of the tools that are constitutive for this role, like for instance a stethoscope for a doctor, measuring devices and numbers for a gardener or carpenter, symbolic representations for mathematicians. Tools are unavoidable for enacting a role in a cultural practice. As Vygotskij has pointed out, language is an essential tool for executing context-based actions and cognitive functions and for communicating with others and oneself about tool use in specifc situations (e.g., Vygotskij, 1982/1934).

In this article, I want to focus on the emergence of mathematical thinking as a meaningful activity, that is to say as a cultural activity that has cultural relevance and also makes sense for the learner. Studying the genesis of mathematical thinking (as in this article) frst of all implies that we have to explain how 'mathematics' enters into the child's mind in the frst place. In one of his books on child development, Vygotskij (1984, pp. 226–227) explains this in general terms as steps in the process of development of children's consciousness. Freely summarized here in terms of an emerging mathematical consciousness, this process can also be sketched as a result of the interaction of a child and an adult. As a frst step, the child is **acting spontaneously** with objects in his environment, often on the basis of exploratory needs but also on the basis of imitations of signifcant others. In this stage the children may be reciting number words, but this action is not yet 'mathematical' for them, although it may look like that from an adult perspective (Munn, 1997, p. 16). These 'counting'- acts and included objects at frst do not have specifc intellectual meanings for the child beyond the fact that she/he wants to manipulate the objects and by so doing look like the surrounding adults. The observing adult then may **interpret this** in terms of her/his own cultural cognitive system and (for example) mention the name of the object or of the act carried out on this object. This is the second step: the act and object by themselves are then turned into a shared topic that can be enriched by the adult by adding new meanings ('predicates'). This is the moment that mathematical qualifcations enter the consciousness of the child. The child hears and may repeat 'ten', 'count', 'number', or 'more' etcetera. Finally, in due time, the child may begin **using these new meanings for her/himself** and for communication with others. This third step is the real starting point of genuine mathematical thinking *of* the child ('mathematising').

From the perspective of the cultural-historical activity theory, we conceive of mathematising as a motivated cultural activity with specifc tools and goal-oriented actions to solve problems that participants in this activity encounter. However, in our own observations in primary schools, we discovered that the goals for the use of mathematical tools (like measuring, adding, subtracting, etc.) could be of different kinds: frst children used mathematical tools for solving problems in the context of everyday settings and cultural practices (like: fnding out how much to pay in the shoe shop, or determining number of blocks one needs for building a wall around a castle). We can call this *extra-mathematical* use of tools. On the other hand, when growing older the children sometimes also become interested in the structure of the mathematical tools and the operations themselves, and start using the tools they have acquired for solving problems of the activity of mathematising itself, like 10 year old children who explored the structure of the number line and discovering (with the help of the teacher) even and uneven, the features of '10', or even what happens when we go endlessly to the left on the number line and arrive 'under zero'? We call these *intramathematical* reasoning. For young children (until eight), however, we never witnessed spontaneous intramathematical reasoning in the classroom. Their involvement with mathematics was only extramathematical: the use of mathematical tools for exploring concrete situations and for the achievement of practical goals in the cultural activities they play. This is the type of activity I will

focus on in this article. From these general theoretical assumptions about the enculturation of children in school, I have studied with my colleagues from the *Vrije Universiteit Amsterdam* (VU University Amsterdam) the development of cultural abilities like literacy and mathematising from preschool on. In this article I will review some outcomes of our research program on the development of mathematical thinking in 4–8 years old children over the past decades. This research program promotes research that is at the same time essentially theory-driven, evidence-based, and practically implemented. Methodologically, this means that we are not satisfed with positive empirical evidence alone, but also examine how this empirical result can be conceptually explained in terms of our CHAT theoretical framework, and how this outcome can be implemented by teachers in their everyday classroom. Hence, an important part of this research is accomplished in collaboration with teachers in their classrooms as long term quasi-experimental research or case-studies. The implementation process is most of the time also guided by teacher trainers who are expert in the (theoretical background of) *Developmental Education* (van Oers, 2012a, 2013a; van Oers & Pompert, 2021)*.*

### **Productive Conditions**

Over the past decades, we have discovered and researched a number of conditions that have turned out to be productive in promoting some form of mathematical thinking in young children. Such productive conditions (both interactional and situational) are assumed to be effective for the promotion of mathematical thinking, as they call for specifc actions or imply affordances for such actions. From the CHAT perspective on human practical, perceptual, verbal and mental actions, we assume that all these actions are object-oriented, goal-directed and tool-based. This latter implies that all mathematical actions require the use of symbols (tools) that can be used to discover (intramathematical or extramathematical) new knowledge of a situation or – more often than not – the use of the operations that can be carried out on these symbols. The organized system of mathematical actions is called a mathematical activity, or just 'mathematising' (using a term from Freudenthal, 1973, p. 134). In an earlier overview of our research on young children's mathematical activity

(van Oers, 20141 ), I circumscribed 'mathematising' as 'the activity of producing structured objects that allow further elaborations in mathematical terms through problem solving and (collective) reasoning/argumentation' (van Oers, 2014, p. 112). The relevant objects children fnd in their direct environment can be collections of things, situations, patterns of events, and the like. In the above-mentioned 2014-article, I also argued that mathematising is a complex cultural practice which can only be accessed by newcomers (like young children) when they are allowed to play it. In play, children know and want to follow some of the *rules* of this practice, children are *involved* participants (i.e., they participate on their own willful accord), and enjoy some *degrees of freedom* in the choice of object, rules and tools. An essential condition to safeguard the mathematical nature of the activity, is the coparticipation of an adult or more knowledgeable peer in this activity, provided she/ he does not disturb the qualities of play (rules, degrees of freedom, involvement) (van Oers, 2004, 2012b).

A seminal insight of Vygotskij is his conception of social origin of psychological functions (including mathematical thinking). Psychological functions show their social origin in the fact that they maintain their communicative function when they are verbalized, and explain to the Self or someone else what the thinker has in mind. But even when they are internalized, we may theoretically assume that this inner speech is preserved in moments of orientation and control or evaluation (see for example Vygotskij, 1982/1934). It was Gal'perin in particular who elaborated this idea in a cohesive argument and empirical research (see Gal'perin, 1969, 1976).

In our research program, the above described tenets turned out to be powerful ideas to lead our research. Van Houten et al. (2013) demonstrated a positive correlation between children's narrative competence (as measured by a standardized test) and their abilities in early arithmetic. This correlation may be caused by intermediating factors like sociocultural background and cannot by itself be interpreted as proof for a causal relationship between arithmetic and narrative competence. Nevertheless, it strongly suggests that the ability to communicate coherently about aspects of a situation (including mathematics) is a core factor in both types of competences. In another study of mathematical thinking in young children in primary school, we could further support this assumption about the importance of communicating in the process of promoting mathematical thinking (van Oers, 2013b). Characteristically, this thinking is based on 'connected discourse' (Luria, 1969, p. 138*ff*). That is to say, according to Luria, it shows that coherence is linked to the actual speech situation, and is syntagmatic, i.e., combines different terms into a meaningful proposition (Luria, 1969). In different previous studies we found that children like to use drawings in their play and integrate these with symbols representing number or quantity(−change), often to make sure to others that their drawing communicates extra information about the drawing and the context of its use. The use of drawings (and schemes) combined with symbols contributes to a

<sup>1</sup>Based on a presentation at the POEM 2012 conference in Frankfurt: Mathematics Education **P**erspective **o**n **e**arly **M**athematics – Learning between the Poles of Instruction and Construction

further mathematisation of young children's language (van Oers, 2002). As a matter of fact, these studies convergently support the productive role of connected discourse for the formation of emergent mathematical thinking.

Pondering further on the idea of communicative activity as a way to promote coherent mathematical thinking in children, we also wondered whether the use of more structured communicative tools (like schemes) could further support young children's thinking on mathematical objects, like number, quantity, addition or subtraction, spatial relations etc. For the study of this question, we guided teachers in early years classroom (5–6 year olds) to participate in children's play activities and introduce useful schematic representations (schemes like ground plan, construction plan, diagram) that were recognized by the children as interesting and helpful for solving problems in their activities (see for instance, van Oers, 1994, 1996). In their shoe-shop in the classroom, children for example piled up different sizes of shoes and copied this contraption by drawing a kind of histogram (see Fig. 1a); the teacher also offered them a so-called "feet measuring thing" (see Fig. 1b), which was appreciated by the children as a handy thing to fgure out the size of their and the teacher's shoes. In our studies we always took care that the presented schemes made sense for the children, were linked to a current and meaningful activity of the children, and were integrated into their own speech.

In one of our further studies on this topic we conducted a longitudinal, quasiexperimental study with a pretest-post-test control group design (N = 133). In a 5-year olds classroom the teachers provide different types of schemas that ftted in the current activities of the children's play. During the whole year the children had a variety of schematizing experiences (reading schemes, construction schemes, action plans). In the next year these children started their frst mathematics lessons and their scores on mathematics tests (Post-test and delayed post-test) were compared to a randomly chosen control group of the same age that took the same tests. Our experimental groups scored signifcantly better on post-tests (both for arithmetic and schematizing) than the control group which had not got any experience with schematizing (Poland, 2007; Poland & van Oers, 2007; Poland et al., 2009; van Oers & Poland, 2007). However, in a delayed test after one-and-a-half year the

**Fig. 1** (**a**) histogram of shoe boxes, (**b**) "feet measuring thing"

children's scores on an arithmetic test were not signifcantly different anymore. Obviously, good arithmetic scores can also be achieved by drill and practice.

In sum, in a series of theoretically connected studies on children's manifestations of acts that we as adults could identify as productive conditions that could be linked to the meaningful promotion of mathematical thinking in the context of children's play: playful enactment of the use of numbers and quantities, the use of symbolic tools like schematic representations, narrative competence, presence of an helpful adult or more knowledgeable peers for learning how to communicate about number, quantity and change. However, most of the studies mentioned above were carried out in classrooms with young children in the age of 5–8. We now have reasons to assume that mathematical (co-)thinking starts earlier than at the age of 5 years (see for example Carruthers & Worthington, 2006; Worthington, 2021; Worthington & van Oers, 2017).

### **When and How Do We Start?**

In the educational literature, it is an empirically well-established fact, that children spontaneously engage during everyday situations with symbols, patterns and objects that are generally recognized in our culture as mathematical, long before they enter formal schooling in arithmetic (McMullen et al., 2019; Ramani et al., 2015; Rathé et al., 2016a, b; Wijns et al., 2020). The children (4–5 years old) encounter these symbols, etc. in their cultural environments: on the streets (e.g., numbers on cars, busses), supermarket (price list), in the newspapers, picture books etc. Children notice these numbers without directions from adults (see Rathé et al., 2022).

Despite the empirical evidence for young children's awareness of numerocity and numbers, a more detailed description of the course of this evolution from spontaneous use of notions of numerocity and number to number concepts and operations, remained unspecifed. My basic hypothesis was that the previously described Vygotskian three step approach to the development of mathematical thinking may be applicable in the younger ages too. The frst interest in exploring the tenability of this hypothesis was to fgure out how adults could help younger children improving their communicative ability regarding numerosity, number and changes in this intellectual domain, without direct instructions and/or impairing the quality of children's spontaneous play.

In our own research group, it was Maulfry Worthington (UK), who addressed this problem with a well-balanced series of studies into the emergence and development of young children's personal mathematical inscriptions in the context of their spontaneous play. This work was rooted in the work that she did before in collaboration with Carruthers in the UK (see Carruthers & Worthington, 2005, 2006). In their collaborative previous work, these researchers aimed at discovering and describing the range of these mathematical marks from early play exploration to later written

calculations. Worthington's research2 consisted of longitudinal (1 year), ethnographic research in case-studies taken from the contexts of children's homes and nursery classrooms, particularly formatted as pretend play (see Worthington, 2021). In these play contexts, she observed how these young children built up their ability to *communicate* about quantity and number and to *construct* the types of graphical means they used. Her data collection was based on interviews with children, teachers and parents, and on participatory observations. The analyses were mainly qualitative, but in some cases data could be categorized into different classes of graphical means which permitted quantitative analyses too (e.g., regarding the use of different types of symbolic representations – derived from the work of Pierce's semiotic view, Buchler, 1995 – in different groups of children).  This school was located in a large multicultural city in the southwest of England.

Worthington's research project yielded a number of very interesting fndings. Without trying to be exhaustive, a few of them must be mentioned here as they ft perfectly well into the body of my present argument regarding the development of young children's mathematical thinking. First of all, Worthington could identify many cases of children who attempted to communicate about numbers and counting in their play setting, like the boy who had constructed a parking place completely with tickets and prices on each ticket (40 p., 60 p. etc.). It also turned out that the numberings were not suggested or imposed by the teacher, but were clearly drawn from children's 'funds of knowledge', built up spontaneously in their home contexts, the adult world of his parents, events in the outside world, observations of the teacher's behavior in her daily classroom activities (see Worthington & van Oers, 2016, 2017). Furthermore, Worthington could identify different types of graphical means constructed by the children, at frst using their own words, scribble marks, drawings, or even alphanumerical number-like signs copied from their environments. Later on, the children started using iconic (like wavy lines for written texts), indexical (like arrows), and symbolic means, like tallies or numbers (categories borrowed from Pierce), and a new category referring to culturally developed symbols (like numbers, diagrams, plans, etc.).

Finally, the symbols evolved into the culturally accepted mathematical symbols. The observations of the nursery children's spontaneous play and communications about number revealed that their ability to read the intentions of abstract mathematical language and recognize the patterns of their combinations, was growing step by step with the help of the teacher (Worthington et al., 2019). By communicating about numbers and quantities, it also turned out that the communicative signs for the children actually represented 'texts' about number, i.e., things that they could tell about or explain in a coherent narrative about numbers.3 More detailed analyses of the communications between the children and the teacher revealed another

<sup>2</sup>The research was for a doctoral Thesis at the VU Amsterdam, under my supervision and dr Marjolein Dobber's.

<sup>3</sup>This narrative nature of symbolic representations was previously also demonstrated in other situations, see van Oers, (1997).

mechanism of utmost importance in this respect. When playing with the children, teachers often provided new symbols or proper forms of the signs and spontaneous symbolizations of the children, but they did so only when the children showed need for such signs and symbols. No direct instructions. In due time, the children adopted the suggestions of the teacher and used them to expand their own texts. Hence we can say that the children's shared numerical notions were actually *intertextual*. Constructing such intertextuality with the children should be regarded as an essential dynamic element for the emergence of mathematical meanings in young children (Worthington et al., 2023). Adult supervision of these intertextual mathematical constructions (e.g., when using diagrams, graphs, or refecting about number: 'what can we legitimately say about number?') was essential for the correct ways of linking symbols in propositions about mathematical objects (a quality correctly named 'grammaticity' by Worthington, 2021, p. 106ff). Adults' supervision that doesn't reduce the value of the parameters of children's play (rule-based, free to some extent, and involvement), is an essential productive condition for the communication about numbers, numerosity, quantity, etc. for the development of mathematical thinking and the construction of mathematically acceptable topic-predicate structures.

### **Discussion and Prospect**

How can we conceptually understand the above presented productive conditions? What are the psychological dynamics of these processes? The potentially seminal relationships between the emergence of mathematical thinking and children's ways of communication and narration about number, has been discussed by several academics (Pimm, 1987; Krummheuer, 1997; Lorenz, 2012; Maier & Schweiger, 1999; Sfard, 2008). As Pimm (1987, p. 76) pointed out: 'Part of learning mathematics is learning to speak like a mathematician'. A lot of theoretical argumentation and empirical evidence has been accumulated since then. However, it is still important nowadays to clarify how this understanding is to be implemented into everyday classrooms with young children.

In our own research program we have spent serious efforts in adding to this understanding of the development of mathematical thinking with the help of appropriate communicative tools. It is clear from our observations that each of the productive conditions mentioned (play format, schematizing, narrative competence, intertextuality, interaction with adults) engages children in **communication** about quantity, numbers, relationships, patterns of signs, representational processes. Such communications not only invite children to think again and refect, encouraged by questions like 'Are you sure?' (van Oers, 1996), but also to build topics of joint attention (Tomasello, 1999), which can be enriched in the ongoing discourse by new predicates produced by the child's own thinking or offered by peers and adults. So, like Vygotskij already pointed out in the last chapter of his *Thinking and Speech* (Vygotsky, 1982/1934), the construction of *topic – predicate structures* belongs to the core of conceptual thinking (see also van Oers, 2006 for further argumentation and examples). Communication ('speech') is to be considered an essential means for conveying and analyzing meanings, according to Vygotskij. With respect to mathematical thinking, this is consistent with Sfard's (2008) conceptualization of mathematizing. In such collaborative (communicative) constructions of increasingly elaborated and more sophisticated mathematical topic – predicate – structures, (young) learners get involved in *intertextual interactions* in which they weave together their own texts and those of others, provided they make sense for the learning child.

This ought to be a basically playful endeavor that highlights the rules of the game, but also allows the player enough 'degrees of freedom' in his actions and choices for explorations of the meanings and of the (intramathematical or extramathematical) limitations of the (mathematical) rules. Like in the explorations of the number line when children examine what it means to go 'under zero'; does this make sense?

The role of the teacher is essential here in order to guarantee the cultural relevance of the children's inventions for the enrichment of the (mathematical) topicpredicate structures. It is important to realize that this does not prohibit the teacher to teach the children. However, this should always be *embeddedteaching,* embedded in the context of the children's playful engagements with mathematics, trying to answer their own questions. All assistance of the teacher should be connected to the children's questions and interests in the context of their playful mathematics.

How to implement these ideas into the classrooms? In our own work in schools in the Netherlands for the implementation of the educational concept of *Developmental Education* (van Oers, 2012a, c; Slob et al., 2022; van Oers & Duijkers, 2013) we have elaborated a play-based curriculum approach in collaboration with teachers, teacher trainers and researchers, which helps teachers to organize their classroom work in the context of playful sociocultural practices. In the context of such practices, children encounter different kinds of problems (including mathematical ones) that they can try to solve with the help of teachers and peers. In this implementation strategy the main focus is on the innovation of teachers' thinking about curriculum and meaningful learning. In our research thus far we may legitimately draw the conclusion that teachers can indeed initiate and promote dialogical discourses with young children about events and phenomena in their daily life (van der Veen, 2017). From this perspective it is also plausible to assume that such dialogical discourses concerning mathematical topics in everyday practices can indeed promote mathematical thinking in the context of playful participation in varying sociocultural contexts. We have strong reasons to continue along this path in the future.

### **References**

Buchler, J. (1995). *Philosophical writings of Pierce*. Dover.

ment? *Mathematical Thinking and Learning, 24*(1), 38–51. https://doi.org/10.1080/1098606 5.2020.1818468

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### **A Child Perspective and Children's Perspectives on Mathematics Learning in Early Childhood Education**

**Ingrid Pramling Samuelsson**

### **Children's Education Worldwide**

The United Nations Agenda 2030 points out in Goal 4.2: "By 2030, ensure that all girls and boys have access to quality early childhood development, care and preprimary education so that they are ready for primary education". Globally, the gross pre-primary (1 year before school) enrolment rate increased by 27% points in the last 19 years, from 34% in 2000 to 61% in 2019. Despite this progress, as of 2019, there were at least 175 million children aged 3–6 years old who were not enrolled in education, according to a UNICEF (2022) global report on early childhood education (ECE). It is great that ECE is on the agenda and that more and more children get a chance to participate in education, but the whole feld of education and care before schooling is very complicated, and it is diffcult to compare fndings among countries. For example, the names for these kinds of activities vary: day-care, crèche, preschool, play school, infant school, kindergarten, toddler group, early child development, early childhood care and education, child-minding services, etc. There is no uniformity whether on what settings before school are called or what they contain. Further, the same notion can have different meanings in various countries (Pramling Samuelsson et al., 2018). In Sweden, preschool stands for all education from toddlers to school entrance, whereas preschool in, for example, the Netherlands represents part-time education for children between two-and-a-half and four as a preparation to start school, while a full day programme is called daycare (Preschool and day-care in the Netherlands, 2022). The reason for the large variation may be that children's learning before school is most often linked to family policy and not to educational policy. Nevertheless, the common rhetoric remains:

I. P. Samuelsson (\*)

Department of Education, Communication and Learning, University of Gothenburg, Gothenburg, Sweden e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2024

H. Palmér et al. (eds.), *Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives*, https://doi.org/10.1007/978-3-031-37663-4\_2

"Education starts at birth! Parents are the child's frst teachers; ECE is only complementary or compensatory to family experiences, depending on how the family functions".

However, all children should mean ALL—all ages and wherever they live. Agenda 2030 is for all children, and even though Goal 4.2 states readiness for school, it does not indicate whether early years education should be like school education. Professionals have to decide what is meant by preparing children for school and life. Goal 4.7 points out the knowledge and skills needed to promote sustainable development. One of these notions is *global citizenship* as a process in which evidently language and mathematics become necessary for frst becoming a local citizen and then becoming a global citizen. Thus, although the goal may not point out mathematics or literacy in some form, the target aspect is citizenship.

### **How Can a Child Perspective and Children's Perspectives Be of Use in Young Children's Learning?**

Agenda 2030 is not the only central aspect of early education; the UN Convention of the Right of the Child (UNICEF, 1989) is equally crucial. Specifcally, two articles of the Convention stand out:

• *Art 3: In all actions concerning children, whether taken by public or private social welfare institutions, courts, administrative authorities or legislative bodies, the best interests of the child shall be a primary consideration.*

This article posits that adults' thoughts are the best for a child, based on both general knowledge of child development and knowledge of the child as a person. This can be viewed in ECE mathematics as the adults' plan according to the age of the children and the personal knowledge of the individual children in the group. This is what Sommer et al. (2010) called *a child perspective –* interpreting what is best for each child based on general knowledge about the specifc child and earlier experiences. This may be what is often labelled "developmentally appropriate" education in the United States (Sanders & Farago, 2018).

• *Art. 12: States Parties shall ensure that children who are able to form their own opinions have the right to express them freely in all matters affecting the child, in which case the views of the child shall be given importance in relation to the age and maturity of the child.*

Sommer et al. (2010) perceived this as the *children's perspectives*—listening to and interpreting the child's expressions. Here, the child is involved in an active way, giving his or her thoughts, ideas, etc. Children are given agency for their own experiences, which makes it necessary to create opportunities to listen to children.

In Article 3, the child is viewed as an object, even though adults intend to meet children's needs, and in Article 12, each child is viewed as a subject who has agency. In the frst case, teaching a child mathematics refers to giving children tasks that are appropriate for their age, according to what child development tells us that children are capable of doing. In the second case, mathematics teaching in ECE is a question of giving tasks and communicating with children to capture their ideas and learn from that to take a new step towards challenging children (Shier, 2001). From this perspective, solving a problem is a joint negotiation between the teacher and children, in which there is an opportunity to solve the problem in various ways and not always according to what the teacher thinks is correct. An example of the last case could be when children get the task to divide 10 buns between three people, and some children take an ethical perspective instead of a mathematic one, claiming that it is fairer for the father to get fve buns because he needs more food, or for the mother to get only one bun since she always is on a diet (Doverborg & Pramling Samuelsson, 1999). Here, considering the children's perspectives is a key component of teaching!

Listening to children is important since it also has become both related to democracy and participation, but studies have shown that participation from the teachers' perspectives ranges widely from children participating in their own play, since they decide on what to do by themselves, to voting about what to do in various learning situations, etc. (see Williams et al., 2016). Listening to children has been used as a pedagogy in terms of "The Pedagogy of Listening" (Åberg, 2018), which originally emerged from the preschools in Reggio Emilia but has also been used as a rhetoric for all good work in preschool.

Data from an empirical study based on open questions about sustainability work in preschool in Sweden by Engdahl et al. (2021) were re-analysed with a focus on how the preschool teachers use the often-intertwined notions of *listening to children* and to *follow children's interest,* which in everyday work is related to taking children's perspectives. The new analysis revealed three categories of conceptions, representing various ways of thinking and talking about how teachers say they take advantage of listening to children or meet their interests (Björklund, 2020) which will be described and illustrated by quotations from the teachers in the following segments.

### *Listening Is Central to Discussions About Democracy and Participation*

In this category, teachers relate the perspectives of listening to children and following children's interest to the social developmental aspects as a representation of making children participating in various activities, and by that giving them possibilities to be involved in a democratic process, becoming able to give their views.

During the past year, we have chosen to focus on collaboration between children in our activities. We have focused on abilities such as feeling responsible, listening, following, leading, expressing thoughts, and explaining. This approach increased awareness of democracy.

We educators have a responsibility to listen to the children, give them infuence, and create participation.

Everyone's opinions are equally valuable!

We practice listening to each other and respecting each other.

From the teachers' view, taking the above perspective, the reason for listening to children is to develop the social aspects—that is, children's rights to become a participant in matters about themselves, which today is clearly spelled out as a goal in the Swedish curriculum (National Agency for Education, 2018).

### *To Follow Children's Interests and to Listen to Children for Their Learning*

In the following category, we can see the perspective of a theory of learning in which children learn when they are active in thinking and refecting, which has always been a perspective in Swedish preschool, where play and learning both have been central as well as care and education as two sides of the same coin. In this analysis, the teachers further expressed the fact that the child's interest or idea should always be the beginning of learning about something. The teachers' task is to listen to what children are interested in to know what to work with.

In addition, there are expectations and faith in the future that will characterise the meeting between the children's questions and challenges about climate and sustainable development. Therefore, the preschool should listen to and meet the children where there is interest.

The children's interest and curiosity control the content of the project, and the children become involved.

To slow down, see/feel the joy in what we have and are, dare to stop and start from children's questions, thoughts, dilemmas, and problems.

The project started with us being out and discovering and exploring to capture children's interest in sustainable development.

Children's experience is that projects become the best if we take care of the children's interests. Then you can really see a genuine interest and a desire to learn. That's why I start with the children's ideas.

The role of play in young children's learning and wellness is central to teachers becoming aware of their interests.

In this category, children's interests are central to learning. Teachers indicate that they have to listen to children or observe play to understand what topic/theme to work on, since children's world is the starting point for starting with a specifc content area.

### *Negotiation as the Beginning of Both Learning in and About Democracy and Various Content Areas*

In the following category children's perspectives are always related to communication and negotiation with the teacher. If the ideas or interest comes from the children or not is not the main question, but how the teacher managed to get children interested in something or to share their ideas in the education setting.

We should of course use children's interest, but you cannot request that exactly all children who come to preschool are interested in just this; therefore, it may sometimes be necessary that we who work in preschool arouse an interest by challenging …

When we try to listen to children's interests, it usually becomes a Matthew effect—the more you have, the more you get—and the children who already have a lot of infuence get more infuence. In addition, we educators have a tendency to pick up what the children are interested in that we recognise. This means that we rarely start from the interests of the children who come from other cultures and have different experiences than we educators.

One could claim that listening to children and using their interest in pedagogy is more problematised than the two approaches above. The teacher and the children are both necessary for educating children. The teachers know about the curriculum and have to be active in pointing out content areas. If children should always decide, nothing new may come up; instead, all children need to be challenged. This third category more clearly shows that the teacher's role is a pedagogical approach to learning and development, which is more in line with *developmental pedagogy,* the approach to preschool developed in the research group at the University of Gothenburg (see, e.g., Pramling & Pramling Samuelsson, 2011). Teaching in ECE is always a joint venture between children and teachers, in which the teachers know the direction to point out to children for learning (Doverborg et al., 2013), and the children's experiences and ways of talking about the content are key questions for the teacher to challenge further development of understanding. In developmental pedagogy, the content can be described in two steps: an area, such as the content of mathematics, and the learning objects—that is, the specifc aspect of mathematics addressed in every new teaching situation that could, for example, be number conceptions, sizes, and patterns.

All these aspects that teachers bring up are, of course, relevant. Sometimes, children become occupied with something to which the teacher could link his or her planning, but most of the time, the teacher is the one who knows the experiences or the curriculum goal the children need to work towards. However, democratic and negotiating lenses are both necessary for infuencing children's learning. Most importantly, taking children's experiences and ideas into consideration can be viewed as a touchdown in time regarding the status of the children in their learning process of the aspect of mathematics work right now on the spot (Pramling Samuelsson & Pramling, 2009).

### **Empirical Research Paves the Way to Education Based on Science?**

ECE has a tradition based on child development, a question of children's more general personal development, which is important in the early years. However, as Sommer (2006) pointed out in his description of childhood psychology, the large general theories about child development have been exchanged with many minitheories in various areas, with children's knowledge of mathematics learning being one area (Björklund & Palmér, 2018; Björklund, 2019). This means that content has become important in ECE, which was not previously the case. In the past, the approach of teaching was the key question, which is still central but related to various content areas and to children's sense-making of the content (Björklund & Pramling Samuelsson, 2020; Pramling, 1983). Hence, empirical work in various content areas is so important for developing didactics for early years that it takes children's agency into consideration, rather than relying on child development in relation to various content areas.

In what follows are two empirical examples of aspects of mathematics in which the children's worlds of understanding a specifc task vary and become visible in their acting and communication. The scenarios provide the teacher with an understanding of how children make sense and offer opportunities to understand how he or she has to challenge children for further development of understanding a concept. The frst scenario evaluated young children's understanding of the *frst and the last*. The teacher and each child in the study played with animals, and the teacher suddenly asked, "Should we let the animals go for a joint walk?" She then took one animal at the time and said, "First comes the cow, then the panda, followed by the lamb and last comes the pig". After that, she asked the child, "Which animal is the frst one, and then which is the last one?" In the study, 240 children between ages 1 and 3 years participated, and the analyses of children's understanding were categorised into four categories: (1) know both frst and last, (2) know frst, and say that the last is the second one (which may be because we had 4 items, 3 may have been easier for the youngest), (3) know only frst, and (4) did not bother at all, continue to play (Sheridan et al., 2009).

The second empirical case tried out an example of representation in mathematics. The teacher and a boy (3.4 years) counted animals—two tigers of various sizes—and the teacher posed the question, "Can you write on this paper, how many tigers you have, so we can remember?" The boy had already counted the tigers. He looked at the tiger and said, "I cannot draw the ears of the tiger!" Teacher: "It is not important!" The boy continued, "I cannot draw the mouth of the tiger, either". The teacher, realising that the boy thought he had to draw the tigers, responded by saying, "But do you think there is any other way to write on your paper how many tigers you have?" Then the boy drew one long line and one short, and then he took each tiger and put it on the respective line, according to the length of the tigers.

These two examples can be viewed both as making children's way of experiencing these mathematical questions visible from their perspectives and as touching down in time regarding the status of the child's knowledge development at the present moment. However, we do not know whether these tasks created new knowledge in children's minds. Thus, there is a need for research on how the process of children's learning can become visible to the teacher.

Let me exemplify with one more example that has very little to do with mathematics (that is, indirect distance): *how to fnd out something.* The participants are 300 children between 2 and 8 years of age. The researcher's question is, "If you want to fnd out how far it is to the moon, how would you go about that?" The four categories of conceptions analysed from the data are: (1) I would build a space

shuttle and go there, (2) I would ask someone who has been there. (3) I would ask my parents or my teacher, and (4) I will fnd it out via media, books, the internet, etc. We can see that in children's minds, conquering knowledge ranges from fnding out by independent enquiry to acquiring free knowledge from others (Pramling, 1983).

In a group of children, one can always discern numbers of qualitatively different categories of conceptions. These categories become visible by making children's perspectives visible. Sometimes these categories represent a step towards a more advanced understanding of something in relation to culturally accepted conceptions; other time conceptions may be horizontal, just representing various ways of expressing the same understanding. The example of how to fnd out something represents more or less advanced understanding by the children, which helps the teacher see what the next step of understanding for children might be.

### **How Is Play Related to Learning?**

All ECE curricula mention play as a key factor in young children's learning. However, it is seldom problematised but is just taken for granted. Article 35 in UNCRC (1989) *also recognizes the child's right to rest and leisure, to play and recreation adapted to the child's age and the right to participate freely in cultural and artistic life.* If there is anything children relate to, it is play. All children play if they are not hindered from doing so! The play, as such, is what children are mostly interested in, as well as coming to ECE to meet their friends. Some children are ultimate players, whereas others do not depend on their early experiences. Fleer (2015) claimed that early interaction in families sometimes lack the interactive ground for developing playful children. Thus, even though there may be aspects of play that children are born with, acting in imaginative play is learnt from the environment. Further, by being included in imaginary play, children also have to have cultural knowledge to take part (Mauritzson & Säljö, 2003).

Some researchers claim that we, as adults, destroy children's play if we intervene, and that we should not relate play to learning (Steinsholt, 1999; Øksnes & Sundsdal, 2018; Hangaard Rasmussen, 2016), whereas others consider it advantageous to relate play and learning in ECE (Lillemyr, 1995; Pramling Samuelsson & Asplund Carlsson, 2008). Therefore, how are preschool staff dealing with play in practice? Studies about what teachers say as well as what they do can be described from three perspectives. The frst and most common is that play is the child's own world, and teachers do not get involved in children's play. Second, teachers are external observers of the play who then try to help children expand their play by giving them new props, reading books on the topic of their play—they try to inspire but not take part. Third, teachers become involved in children's play as playmates. The debate of whether or not to participate in children's play is often based on ideology—what teachers think is best for children (Pyle & Danniels, 2017). With this debate as rhetoric, we began to work in our research group to fnd empirical evidence of what happens if teachers go into children's play in a praxis-oriented project

called *Play-based didactics: Developing early childhood didactic theory in collaboration between researchers and preschool teachers.* The following is a summary of the application:

Discussions about the preschool's activities often end up in an advocacy of either teaching or play (in preschool often referred to as 'free play'). A premise for this project, however, is that such a dichotomy is not fruitful for understanding and promoting children's development. Instead, a central challenge for contemporary Swedish preschool is to develop a form of teaching that is in line with the preschool's tradition and history, i.e. to design play-based preschool didactics. In this project, we further develop the preschool's own theory formation: Developmental pedagogy.

The journey from developmental pedagogy (Pramling, 1994) to the theoretical foundation of this last project has investigated play in various ways. In our earlier studies, we took play for granted; we described it but did not problematise it—play was self-evident therein. However, in 2003, when we published a meta-study of our research in the book *The Playing Learning Child* (Pramling Samuelsson & Asplund Carlsson, 2003/2014), it became obvious that we looked at children as playing learning children; we did not separate play from learning. As teachers, we have to learn that this is the case, and we have to integrate play and learning in praxis. Then came the question: Is it possible to integrate play and learning into a goal-directed practice? However, this seems to involve two different views: play as children's world and learning as the teachers' and the curriculum's intention. The simple answer to this question became: It all depends on the teacher's communication whether they could allow both themselves and children to use both fantasy and reality in the communication (Johansson & Pramling Samuelsson, 2006; Pramling Samuelsson & Johansson, 2006). The latest way to look at play and learning developed from the project mentioned above is through *play-responsive teaching* (Pramling et al., 2019).

These changes in how play and learning are conceptualised have been viewed more carefully and extended by Pramling Samuelsson and Björklund (2022). The latest publication in process is one in which teachers who participate in a network based on this last research project have developed agency to become authors of their own work, as they have interpreted play-responsive teaching in their everyday work with children. The book is called *The Teaching Playing Preschool Teacher.* The teacher becomes both the one who leads children towards understanding various mathematical notions and one who learns from children's ideas all the time. Examples of mathematics in play-responsive teaching can be found in Pramling et al. (2019), specifcally Chaps. 8 and 10.

*A short summary of what play-responsive teaching means*

stand what is the one and the other perspective – but communication becomes enjoyable.

### *What Did We Learn From the Project?*

First of all, the participating teachers became quite skilled in participating in children's play, even though it was hard for some of them in the beginning, as they video-recorded children play without being involved, or tried to focus on the social aspect to get all children involved. In the end, they said that the children often asked them to come and participate in their play. They could, in other ways, participate without destroying the play! Further, the children enjoyed and asked them to participate when they got used to having teachers involved.

The notions of *intersubjectivity, narrative, and meta-communication* became important to talk about, and the teachers used them to interpret their video recordings. Intersubjectivity clarifes what communication with a joint focus or content can look like, and allows for understanding the intersubjectivity in short situations of shared focus. Narratives seem to constitute a frame for play content. Especially for young children, it was often related to stories or songs they were familiar with as the focus of play. Meta-communication is known as a source for developing awareness in children by putting words and refections into what they are doing. Children do not always create an understanding of just doing something; thus, engaging the mind of children also means refecting and communicating about it.

As *didactic consequences*, we observed that teachers had to plan to become able to get involved in children's play. This does not happen on the spot, and the other staff must be aware that the teacher now intends to be involved in children's play and cannot do everything else, such as answering the phone, welcoming late children, or taking care of the dishes. Further, teachers must be convinced that play and learning can infuence each other and that there are possibilities to focus on questions in play that are related to the curriculum goals. This means seeing themselves as active in communication with some children all the time, and not only in the thematic work in which they intend to distribute knowledge. Teachers must be responsive to children's actions (verbally and in acting) and give children preconditions for being responsive to the teacher's activities. Some of these key notions are the exchange between "as if" (fantasy) and "as is" (culturally established) knowledge. Teachers must become able to talk in play or be different characters, as well as meta-communicate about the play or the content in the play activities in which both the teacher and children are involved. Just as teachers have to be skilled in communicating with children about various content areas, they have to learn the characteristics of play to be play-responsive.

### **Conclusions**

In a recent analysis of documents during the history of Swedish preschool's development, it becomes obvious that preschool has developed through a hard political battle, frst as a struggle between men and women and later between different political parties. It also became visible that changes in both access to preschool and pedagogy depend on political decisions (Klingvall & Pramling Samuelsson, 2022). Hence, lobbying is an important aspect of being a researcher if one intends to infuence change. Bringing play into children's learning has support in the text of the Swedish curriculum (National Agency for Education, 2019, p. 9):

Children should be given the conditions both for play, which they themselves take initiative and play introduced by someone in the work team. All children should be given opportunities to participate in shared games based on their conditions and abilities. When someone in the work team follows or leads play appropriately, either outside the games or by participating themselves, factors that limit play can be noticed and work methods and environments conducive to play develop. An active presence makes it possible to support communication between the children and to prevent and manage confict.

As play has been more extended and related to learning in the curriculum, Swedish preschools have become more knowledge-oriented. The reason for this is, of course, that we have so much more research today showing the capacity of the child under appropriate conditions. Education has, by tradition, been related to school, where in many places in the world, knowledge is distributed to children. Changing this conception is diffcult, even though research reveals otherwise regarding young children's play and learning.

One aspect concerns the contemporary emphasis in ECE on paradigmatic (categorical, scientifc) knowing at the expense of narrative knowing (Bruner, 1990; Singer & Singer, 2005). Due to fear that some children may be falling behind in their knowledge (language development, subject-matter expertise, etc.), increased focus has been put on paradigmatic knowing. However, narrative knowing (typically nurtured through play) is arguably more important and certainly more foundational regarding mathematizing. Through narrating, people communicate, make sense (of themselves, each other, and the world), remember, form identities, and imagine other possibilities (as if and what if?), and it is therefore critical also to a sustainable future—imagining other ways of living and solving problems. Communication is central in all early learning, and a recent research overview, *A Systematic Review of Educators' Interactional Strategies that Promote Rich Conversations with Children aged 2–5 years* (Houen et al., 2022), shows that it is

not only open questions that contribute to learning; play-responsiveness does as well.

The formal education of ECE is non-formal, in which the teacher needs to have professional knowledge of all content areas, such as mathematics, to know what to focus children's attention on, thereby giving children opportunities to learn. Putting children's attention towards various mathematical notions can be achieved by capturing the situation in which a child shows interest, or planning tasks of various kinds. These have been done with great success by designing learning situations based on variation theory (Björklund, 2016). Challenging children in a playresponsive way is important, since play-responsiveness does not have to only take part in imaginative play—it can be used in any situation or with any content area. Play and learning should always be integrated into each other in ECE.

With mathematics, similar to other content areas, children need to both live and experience mathematics in meaningful situations in daily life and communicate mathematics. It is in this communication that teachers can direct children's attention to aspects that can help them develop an understanding. Even though the intention during preschool years is to develop children's understanding of various notions in mathematics, the teacher must always be aware of the fact that children need concrete objects that can be used as support to explain or argue for something (Nergård, 2022).

Regarding both a child perspective and children's perspectives in ECE, both are necessary in practice. For the teacher to have appropriate professional knowledge in mathematics and child development is the basis for early education (child perspective). However, within this frame are the children's perspectives of, for example, numbers, patterns, and forms, aspects that need to be met and challenged in teaching and infuencing children's learning. This call for specifc knowledge by the teachers in didactics in various areas.

### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Preschool Teacher Practices During a Collective Play-Related Thinking: Dialectic Between Pretend Play and Mathematics**

**Linda Amrar, Anne Clerc-Georgy, and Jean-Luc Dorier**

### **Introduction**

From an historico-cultural perspective, pretend play holds an important place and is thought to create a zone of proximal development (Vygotsky, 1966/2016). In fact, when children play, they "jump above the level of [their] normal behavior" (Vygotsky, 1966/2016, p. 18) by taking advantage of the degrees of freedom to manipulate and explore their environment (van Oers, 2014). Vygotsky (1966/2016) suggests that during pretend play, children create an imaginary situation in which they substitute objects meaning, take roles, and defne rules associated with these roles. Furthermore, pretend play constitutes the leading activity for children aged between 3 and 7 (Vygotsky, 1966/2016). Meaning that even if children do not engage most of their time in pretend play activities, it is thought to drive their development.

Pretend play is associated with the development of different cognitive and socioaffective competencies, some of which are associated in particular with the development of mathematical thinking (Amrar & Clerc-Georgy, 2020). During free play, more than half of the play time is devoted to the exploration of mathematical concepts (Seo & Ginsburg, 2004). During pretend play, a wide range of mathematical

L. Amrar (\*)

University of Teacher Education of Lausanne, Lausanne, Switzerland e-mail<EMAIL_ADDRESS>
A. Clerc-Georgy University of Teacher Education of Lausanne, Lausanne, Switzerland e-mail<EMAIL_ADDRESS>
J.-L. Dorier University of Geneva, Geneva, Switzerland e-mail<EMAIL_ADDRESS>
© The Author(s) 2024 H. Palmér et al. (eds.), *Teaching Mathematics as to be Meaningful – Foregrounding Play and Children's Perspectives*, https://doi.org/10.1007/978-3-031-37663-4\_3

University of Geneva, Geneva, Switzerland

activities are explored (van Oers, 1996). According to van Oers (1996), when mathematical activities initiated by children are seized by the teacher, they become mathematical teaching opportunities. Together, these studies show the importance of scaffolding the pretend play components dialectically with the mathematical activities initiated by children during play.

However, many preschool teachers restrain from intervening during childinitiated activities, by fear of interrupting the children's play. As a result, many opportunities to sustain the development of children are overlooked, which could explain the dichotomy between play and learning amongst teachers and children. Pyle and Danniels (2017) identify two preschool teacher's profles depending on whether the play is investigated or seen as separate from learning. Clerc-Georgy et al. (2020) propose an alternative view to overcome this dichotomy. In their paper, the authors support the idea of a dialectical relation between play and learning, highlighting that scaffolding should be directed towards the development of play and curriculum at the same time. Scaffolding interventions promote learning and can take place either inside or outside the play (Fleer, 2015, 2017b; Wassermann, 1988). To date, few studies have investigated what preschool teachers do when they seize dialectically mathematical teaching opportunities and pretend play. This will be the focus of this case report describing the practices used by a teacher during a CPRT which takes place in the aftermath of a children-initiated pretend play activity.

### **Pretend Play and Mathematics From an Historico-Cultural Perspective**

Pretend play and mathematics are both cultural and semiotic activities. According to Ernest (2006), a semiotic activity involves the use of signs, the use of rules associated with these signs and the attribution of meaning to the signs. Mathematics share all the component of a semiotic activity as signs are used continually during a mathematical activity (Dijk et al., 2004). Regarding play, Vygotsky points out: "Play is the main path to cultural development of the child, especially for the development of semiotic activity" (Vygotsky, 1930/1984, p. 69, as cited in van Oers, 1994). However, Vygotsky (1966/2016) highlights the need to be cautious with the comparison between pretend play and mathematics through the lens of signs. Indeed, the signs used in play are not specifc and identical across the play of children, however, in mathematics, a consensus around the use and meaning of signs has been reached across a community of mathematicians. Hence the importance of not intellectualizing children's play when considering their relation to semiotic activities.

The change in the meaning of objects and actions is specifc to the substitution component of pretend play and is used as an indicator of the maturity of the play (Bodrova & Leong, 2012). The role of the teacher is to sustain the development of this component bearing in mind to decrease the mount of support, allowing the children to take responsibility for this change in meaning (Kravtsov & Kravtsova, 2010). Furthermore, the substitution component of pretend play is the best predictor of performances in mathematics and reading (Hanline et al., 2008). More specifcally, results of this study have shown that the more the children are able to substitute objects and actions while pretending, the higher their mathematical performances will be when they reach the age of 8. The hypothesis of the authors is that both activities engage the use of signs. This result shows the importance of developing the substitution component of pretend play, considering the long-term effect associated with the maturity of this component.

The close link between pretend play and mathematics makes it a particularly interesting activity to sustain the development of the mathematical thinking of children. In a study by van Oers (1996), teachers were supposed to ask semiotic questions to children (e.g., "Are you sure?", "How could you be sure?") in order to stimulate the mathematical actions of children during a pretend play activity based on a shoe shop scenario. The results show that children initiated different types of mathematical actions such as classifcation, 1–1 correspondence, measuring, and schematizing. Another result of this study is that teachers can use semiotic questions to seize the mathematical activities of children, which then become mathematical teaching opportunities. This study is of particular interest as it shows that teachers can sustain the mathematical actions during pretend play.

These studies highlight the semiotic nature of the link between pretend play and mathematics. They also reveal the fundamental role of the teacher in seizing opportunities to scaffold mathematical actions and pretend play activities initiated by children. However, they do not take into account how teachers can scaffold dialectically the development of pretend play and the mathematical actions initiated by children simultaneously.

### **Scaffolding During Pretend Play**

According to Wood et al. (1976), scaffolding is the process at stake when a teacher supports a student during the completion of a task too diffcult to be solved by the student alone. The three main components of scaffolding are: transfer of responsibility, contingency and fading (van de Pol et al., 2010). Scaffolding refers to the progressive transfer of the amount of responsibility to perform a task that is gradually transferred from the teacher to the student. Meaning that scaffolding is aimed at progressively giving ownership of the task completion to the student. Contingency refers to the adjustment of the support to the current level of the student. According to the clues provided by the students' answers, the teacher will elaborate an idea on the current understanding of the student and adapt the scaffolding. Fading refers to the decreasing amount of support provided by a teacher to a student. When scaffolding fades, the assistance provided is progressively lessened to allow the student to succeed without support. In order to be characterized as scaffolding, an interaction needs to bring into play these three components (van de Pol et al., 2010).

Scaffolding techniques can be used during an interaction aiming at communicating about the play. It has been shown in the work of Wassermann (1988) and Truffer-Moreau (2020). In the Play-Debrief-Replay (PDR) model, Wassermann (1988) holds the view that play is an area of explorations which can be shared and expanded during a debriefng session. During play, different groups of children explore a scientifc material proposed by the teacher. The explorations carried out by the children are used as the basis for the debriefng part. After the debriefng session, a replay session takes place where children can play again, bearing in mind the information discussed during the debriefng. The debriefng session share characteristics with the CPRT described by Truffer-Moreau (2020). A CPRT is a refexive interaction between a teacher and a group of children, elaborating on the scientifc concepts investigated during a child-initiated activity. The author highlights that the CPRT "acts as a pivot between children-initiated activities and adult-initiated activities" (Truffer-Moreau, 2020). The CPRT focuses on scientifc concepts explored by children during a free play activity. Compared to the PDR model, the material and the group of children are not restricted during the play activity and the replay session does not necessarily take place directly at the end of the debriefng session. A teacher can decide to set up a CPRT at any time during the play. Thus, the teacher is required to be sensitive to children activities as well as to the dimensions of the curriculum associated with the scientifc concept explored. During a CPRT, the teacher scaffolds children's thinking on the basis of what happened during the childreninitiated activity. The teacher guides an interaction involving the group of children based on the scientifc concepts explored during the play activity. The CPRT is part of a broader systemic structure called pedagogical structure including training activities and two leading activities, which are play activity and learning activity (Elkonin, 1999; Vygotsky, 1935/1995). In the pedagogical structure, the knowledge holds a central place as it constitutes the core and the binder of all the structure's components. In the PDR model, a replay session takes place directly after the debrief session. The aim is to allow children to replicate their fndings or to test other hypothesis discussed during the debriefng. In the CPRT model, play is included in a broader structure. The teacher can thus decide to put in place a structured activity on the scientifc concept seized during the CPRT shortly after the CPRT.

### **Research Question**

The studies presented thus far provide evidence that pretend play constitutes a central activity for children's development during which mathematical contents appear. These contents can be seized by the teacher to foster their appropriation through scaffolding techniques such as CPRT. However, little is known about the teacher's practices at stake during a CPRT dealing dialectically with a pretend play activity and a mathematical content seized by the teacher. The aim of this research is to explore in terms of scaffolding the practices used by a teacher who seize

mathematical teaching opportunities from pretend play activities during a CPRT. The research question is: What are the practices associated with scaffolding used by a teacher during a CPRT in order to seize mathematical teaching opportunities?

### **Data**

The data are part of a larger study exploring the mathematical teaching opportunities arising from children-initiated activities. This extract has been selected as it is considered to be a CPRT. The classroom is situated in the French-speaking part of Switzerland. The teacher has 7 years of experience in teaching. Eleven children aged between 5 and 6 took part in the activity. The teacher and all of the children's parents signed the ethic form and the data were recorded according to ethical considerations. For ethical reasons, fctitious names are used in this case report. The researchers acted as observers and did not interact with the children.

Before the free play activity begins, the teacher provided materials (e.g., X-ray pictures, fabrics, big jigsaw pieces made of foam, paper, pens...). During a free play activity, the children initiated a hospital pretend play. They undertook different roles (e.g., receptionist, doctors, patients) and recreated different areas in the classroom (e.g., reception, patient's house, waiting room). The teacher acted as the doctor and moved between being inside and outside of the play. After approximately 1 h, the teacher gathered the children to do a CPRT.  In Fig. 1, the drawing of the plan of the hospital partly recreated during the CPRT based on the hospital play activity initiated by children is shown.

**Fig. 1** Plan of the hospital

### **Methodology and Analysis**

In this case report, a qualitative approach will be used based on the interaction analysis of Jordan and Henderson (1995) and the scaffolding framework analysis by van de Pol et al. (2010). The interaction analysis investigates the orderliness and projectability through the decomposition into units of coherent interactions. This method is particularly useful in studying interactions in a school context as it allows the identifcation of the different practices at stake during teaching/learning interactions. Derry et al. (2010) suggest focusing on "a particular pedagogical or subject content". In this case report, a CPRT focusing on the elaboration of the plan of the hospital created by children during a pretend play activity is analyzed. Jordan and Henderson (1995) highlight the importance of identifying the beginning and the end of the interaction to analyze. The defnition of a CPRT is used to identify the boundaries of the interaction analyzed. The beginning of the activity takes place when all the children sit down together. The end of the activity takes place when the modality of the activity changes from a collective to an individual activity. The interaction has been transcribed including the gestures of the teacher and the children prior to the analysis.

### **Results**

The analysis reveals four practices used by the teacher to seize mathematical teaching opportunities during a CPRT, which will be detailed below: (1) guiding a twolevel intersubjectivity, (2) fostering the development of an imaginary situation, (3) raising awareness on challenges and (4) creating meaning between symbolization in the play and symbolization in a cultural tool.

### *Guiding a Two-Level Intersubjectivity*

At the beginning of the CPRT, the teacher guides a two-level intersubjectivity in order to create a common understanding: an intersubjectivity between the teacher and the children (Björklund et al., 2018), and an intersubjectivity between the children themselves. At the teacher-children level, the teacher is responsive to the children's answers and ensures that a suffcient intersubjectivity emerges. At the children-children level, the teacher guides the interactions to allow children to share their perspectives, mutualize their knowledge, and reach a common understanding. This two-level intersubjectivity (teacher-children; children-children) is directed towards the adjustment of all of their perspectives on the discussed play activity. The teacher makes room for perspectives to be expressed and orchestrates the discussion to establish a suffcient intersubjectivity at these two-level.

Excerpt: "location of the waiting room"

In this excerpt, the teacher asks questions about the location of the hospital's waiting room created by the children and documents the plan by drawing a rectangle on the paper sheet. This is done systematically for each area every time an agreement is reached. The teacher asks about the location of the waiting room. John points in the direction of the area dedicated to the waiting room during the play activity. As all the children agree, the teacher adds the delineation of the waiting room on the plan. In this excerpt, the plan supports the two level intersubjectivity as it contains information on what is agreed on between the teacher and the children but also between the children themselves.

### *Fostering the Development of an Imaginary Situation*

The second practice requires fostering the imagination of the children while refecting on the play. To expand the learning at stake, the teacher builds the CPRT on the basis of what happened during the play activity and fosters the imagination of children to encourage them to go further. While doing this, the teacher creates a bridge between play activity and learning (Fleer, 2017a).

Excerpt: "adding a school to the hospital"

In this excerpt, Veronica adds a school in the hospital. During the play activity, the children did not create a school which could explain Oscar's objection. Instead of asking if the school was set up during the activity, the teacher encourages Veronica to give more information about the school. Tom makes a link between the function of a hospital and a school to make a case for Veronica's proposition. Once the agreement is reached, the teacher adds the new area of the hospital on the plan. The plan thus becomes more complex and supports children as they create meaning. The teacher fosters the imagination of the children by being responsive to their propositions and by encouraging them to capitalize on their knowledge to explore further through imagination.

### *Raising Awareness on (Potential) Challenges That Can be Resolved Using Mathematical Tools*

The third practice used by the teacher while talking about the imaginary situation, is to make children aware of (potential) challenges that can be resolved using mathematical tools. The teacher can also mention an event experienced by the children during the imaginary situation, and discuss the challenges through a mathematical perspective.

Excerpt: "shed light on a challenging situation"

In this excerpt, the teacher anticipated the diffculty that children would face when trying to match an area on the plan and its corresponding area in the classroom, when rebuilding the complex hospital they created, if they do not add a coding system to the plan. The teacher raises the awareness on the limit of memory span and the need to fnd a strategy. Charlotte suggests writing it down and the teacher raises another challenge related to the space required to write down long words. Charlotte insists by saying that it is possible to write above the delimited areas on the plan. Finally, the teacher suggests making drawings associated to each delimited area on the plan. All the children agree with this solution. In this excerpt, the teacher raises awareness on a potential challenge and guides the discussion towards the use of symbolization. The teacher takes the responsibility to shed light on a challenging situation and suggests a solution involving the use of a mathematical tool: symbolization.

### *Creating Meaning Between Symbolization in the Play and Symbolization in a Cultural Tool*

This practice requires the creation of meaning between symbolization in the play and symbolization in a cultural tool. During pretend play, children substitute the meaning of objects and actions to assign them a new meaning. During the CPRT, the teacher guides the interaction toward the creation of meaning between the substitution during play and the use of symbols in a cultural tool. Children are encouraged to refect on the symbols they used in the play and expand their abstract thinking to symbols used in cultural tools. Here, the CPRT acts as a bridge between symbolization in the play and symbolization in a cultural tool.

Excerpt: "fnding symbols to represent areas on the plan"

In this excerpt, the teacher encourages children to fnd symbols that could be drawn on the plan to remember the function associated with each area delineated on the plan. The teacher asks questions to make the children aware of the challenges that need to be resolved collectively and transfers the responsibility of creating the symbols to the children. The teacher guides the children to move from symbolization in the play to symbolization in the plan. For different areas, children suggested the props used in these specifc play areas. To represent the reception area, Maya suggests drawing a computer.  On the plan, a keyboard and a computer mouse have been drawn to represent the reception (pink). When the teacher asks about the waiting room, Emily suggests a bed or a chair as they created beds and used chairs from the classrooms to represent the waiting room area during the play activity. As beds are already used to represent the patients' room (yellow), the teacher repeats "chairs". In the Fig. 1, we can see some iconic signs sharing characteristics with props used in the play. For example, the waiting room is represented by a chair and a table (black). When the teacher creates meaning between symbolization in the play and symbolization in a cultural tool, it scaffolds symbolization.

### **Discussion**

The aim of the study was to identify the practices used to seize mathematical teaching opportunities during a CPRT. The results suggest that the teacher used four different practices sustaining dialectically mathematical and play-related thinking of children during a CPRT taking place after a pretend play activity. The teacher ensures that a suffcient intersubjectivity emerges at two level creating a common understanding of the subject at stake: between the teacher and the children and between the children themselves. The teacher fosters the development of the imaginary situation and raises awareness on challenges. The fourth practice is used to create meaning between symbolization in the play and symbolization in a cultural tool.

During the CPRT, children were encouraged to refect on the imaginary situation created at a metacommunicative level and using a mathematical perspective. During a CPRT, the teacher undertakes the role of the expert and guides the children through the appropriation of cultural tools while leading a refexive discussion, which maintains the link between imaginary situation and learning. In this CPRT, the teacher starts a plan of the hospital created by children during a pretend play activity by delineating the areas imaginated by children and then let them individually draw a symbol representing an area. Dijk et al. (2004) found that the frst signs used by children often represent a real object. This result highlights the importance of introducing symbolization activities in the early years as they play an important role in the development of the abstract thinking of children. This is in line with the idea stated by Vygotsky (1966/2016) that learnings occur at a social level before being internalized at the individual level. During a CPRT, the teacher guides a discussion to create this dynamic using different practices.

In our study, the fndings suggest a role for the teacher in sustaining and scaffolding dialectically pretend play and mathematical activities initiated by children in order to build a cognitive structure (van de Pol et al., 2010). Children's cognitive structure will build upon the pretend play and mathematical explorations they initiated, as well as the refexive interactions guided by the teacher. This article adds to the growing body of research highlighting the need to scaffold dialectically pretend play and the scientifc concepts explored during an imaginary situation (Fleer, 2017b). The specifcity of this study is that a pretend play activity initiated by children could provide the basis for a mathematical refection guided by the teacher beyond the play activity.

The major limitation of this study lies in the fact that the fndings come from a case study. Thus, other practices could exist, which have not been found in our analysis. There is, therefore, a defnite need for further studies to explore the practices used by teachers to sustain dialectically the imaginary situation and the mathematical content initiated by children in a larger corpus of CPRT. Although the current study is a case study, the fndings suggest that different practices can be used by teachers to seize mathematical teaching opportunities during a CPRT taking place outside of a pretend play activity. The practices found in this article could be used for the initial training of teachers, guiding them through the appropriation of CPRT as a useful tool to foster children's learnings.

**Acknowledgments** The authors would like to thank the teacher and the children who participated in this project.

### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

### **Supporting One-Year-Olds' Digital Locating: The Mediating Role of the Apps and the Teacher**

**Silje Fyllingsnes Christiansen**

### **Introduction**

Digital tools are becoming increasingly more common all over the globe, with young children encountering them at home as well as in early childhood education (Otterborn et al., 2019). Although there is some research on older children using digital tools, little is known about the youngest children, especially when it comes to mathematics. Yet digital tools receive a lot of political attention, with an OECD report stating that "linking the way children interact with ICT inside of school to the way they already use it outside of school can be a key to unlocking technology's potential for learning" (Schleicher, 2019, p. 10). Similarly, a link between play and learning is commonly made in Early Childhood Education and Care (ECEC) practice and research (Nilsson et al., 2018). For example, Lundtofte's (2020) literature review of children's tablet play found that children's use of tablets is often connected to digital literacy and learning.

In this article, I investigate how two one-and-a-half-year-old children and a kindergarten teacher1 engaged with playful digital apps together in a Norwegian kindergarten. Across the Scandinavian countries, digital practices are part of the national curricula for ECEC (Børne-og Socialministeriet, 2018; Ministry of Education, 2017; Skoleverket, 2018) and, consequently, teachers are expected to facilitate their use. In Norway, although kindergarten is voluntary for children aged

<sup>1</sup>Kindergarten teachers in Norway have a bachelor's degree in kindergarten teacher education, and make up 42.3% of the staff in kindergarten (UDIR, 2022).

S. F. Christiansen (\*)

Western Norway University of Applied Sciences, Bergen, Norway e-mail<EMAIL_ADDRESS>
1–5 years old, in 2021, 87% of children between 1 and 2 years old attended kindergarten (SSB, 2022). Given that toddlers encounter digital practices in kindergarten and elsewhere, more research is needed to explore how this age group engages in digital activities, particularly those which facilitate very young children's mathematical engagement. This article aims to construct knowledge about how the design of digital apps supports kindergarten teachers in doing this. Therefore, the research question is: in what ways can digital apps mediate the way kindergarten teachers support the youngest children's engagement in the mathematical activity of locating?

### **Teaching in Digital Environments**

Although digital tools such as mobile technologies have been around for some time, their entry into early childhood education remains contested, with opinions often divided into two groups: those who see digital tools as a threat, and those who think they can make the world better (Danby et al., 2018; Palaiologou, 2016). Given this contested space, it can be challenging for teachers to implement digital tools in their teaching. Vangsnes et al. (2012) suggested that teachers rarely bring the Nordic pedagogy of valuing play into digital game-playing, indicating that the digital tools' entry into kindergarten might entail a risk of changing the children's pedagogical environment to one with less focus on children's play (Christiansen & Meaney, 2020). Fleer (2014) argued that that there is a need for further research to recognize new opportunities for play in digital settings.

Tablets with multitouch capabilities are often considered an intuitive tool suitable for children's developing coordination skills. For example, Geist (2012) found that when toddlers use digital touch screens, they showed a high level of skills in using the screens independently, and explored options in similar ways to how they used other play materials such as building blocks. Nevertheless, digital apps for children are often given a strong learning focus, being marketed as "fun learning" (Kvåle, 2021). In apps with a linear build-up of tasks, children can often guess their way to the correct answer and seem to be motivated by the digital stardust or confetti they get for fnding the right answer, rather than by learning ideas or concepts (Nilsen, 2018). Children can also change the purpose of digital activities from what was planned by teachers. Lafton (2019) found that children playing a digital memory game ignored the intended rules of the game, and instead created their own game with different rules. Digital apps with strong framing, those with clear control over the communication and the pace of the app, can sometimes limit children's participation (Palmér, 2015). Lembrér and Meaney (2016) suggested that children's agency could be strengthened by giving them control over the game, which could lead to them needing or wanting to explain what they were doing. This need could support children's refective mathematical thinking. Playful apps with a weak framing, where the app provides the children with control, could engage children in mathematical language and thinking when they played by themselves (Christiansen, 2022).

### **Theoretical Framework**

This article is part of a wider study where I have used Bishop's (1988) six fundamental mathematical activities to describe children's mathematics. In this article, I focus on the mathematical activity of locating as this is mostly what these very young children engaged with when using digital apps. To investigate how digital apps mediate kindergarten teachers' support, I use Ladel and Kortenkamp's (2011, 2014) artefact-centric activity theory (ACAT) which has been developed to capture the complexity of children's use of digital tools.

### *The Mathematical Activity of Locating*

More than 30 years ago Bishop (1988) identifed locating as one of six mathematical activities that are present in all cultures. In Norway, the curriculum known as the Framework Plan for Kindergartens (Ministry of Education, 2017) includes the learning area of quantities, spaces and shapes, which is based on Bishop's (1988) activities (Reikerås, 2008). Researchers have also found that situations involving Bishop's (1988) activities are present in children's culture (for example, Helenius et al., 2016; Johansson et al., 2014).

Bishop (1988) described locating as more or less sophisticated activities connected to navigating and communicating about the environment which are closely linked to language as language develops through taking part in these situations. Lowrie (2015) found that video games for 5-year-old children required visuospatial skills similar to those needed in real life-situations, suggesting that digital games can provide opportunities to engage children in locating activities. Although verbal language can both make locating ideas easier to spot and focus the children's attention on the mathematical aspects of a situation, children are also able to engage in the different mathematical activities without using verbal language (Flottorp, 2010; Meaney, 2016). Children's engagement with locating ideas does not have to be done through verbal language. What remains unknown is how apps might mediate the kind of support, including the development of verbal mathematical language, that teachers could provide to increase young children's opportunities to engage in locating tasks.

### *Artefact-Centric Activity Theory*

Children's engagement with multitouch screens such as tablets is complex. To deal with this complexity, Ladel and Kortenkamp (2011, 2014) developed artefactcentric activity theory (ACAT), based on Engeström's (2015) activity theory. ACAT theorises the relationship between the child, the app, the mathematics and others who participate in the activity. ACAT describes how the app mediates the processes the children engage in, rather than assessing their learning outcomes (Ladel & Kortenkamp, 2014). In this theory, the artefact (in this case the digital app) is at the centre and is affected by and affects four components: subject, group, rules and object. The model describes how the interaction between the subject (child) and the object (mathematics) is mediated through the artefact (the digital app). By using ACAT, it is possible to show how the app's design affects children's mathematical involvement and to discuss how digital apps mediate the way kindergarten teachers support the youngest children's mathematical engagement (Fig. 1).

The ACAT model consists of two triangles. In the frst triangle, the artefact (the digital app) is connected to the nodes for rules (how the app is designed) and the object (the mathematical content). The triangle describes how the mathematical object is internalized in the app and then externalized through the tablet's design. The rules (the design principles of the app) describe how the mathematical object is presented in the app (what kind of activities and feedback are available).

In the second triangle, the focus is on how the artefact (the digital app) is used by the subject (the child who is playing) and the relationship to the group (the social group around the child). The focus is on what the subject internalizes from engaging with the app, and how the subject externalizes that engagement through their words

**Fig. 1** ACAT from Ladel and Kortenkamp (2011, p. 66)

and actions. The group infuences the child's engagement because it can facilitate or hinder, or be facilitated or hindered, by the way that the app presents the mathematical content (as a basis for what happens in the frst triangle).

### **Methodology**

As part of a wider project, video recordings were made of children engaging with digital tools at a Norwegian kindergarten over a 2-month period. For this article, I analyse 20 min of video, where two one-and-a-half-year-old children, Ole and Trine, played with three digital apps with a teacher.

### *Informants and the App*

The teacher was informed about the wider project in a series of staff meetings before flming began. She gave her informed consent to be part of the project. The parents also gave written consent for their children's participation after a parents' evening was held to inform about the project. The children were given time to get to know the researcher before the video recordings began. The video recordings were carried out in the rooms the children usually used in the kindergarten, and they were always with one of their permanent teachers. There was an agreement to turn off the camera if the children seemed uncomfortable with being flmed. The two children were learning Norwegian as their frst language. In the videos, the children engage with three different apps which they chose from what the kindergarten staff made available on the tablets. These apps were: My PlayHome, an app designed to excite and captivate children (My PlayHome, 2022); Toca Kitchen, an app designed for play (TOCA BOCA, 2022); and Crocro's Friends Village, an app designed to "both entertain and encourage them [children] to learn, develop, and fourish" (Samsung, 2022). Both the children had previously tried all three apps.

### *Analysis*

Transforming the video data into written text was the frst step in the analysis. I started by transcribing verbal utterances and adding in screenshots of the actions connected to what was happening on the screen, inspired by Cowan (2014) (see Table 1). To identify how the app mediated the kindergarten teacher's support of the children's locating, I examined the transcripts from the videos carefully many times, and returned to the video if I was uncertain about something.

I then organized the analyses into two parts, each part focused on one of the triangles in ACAT. Everything related to how the children used the app to engage with

**Table 1** Transcript of datamaterial

the mathematical activity of locating (actions and vocabulary linked to locating and navigating in the virtual space of the app) was coded as part of the rules–object– artefact triangle.

Following earlier research on children engaging with apps (Christiansen & Meaney, 2020; Geist, 2012; Kvåle, 2021; Nilsen, 2018). I wanted to examine apps which did not have (mathematics) learning as a stated aim of the app. Therefore, the chosen video recordings did not include the children using apps that were described by their developers as educational.

After watching the children's engagement, I identifed locating as a mathematical object that was made available to the children through playing with the app. It may be that the teacher did not think that she was focusing on mathematics or locating when engaging with the apps with the children. However, this way of engaging in mathematics through playful participation is in line with the Norwegian framework plan which states that children should learn through play (Ministry of Education, 2017). In the video, the teacher appeared to focus on communicating about the children's navigating, and communicating about the environment, which is in line with Bishop's (1988) understanding of locating.

Everything related to how the child and the group around the child interacted with the app was coded as part of the subject–group–artefact triangle. In ACAT, the role of the teacher is not explicit, but the teacher is part of the "group" in the subjectgroup-artefact triangle. In the recordings, the children each had a tablet, but they mostly focussed on Ole's screen. Therefore, I have made Ole the subject of the analysis. Trine and the teacher are part of the social group around Ole. The choices for this coding are discussed in more detail in the fndings and discussion section.

### **Findings and Discussion**

To answer the research question, I describe how the three digital apps supported the kindergarten teacher in facilitating the youngest children's locating. I found that the pace of the apps appeared to have an impact on what kind of conversation the teacher and the children engaged in. If an app operated at a slow pace, there was an increase in opportunities for the teacher to facilitate children's engagement with aspects of locating. Lembrér and Meaney (2016) had noted a similar result with preschool children using an interactive table and a balancing app.

### *Crocro's Friends Village*

The frst triangle of ACAT focus on rules-object-artefact. In the video, I identifed aspects of locating externalised through the app as a result of its design, especially the feedback provided to the child when they interacted with the app. Ole touched the screen on the sloth's foor (see Fig. 2, left). The app provided feedback in that it zoomed into this foor of the building. Ole was then in the sloth's virtual home. It was unclear if this was a deliberate move by Ole. The reaction of the app provided Ole with the opportunity to see how his actions allowed him to move between the outside and the inside of the building. This indicates two kinds of locating; a virtual one where Ole moved inside a building from being outside, and a physical one in which moving his fnger around the screen changed what appeared on the screens. At this point, neither the teacher nor Trine made any comments verbally or with gestures about either aspect of locating.

On the next screen, there was a fshbowl in front of the sloth and fsh were fying from left to right, over the sloth's head, along with some boots (see Fig. 2, centre). Ole dragged his fnger across the screen several times where the fsh were fying, but the app did not respond. Ole seemed aware of the movement of the objects, suggesting that he had an opportunity to engage with aspects of locating such as moving from left to right. He also seemed to expect that touching the moving fsh would result in something happening. The object fying over the app appeared to fy at a pace which required visuospatial skills beyond what Ole had at this time.

After a few seconds, Ole stopped touching the screen. The teacher asked, "What will happen if we press the apple?" and pointed to the apple at the bottom of the screen. In this way, the teacher supported Ole in locating other items which may have been diffcult to see because of the many small items on the screen. Ole pressed the apple and the fshbowl was replaced with a salad bowl (Fig. 2, right), where

**Fig. 2** Screenshots from Crocro's Friends Village

different fruit and footballs appeared and disappeared. Ole tapped the fruit but again the app did not respond. Then the teacher suggested, "Maybe he wants to have it inside his stomach"? Ole dragged the fruit towards the sloth's mouth: the sloth ate it, smiled and gave the 'ok sign'. The design of the app provided feedback to Ole which supported him to see that feeding fruit to the sloth would make it respond, unlike tapping on the fruit. He then engaged with locating aspects of moving items around the virtual space. The actions of the sloth seemed to motivate Ole to continue exploring by pulling something else from the salad bowl. The screen changed back to the fshbowl. Ole went back to tapping the screen, which did not result in the sloth responding.

The second triangle focused on how the app (the artefact) was used by Ole (the subject) and his interactions with Trine and the teacher when playing with the app (the group). When the teacher suggested that "maybe he wants to have it inside his stomach" Ole was guided towards trying something new, and the teacher was able to focus on locating by asking, "Should he (the sloth) put it in his mouth?"  Trine did the same and the teacher said "Now we will be full up. In our stomachs" and rubbed her stomach. This exchange would have helped the children to understand about different parts of their own bodies, such as their mouth where food goes in, and their stomach where food ends up. Locating different parts of their bodies and knowing their names, such as stomach, are new and relevant aspects for many young children to learn.

In Crocro's Friends Village, there was a lot of quickly-moving objects. Ole did not have the fne motor skills needed to navigate the app, and consequently he did not always get feedback from the app that supported his engagement with aspects of locating, apart from watching the items move from left to right and tapping on the fsh on the screen. The teacher's communication, even if it was not explicitly about aspects of locating, appeared to support Ole to keep exploring. However, it seemed as if the somewhat fast-moving items, and a large number of small items on the screen, made it challenging for the teacher to engage in a discussion about this with Ole.

### *Toca Kitchen*