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Article

A Play-Responsive Approach to Teaching Mathematics in Preschool, with a Focus on Representations

by
Maria Lundvin
1,* and
Hanna Palmér
2
1
Department of Mathematics, Linnaeus University, 391 82 Kalmar, Sweden
2
Department of Mathematics, Linnaeus University, 351 95 Växjö, Sweden
*
Author to whom correspondence should be addressed.
Educ. Sci. 2025, 15(8), 999; https://doi.org/10.3390/educsci15080999 (registering DOI)
Submission received: 30 May 2025 / Revised: 25 June 2025 / Accepted: 29 July 2025 / Published: 5 August 2025
(This article belongs to the Special Issue Early Mathematics Education with a Focus on the Teacher and the Teaching)
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Versions Notes

Abstract

This article reports on a Swedish study investigating how children aged 2–3 years experience mathematical concepts through representations in play-responsive teaching. Drawing on the semiotic–cultural theory of learning, this study examines how representations, such as spoken language, bodily action, and artifacts, are mediated. Video-recorded teaching sessions are analyzed to identify semiotic means of objectification and semiotic nodes at which these representations converge. The analysis distinguishes between children encountering concepts expressed by others and expressing concepts themselves. The results indicate that play-responsive teaching creates varied opportunities for experiencing mathematical concepts, with distinct modes of sensuous cognition linked to whether a concept is encountered or expressed. This study underscores the role of teachers’ choices in shaping these experiences and highlights bodily action as a significant form of representation. These findings aim to inform the use of representations in teaching mathematics to the youngest children in preschool.

1. Introduction

This article reports on a Swedish study investigating play-responsive teaching in preschool, focusing on the teaching and learning of mathematical concepts. Mathematics as a field of knowledge can be described in various ways, each of which has different implications for how mathematical concepts, teaching, and learning are perceived. This study is based on a view of mathematics as a social construct, within which mathematical concepts are the results of human activity and interaction. From this perspective, learning mathematical concepts is not solely about understanding formal definitions and rules but also interacting with the concept while engaging with the surrounding world (Ernest, 2018). Although children’s desire to discover and investigate is natural, learning should not be taken for granted (Siraj-Blatchford & MacLeod-Brudenell, 1999). To promote learning, teachers have a central role in providing children with experiences where mathematics can be noticed and communicated in interaction (Björklund & Palmér, 2022; Siraj-Blatchford & MacLeod-Brudenell, 1999).
Understanding mathematical concepts is fundamental to developing mathematical competence (Niss & Højgaard, 2019; Kilpatrick, 2020). Several studies emphasize the importance of children engagement with mathematical concepts at an early age (see, for example, Palmér & Björklund, 2016; Clements & Sarama, 2014). This study is among the first to closely examine how these distinct modes of engagement—encountering versus expressing—manifest in children under the age of three within a play-responsive educational setting. According to Vygotsky (2001), it is important that young children meet and engage with mathematical concepts in meaningful contexts, in which interaction and language are fundamental components. In the study described in this article, play is explored as a meaningful context. Play-responsive teaching, as outlined by Pramling et al. (2019), is a pedagogical approach that emphasizes the interplay between play and teaching by preserving the playfulness while simultaneously directing attention toward a specific content. Teaching is situated within the “as is” (reality) and “as if” (fantasy) dimensions of play, recognizing that children shift fluidly between the real world and imagined scenarios. Play is not seen as contradictory to the goal-oriented aspects of preschool, play can be both planned and purposeful. Responsiveness is central to this pedagogical approach, meaning that teachers attentively engage with the children’s initiatives and expand on their interests without interrupting the flow of play. Rather than directing play, teachers act as co-participants who scaffold learning by introducing content in ways that are meaningful within the play context. This requires sensitivity to the children’s intentions and the ability to build upon their sense of agency.
In this study, within the context of play-responsive teaching, activities in which children can engage with mathematical concepts were developed. This article focuses on how the use of representations in these activities may influence the opportunities for children to learn mathematical concepts. Studies have shown that representations in the form of manipulatives, symbols, and pictures are significant to mathematical learning (Björklund et al., 2018; Franzén, 2015; Danish et al., 2020). However, according to Hundeland et al. (2020), there may be an unexplored difference in encountering versus expressing representations. Studies by Hundeland et al. have shown that to develop a deeper understanding of mathematical concepts, it is crucial that children are given opportunities to express their ideas and understanding. Teachers’ active presence and responsiveness to the children’s initiatives are essential to supporting such communication. Furthermore, research has demonstrated that the environment, materials, and embodied experiences each play a significant role in children’s mathematical learning (Franzén, 2015; Danish et al., 2020). Sinclair and Ferrara (2021) investigated how children relate to and experience mathematics using their entire bodies and in interactions with various types of materials. Their findings indicate that children’s bodily engagement in mathematical activities is central to their learning of mathematical concepts.
The setting for this study is a Swedish preschool, for which there is a curriculum that outlines goals and guidelines for activities. Play is emphasized as the foundation of children’s development, learning, and well-being and is thus central to preschool education. Further, the curriculum states that each child is to develop their ability to distinguish, express, explore, and use mathematical concepts and relationships between concepts (The Swedish National Agency for Education, 2018). Within this setting, this study takes its starting point in mathematics as a process shaped through interaction, exploring how the youngest preschool children (aged 2–3 years) experience mathematical concepts within the framework of play-responsive teaching. There is limited research on mathematics in preschool that specifically targets the age group of 1–3 years (Svane et al., 2023; Tirosh et al., 2020). However, according to Johansson (2011), theoretical models and activities developed for older children cannot be easily transferred to younger children, so studies on 1–3-year-olds are needed (e.g., Lökken et al., 2006). In this study, the explicit focus is on representations, with the aim of contributing knowledge about how teachers can use representations when teaching early mathematics. This study addresses the following research questions: (1) What experiences of mathematical concepts are provided in a play-responsive approach to teaching mathematics, with a focus on the use of representations? (2) How might encountering versus expressing representations of mathematical concepts influence children’s experiences of these concepts?

2. Literature Review

In general, play is highly valued in preschool; however, there are different views on how play should be understood and its connection to teaching and learning. Sutton-Smith (1997) emphasizes that play is not a univocal concept but involves a diversity of actions, objects, and forms, as well as a variation in perspectives on its function. Some perspectives highlight the potential of play to promote children’s cognitive development and serve as a foundation for teaching, while others regard play as the “property” of children as an ex-pression of enjoyment and well-being. Regardless of how play is understood, it is a multi-faceted phenomenon where imagination and hypothetical thinking are central characteristics. These features are expressed in ideas such as “as if” and “what if,” which are also found in children’s mathematical exploration. Further, Ginsburg (2006) shows that children often apply mathematical knowledge in play, thereby creating an important foundation for cognitive development. Vygotsky (2004) describes play as a creative and reconstructive process through which children process and transform their experiences. Given this, learning can be understood as a process in which children, for example, through play, are guided by the experiences of others, thereby enriching their own experiences. Play can thus be seen as a context that offers children relevance and meaning, in which mathematics can be integrated (van Oers, 2013a, 2013b).
Teaching being responsive to play does not mean that teachers should constantly participate in play or teach continuously; rather, the intention is to encourage teachers to occasionally join play for the purpose of teaching (Pramling & Wallerstedt, 2019). In relation to preschool, teaching is complex and multifaceted and can take a range of forms where a perceived opposition between play and teaching according to Clements et al. (2023) is a misconception. Preschool mathematics teaching should not be understood as a specific form or method of instruction, but rather as a practice in which different pedagogical strategies are integrated depending on context, purpose, and children’s needs. In connection to play, teaching can, for example, be conducted as guided play or as playful teacher-initiated instruction. However, a national review has shown how opportunities for guided play and playful teacher-initiated activities are missed by preschool teachers. These shortcomings are explained by teachers’ uncertainty about how teaching in pre-school can be carried out (The Swedish Schools Inspectorate, 2016).
Conceptual understanding involves recognizing relationships and structures within and between mathematical ideas, enabling children to apply knowledge flexibly across contexts (Baroody, 2003). Concepts such as quantity, number and spatial relations are fundamental not only to mathematics but also to children’s broader cognitive development (Clements & Sarama, 2014; Ginsburg et al., 2008). This is why early mathematics education (ages 1–5) should emphasize meaningful engagement with core mathematical ideas through play, exploration, and guided interactions (Clements & Sarama, 2014). In this context, meaningful engagement implies an environment that supports children in developing their understanding of concepts through language, manipulatives, and everyday problem-solving situations (Verdine et al., 2017). It also implies that teachers are intentional in designing mathematical activities that are both developmentally appropriate and conceptually focused. In these activities, the teacher’s way of responding to children is emphasized as a central aspect of teaching. In particular, the teacher’s sensitivity to the children’s understanding, ideas, and perspectives is highlighted as a key factor in early mathematics education (Björklund et al., 2018).
Thus, through communication and shared understanding with others, children develop an understanding of concepts they encounter in meaningful contexts. Understanding a mathematical concept can therefore be viewed as a process that begins with the child’s first encounter with the concept and continues until their understanding is generalized. A generalized concept involves an understanding that objects may differ (for example, three apples or three fingers) yet share common attributes that correspond to the same conceptual label (three). Such a generalization includes the insight that certain criteria must be met, while certain variations are acceptable (Vygotsky, 2001).
Mathematical concepts are abstract in their nature and need to be represented to be discerned and communicated. Duval (1995, 2006) and Lesh (Lesh, 1981; Lesh et al., 1987) both emphasize that mathematical concepts are not defined by a single idea or definition but rather form a network of interconnected representations that interact and support one another. Furthermore, representations are not merely tools for illustrating abstract mathematical concepts; rather, they serve as cognitive instruments that support children in understanding mathematics. Both Duval and Lesh emphasize that a deep conceptual understanding implies a capacity to manage and coordinate multiple representations. To this aim, Lesh presents a model with five modes of representation in mathematics education: real-world situations, pictures (both pictographic and iconic), spoken language, written symbols (both formal and informal), and manipulatives, both general and mathematical.
In early mathematics education, it is often necessary to use manipulatives in connection to verbal and symbolic representations (see, for example, Björklund & Palmér, 2022). Severina and Meaney (2020) emphasize that to understand children’s mathematical thinking, teachers need to attend to a variety of modes of expression beyond verbal language. This as mathematical reasoning of young children is often conveyed through an interplay of bodily gestures, material actions, and verbal utterances, highlighting the importance of adopting a multimodal perspective in teaching. According to Norén (2018), manipulatives can support the establishment of a shared frame of reference, which in turn facilitates communication between children and teachers, especially in contexts where language barriers may exist. However, Norén emphasizes that the use of manipulatives is not only beneficial for children with limited linguistic resources but also serves as a strategy for enhancing mathematical understanding in all children. In a review on spatial learning in early mathematics, McCluskey et al. (2023) identified four central themes. First, the manipulation and transformation of objects, emphasizing how children’s physical interaction with objects helps them to understand and modify spatial relationships. Second, bodily engagement, emphasizing that physical activity and movement through various environments contribute to the development of spatial abilities. Third, the representation and interpretation of spatial experiences, where children use bodily expressions and gestures to communicate and make sense of their spatial understanding. The fourth theme addressed contexts for spatial learning, pointing out that the environments and activities children engage in play a significant role in shaping their spatial skill development.
In line with McCluskey et al. (2023), embodied cognition refers to processes of thinking, knowing, and communicating that are rooted in bodily experience. Studies on embodied cognition are based on the premise that understanding is enhanced when bodily experiences are engaged (Kersting et al., 2021). de Freitas and Sinclair (2014) stress that embodied cognition not only involves individual bodily and sensory experiences but is also deeply embedded in social and cultural contexts. Hutchins (1995) introduces sociocultural dimensions to embodied cognition, emphasizing that human cognition is closely intertwined with both physical and social environments. He argues that cognition does not occur solely within the mind but is a process involving interaction between people, artifacts, and their surroundings. Embodiment, in this context, underscores how bodily experiences, gestures, and physical interactions contribute to meaning-making and understanding, especially in collaborative and educational situations.
Kersting et al. (2021) highlight that when children explore and learn (mathematical) concepts through hands-on activities, such as manipulating objects or performing actions, their bodies become an active part of the cognitive process. Thus, bodily interaction with materials and environments supports learning and understanding, while social interaction plays a crucial role in constructing meaning. Smith (2018) compared teaching approaches based on bodily activities versus non-bodily activities. The results indicate that while both groups acquired knowledge, the children who engaged in embodied experiences developed a significantly deeper understanding. Smith links bodily activity to the opportunity for self-referencing through a first-person perspective. In a first-person perspective, objects are related to the self, as opposed to a third-person perspective, in which objects are related only to each other. According to Smith, the majority of traditional mathematics instruction unfortunately promotes only a third-person perspective. According to Smith and Walkington (2019), students’ bodily actions should correspond directly to the mathematical concept being taught. This could involve using the body to represent a geometric figure or performing actions that reveal mathematical properties or relationships. Moreover, the body’s capacity for continuous movement and transformation should be utilized, for example, by using one’s arms to demonstrate length.

3. Theoretical Framework: Semiotic–Cultural Theory of Learning

The semiotic–cultural theory of learning (Radford, 2003, 2009) offers a socio-cultural and multimodal perspective on mathematical thinking and learning. A fundamental premise of this theory is that thinking, emotions, actions, and our relationship to the world cannot be understood as separate components but rather as interwoven elements of a unified whole. In this theory, meaning-making is conceptualized as a process in which children gradually construct an understanding of mathematical ideas and concepts through the use of various semiotic resources in interaction with others. Knowledge emerges through participating in practices in which signs, symbols, language, gestures, and artifacts function as semiotic means of objectification: tools through which learners become aware of, engage with, and express mathematical ideas.
A distinctive feature of this theory is the emphasis placed on bodily actions (including gestures) as a central aspect of participating in a mathematical activity. Through the concept of semiotic means of objectification (Radford, 2003), the theory clarifies how semiotic resources serve as means of expression through which mathematical ideas and concepts can be articulated and communicated. However, even though the theory highlights spoken language, images, symbols, bodily actions (including gestures), and artifacts, among these factors, spoken language is afforded a special status due to its unique capacity to shape scientific thinking, as originally noted by Vygotsky (2001). However, it is the interplay of multiple expressive forms that renders the learning process a meaningful whole.
A key concept in this theory is the semiotic node, a moment in the learning process at which different semiotic systems, for example, artifacts, bodily actions, and verbal expressions, converge to form a coherent mathematical understanding. These semiotic nodes are considered critical points in the process of objectification, that is, when a mathematical idea or concept becomes mentally accessible, thus representing key conditions for learning mathematical concepts (Radford, 2009).
The semiotic–cultural theory of learning enables detailed analysis of how mathematical meanings are co-constructed in interaction. In the study presented in this article, the theory is used to examine how preschool teachers and children explore and communicate about mathematical concepts within the context of play-responsive teaching. Attention is given to how semiotic means of objectification are mobilized and how semiotic nodes emerge. Although this theory was originally developed based on empirical studies conducted in high school contexts, in the present study, it is applied to mathematics education for young children in preschool. Accordingly, the term child is used instead of student.

4. Method and Materials

This study draws on empirical data from an intervention study focused on a play-responsive approach to teaching mathematics in preschool. Over the course of one academic year, the first author of this article collaborated with two teams of teachers at one preschool in an iterative process including the planning, implementation, and evaluation of play-responsive teaching focused on mathematical concepts. As previously noted, the focus was on the youngest children in preschool, those aged 2–3 years. Teaching was provided by the teachers as part of their regular pedagogical practice. Each new iteration was based on a joint evaluation of the preceding instructional activities. Examples of the mathematical concepts (chosen by the teachers) introduced via play-responsive teaching were the circle, over–under, high–low, and symmetry.
Teachers and guardians received both written and oral information about the study, after which informed consent was obtained (Swedish Research Council, 2024). In addition, the children’s participation was affirmed by confirming signals of willingness to participate in the planned activities. In this article, all names have been anonymized, and pseudonyms are used throughout the text. The study was reviewed and approved by the Swedish Ethical Review Authority [Protocol number: 2022-03781-01].
The teaching process was documented using video recordings to capture both verbal and non-verbal communication (Derry et al., 2010). Seven design cycles were implemented, resulting in a total of fourteen documented teaching sessions lasting between 7 and 30 min. Six of these teaching episodes were selected for analysis in this study. These specific episodes were chosen because they included interactions in which temporary intersubjectivity was established between the teachers and children (Lundvin, in press). Intersubjectivity implies a shared understanding and mutual awareness between individuals engaged in communication. It plays a crucial role in meaning-making and social interaction, enabling participants to co-construct reality (Trevarthen, 1998). Thus, the analyzed data represent episodes in which opportunities for learning were theoretically present.

Analysis

The six selected teaching episodes were analyzed through a four-step content analysis (Hsieh & Shannon, 2005), using a deductive approach based on the theoretical framework of the semiotic–cultural theory of learning (Radford, 2003, 2009).
In the first step, episodes were identified according to Radford’s notion of semiotic means of objectification (see Table 1). Following this theoretical framework, one episode could be categorized into multiple categories, as they often involved several semiotic means of objectification. This analysis revealed that the occurring semiotic means of objectification in the data were spoken language, bodily actions, and artifacts.
In the second step, semiotic nodes (Radford, 2009) were identified based on the episodes categorized under multiple semiotic means in the first step. A total of 42 semiotic nodes were identified, each involving spoken language combined with bodily actions, and/or artifacts (no nodes without spoken language were identified). No semiotic nodes including symbols or images were identified. This analysis focused on the first research question: What experiences of mathematical concepts are provided in a play-responsive approach to teaching mathematics with a focus on the use of representations?
In the third step, the analysis focused on whether the children in these episodes encountered and/or expressed mathematical concepts, focusing on one child at a time. This step involved categorizing the episodes from step one into subcategories based on whether the child in question encountered or expressed concepts. Encountering a concept referred to episodes in which another individual (child or teacher) expressed the concept in some representational form. Expressing a concept referred to episodes in which the child themself expressed the concept in some form.
The fourth and final step compared opportunities for sensuous cognition between instances in which children encountered versus expressed mathematical concepts through spoken language, bodily action, or artifacts. This analysis focused on the second research question: How may encountering versus expressing representations of mathematical concepts influence children’s experiences of these concepts?

5. Results: Experiences of Mathematical Concepts

The results are based on the 42 semiotic nodes identified in the analysis: episodes in which the children experienced mathematical concepts through spoken language and bodily action (less observed), and through spoken language, artifact, and bodily action (more observed). This first section focuses on the first research question, that is, what experiences of mathematical concepts are provided in a play-responsive approach to teaching mathematics with a focus on the use of representations. The episodes presented are two examples of the identified semiotic nodes and were therefore not the basis of the entire analysis. In the section below, the semiotic nodes are analyzed based on the children’s opportunities to experience mathematical concepts through sensuous cognition (Radford, 2009).

5.1. Semiotic Node with Spoken Language and Bodily Action

This episode is an example of spoken language and bodily action and thus illustrates sensuous cognition through perceiving and seeing bodily actions and hearing and saying spoken language (Table 2). The spatial concepts focused on are high and low and up and down. Play-responsive teaching is based on interest that the children previously demonstrated in the moon. In one of the preschool rooms, a setting is staged with a spacecraft (made of cardboard) in which the children and teacher gather together (as if) to travel into space. The moon is projected onto the wall (Figure 1), and the children are given the opportunity to reach it using a ladder. One teacher and four children participate in the episode.
Table 2. Sensuous cognition in spoken language and bodily action.
Table 2. Sensuous cognition in spoken language and bodily action.
TurnWhoSayingDoing
1GeorgeCan you wave me?Standing at the top of the ladder, waving down to the others
2TeacherYes, I’m waving.Waving up at George
3GeorgeI’m way up high. Can’t reach me?
4TeacherNo, I can’t reach you!Stretching an arm towards George
5TeacherHey, everyone! George says this: “You can’t reach George.” Can you reach George?Turns towards the other children
6TeacherI can’t reach George!
7RicoBut I reach George!Stretches out and touches George’s foot
8OlaI reach George!Stretches out and touches George’s foot
9TeacherBut you can’t reach George’s hand, only his foot. Right?
10GeorgeNo, not my neck either.Pats his head
11Rico and Ola Rico and Ola look up at George
12TeacherNo, not your head either.
In the episode, the children are given the opportunity to experience the concept of “high” through bodily actions and verbal language. In turns 1–2, George perceives that it is possible to wave when positioned high up on the ladder (Figure 1). His questions (turns 1 and 3) may indicate that he is focused on the consequences of being up high and what he can do and not do while being up high. While he bodily perceives the concept of high, he verbally communicates his position to the others in the room (turn 3).
By repeating George’s utterance (turn 5), the teacher makes it possible for Rico and Ola to engage in the episode, which enables further interaction and allows more children to experience the mathematical concept. Rico and Ola express their experience of high verbally (they say) in terms of “reaching,” in parallel with a bodily action, stretching their bodies (turns 7–8). When the teacher and George add the idea of reaching George’s hand or head (turns 9–10), it provides an opportunity to perceive the concept of high through the experience of being unable to reach him (turn 11).
The bodily actions observed in the episode, in combination with the social context, can be viewed as a foundation for the spoken language used, as the children’s discussion is based on their bodily positions. Even though the concept of high is not expressed through any artifact, the ladder is a crucial condition for perceiving the bodily actions.

5.2. Semiotic Node with Spoken Language, Artifact, and Bodily Action

The second episode includes spoken language, artifacts, and bodily action, and illustrates sensuous cognition through seeing and feeling artifacts, seeing and expressing bodily actions, and hearing and producing spoken language.
In this episode, the mathematical concept of the circle is communicated. The play-responsive teaching approach builds on interest the children previously demonstrated in building houses with blocks. The aim of the activity is for the group to collaboratively construct a circular house using blocks. A circular wooden platform on the floor serves as the foundation for the house, around which the children and the teacher gather. One teacher and four children participate in the episode (Table 3).
Table 3. Sensuous cognition in spoken language, artifacts, and bodily actions.
Table 3. Sensuous cognition in spoken language, artifacts, and bodily actions.
TurnWhoSayingDoing
1TeacherWhat is this? What is this? What kind of shape is this?Moves hand around the wooden platform repeatedly
2Ville, Ronja, and Sara Look at the wooden platform and how the teacher moves her hand around it
3SaraWhat shape is this?Moves her hand around the wooden platform just like the teacher did
4VilleWhat is it?Places his hand on the wooden platform
5Ronja Places her hand on the wooden platform and moves it along
6TeacherIt’s a circle… It’s roundMoves hand along the wooden platform repeatedly
7SaraA circle!
In this episode, the children are given the opportunity to experience the mathematical concept of a “circle” through artifacts, bodily actions, and spoken language. The teacher initiates a dialog by posing an open question (using spoken language) about the shape, while simultaneously offering a bodily action by moving her hand around the wooden platform (turn 1). The coordination of seeing a bodily action, alongside the opportunity to hear spoken language and see the shape in an artifact, appears to capture the children’s attention, as they all direct their attention toward the wooden platform and the teacher’s bodily action (Figure 2).
Sara imitates the teacher’s spoken language and bodily action (turns 3 and 7), thereby providing Ville and Ronja with yet another opportunity to see a bodily action and hear spoken language. Sara’s imitation suggests that she is exploring the teacher’s way of investigating and naming the shape. Ville’s opportunity to see the artifact and bodily action (turn 2), as well as hear spoken language and feel the artifact (turn 4), does not reveal whether the experience is related to the concept of the “circle” or even to the shape of the wooden platform. The experience (turns 2 and 4) may instead focus on the functionality of the wooden platform and the artifact’s properties and thus be entirely non-mathematical. Ronja has a similar experience (turns 2 and 5), as further teacher interaction is needed before it is clear what she is directing her attention to.

5.3. Opportunities for Encountering and Expressing

This section focuses on the second research question, that is: How may encountering versus expressing representations of mathematical concepts influence children’s experiences of these concepts? Similarly to the above section, the episodes presented are examples and thus not the basis of the entire analysis. In the following section, the episodes are analyzed based on the children’s opportunities to experience mathematical concepts through sensuous cognition by encountering and expressing concepts through spoken language, bodily actions, and artifacts (Radford, 2009).

5.3.1. Spoken Language

Spoken language is present in all semiotic nodes. However, the analysis reveals that the extent to which mathematical concepts are encountered and expressed influences the nature of the experiences made possible. The analysis shows that the children are given the opportunity to encounter spoken language (through the teacher speaking) in nearly all episodes analyzed, thereby enabling sensuous cognition through hearing about concepts. One such example is provided in Table 4.
Table 4. Encountering concept through spoken language.
Table 4. Encountering concept through spoken language.
TurnWhoSayingDoing
1Noah Picks up a cylinder from the block basket and holds it up to the teacher
2TeacherWell, look at that! There’s a circle! It’s perfectly round.Traces a circle in the air
Children expressing concepts via spoken language themselves occurs less often. In almost all of the episodes in which the children express concepts, it is based on the teacher’s initiative, for example, in Table 5. The questions posed by the teacher create opportunities for the children to express the concept “down” in spoken language.
Table 5. Expressing concept through spoken language.
Table 5. Expressing concept through spoken language.
TurnWhoSayingDoing
1TeacherHey, my friends! Lisa is up by the moon, but where are we?Looks up at Lisa on the ladder, then turns to the other children in the room
2AylinDown here! Down here!

5.3.2. Bodily Action

Encountering bodily action (as expressed by a teacher or peer) enables sensuous cognition through seeing concepts, while expressing concepts through bodily action enables sensuous cognition via perception. The example below illustrates opportunities for both encountering and expressing bodily action (Table 6). In this example, the spatial concept presented the intended mathematical content, and play-responsive teaching is grounded in the children’s ongoing play, which mimics driving a bus.
Table 6. Encountering and express concept through bodily action.
Table 6. Encountering and express concept through bodily action.
TurnWhoSayingDoing
1Gina and Ralf Argue about who gets to sit in the front and drive the bus. Both try to place their own chair in front of the other’s
2TeacherYou know what! You can sit besides; there can be two bus drivers. You can sit besides, besides.
3GinaNooo!
4RalfNooo!Tries to move Gina’s chair, which is in front of his own
5TeacherOh dear, oh dear! How is this going to work? Who’s going to be the bus driver? Can’t we have two bus drivers? You can sit beside!Shows by patting the floor beside her; Ralf and Gina look
6Gina Looks at the teacher’s gesture. Takes her chair and places it next to Ralf’s
7TeacherThere! Now we have two bus drivers.
8RalfSitting in front!?
9TeacherYou’re both sitting in front!
10Gina and Ralf Pretend to drive the bus
In this episode, the children are given the opportunity to experience the concept “beside” through perceiving and seeing bodily action. They first encounter the concept verbally when the teacher says it, and then, they express reluctance to sit beside someone (turns 2–4). When given the opportunity to see the concept in bodily action (turn 5), Gina changes her mind and expresses it by sitting beside Ralf (turn 6). Ralf ensures he is still sitting in front, despite Gina being beside him, which the teacher confirms (turns 8–9).

5.3.3. Artifact

Encountering artifacts, when they are used by a teacher or another child, enables sensuous cognition via concepts observed in the artifact. When children express concepts using artifacts, sensuous cognition is enabled through feeling or making concepts. “Feeling” or “making” refers to limitations and affordances when expressing concepts through artifacts. For example, geometric shapes can be felt in artifacts, while spatial concepts can be enacted using them, such as in the following episode (Table 7).
Table 7. Expressing concepts through artifacts.
Table 7. Expressing concepts through artifacts.
TurnWhoSayingDoing
1TeacherAstrid, can you help me by putting this under the table?Hands a bottle to Astrid
2Astrid Places the bottle under the table
When Astrid is asked for help, she is given the opportunity to perceive the concept “under” through the artifact. The teacher’s question is more of a prompt than a genuine question. However, the question enables expressing the specific concept through an artifact and thus contributes to Astrid being assigned the role of actor in the activity. Astrid does not respond verbally but with action. Her action, placing the bottle under the table, constitutes a non-verbal response showing how she interprets the concept.
The final episode described below illustrates a case in which a child encounters an artifact by seeing the concept (Table 8).
Table 8. Encountering a concept through an artifact.
Table 8. Encountering a concept through an artifact.
TurnWhoSayingDoing
1TeacherNow we’ve made a circle stove with 1, 2, 3, 4 burners, which are also circles. Perfectly round.Points to each wooden disc (acting as a burner) while counting, the children watch.
In this episode, the teacher is the most active participant while the children have the role of observers. The children are given opportunities to encounter the concept of the “circle” through an artifact, that is, the teacher pointing out the wooden disk. None of the children take initiative to touch the artifact and no prompting is made by the teacher, thereby limiting the possibility of experiencing through encountering the artifact.

6. Discussion

The purpose of this study was to contribute knowledge about how teachers can use representations in early mathematics teaching. The results present opportunities for experiencing mathematical concepts via a play-responsive approach to teaching mathematics. It should be noted that neither Radford (2009) nor the present study seek to evaluate different forms of sensuous cognition. Rather, Radford highlights semiotic nodes as critical in the process of learning mathematical concepts. Based on this study’s results, it is essential that teachers providing a play-responsive approach to mathematics education consciously consider what types of experiences are enabled through encountering and/or expressing concepts, as these lead to markedly different learning opportunities. The findings indicate that whether children encounter or express representations of concepts significantly influences their experiences of those concepts. Encountering refers to a concept being expressed by someone else (a teacher or peer) through some representational form, whereas expressing refers to when the child themself articulates the concept in some form.
The distinction between encountering and expressing concepts can be seen as a nuanced aspect of how teaching enables conceptual experience. Previous studies have emphasized the importance of allowing children to contribute with ideas and express their personal understanding (Hundeland et al., 2020; Björklund et al., 2018), which presupposes that teaching offers opportunities for the expression of concepts.
While studies have shown the value of bodily actions in conceptual learning (Kersting et al., 2021; Smith, 2018; Smith & Walkington, 2019), research in mathematics education has primarily emphasized the use of manipulatives, symbols, and pictures (Björklund et al., 2018; Franzén, 2015; Danish et al., 2020), often excluding embodiment and focusing on concept expression (Lesh, 1981; Duval, 2006). The findings of this study align with Smith (2018), who stresses the importance of a first-person perspective in which objects relate to the child, rather than a third-person perspective in which objects relate only to each other. This supports the idea that embodiment should be recognized as a central form of representation in mathematics education. In line with Radford (2003), representation is not only a way to express mathematical ideas and concepts but also contributes to the process of conceptual understanding.
Previous research has pointed out that every representational form has both strengths and limitations (Duval, 2006; Lesh, 1981). This became evident in this study where different semiotic means of objectification had limitations and affordances in connection to the type of concepts (for example geometric shapes vs. spatial concepts). Verbal language appeared in all episodes analyzed. This is supported by Radford (2009), who, drawing on Vygotsky, emphasizes the significance of language. However, the results show that children were more often afforded opportunities to hear mathematical concepts than to express them verbally themselves. This may reflect an imbalance in experiential opportunities, where the active use of language for expression is emphasized less. As demonstrated in this study, children’s opportunities to express concepts are preceded by an action by the teacher, such as asking a question, preparing the pedagogical environment, or repeating a child’s utterance to invite further participation. In this way, the teacher’s actions are an important part of enabling the children to express concepts.

7. Conclusions

This study demonstrates that a play-responsive approach to teaching mathematics can facilitate children’s experiences of mathematical concepts, understood as semiotic nodes, through spoken language, artifacts, and/or bodily action. The distinction between encountering and expressing concepts is shown to influence the nature of the experience afforded.
Spoken language enables conceptual experience via hearing when concepts are encountered and via speaking when concepts are expressed. Bodily action allows for experience through seeing when encountering and through perceiving when expressing. Similarly, artifacts afford conceptual experience through seeing when encountering concepts and through feeling or making when expressing them.
A key contribution of this study is how the form of representation, and whether it is encountered or expressed, influences sensuous cognition. This adds knowledge to our understanding of early mathematical thinking as a process that is multimodal, embodied, and socially mediated (Radford, 2003; Kersting et al., 2021; de Freitas & Sinclair, 2014). It also emphasizes the critical role of teacher responsiveness in affording varied representational experiences in teaching. The following becomes of particular interest in play-responsive teaching (Pramling et al., 2019) where the responsiveness of the teacher is a central part.
It is important to note that the current analysis is limited to observable opportunities for experience in children’s actions. Determining whether actual learning occurs lies beyond the scope of this study and would require different empirical approaches. Nevertheless, the findings give implications for both theory and practice, particularly in how early mathematics education with the youngest preschoolers can be designed for children to explore mathematical concepts though different modes of representations, and on how encountering and expressing concepts, respectively, influences the nature of the experiences afforded.

Author Contributions

Conceptualization, M.L. and H.P.; Methodology, M.L.; Formal analysis, M.L.; Data curation, M.L.; Writing—original draft, M.L.; Writing—review & editing, H.P.; Supervision, H.P. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by a grant from the Swedish Research Council (no. 2019–03786) to the national research school of Play-Responsive Early Childhood Education and Care for Social and Cultural Sustainability (PRECEC_SCS).

Institutional Review Board Statement

The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board of the Swedish Ethical Review Authority (protocol code: 2022-03781-01 and date of approval: 2022-06-01).

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Education 15 00999 g001
Figure 1. The moon being projected onto the wall.
Figure 1. The moon being projected onto the wall.
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Figure 2. Semiotic node with artifacts, bodily actions, and spoken language.
Figure 2. Semiotic node with artifacts, bodily actions, and spoken language.
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Table 1. Semiotic means of objectification (Radford, 2003) and how sensuous cognition (Radford, 2009) was operationalized in the study for sorting and analyzing episodes.
Table 1. Semiotic means of objectification (Radford, 2003) and how sensuous cognition (Radford, 2009) was operationalized in the study for sorting and analyzing episodes.
Semiotic Means of Objectification Sensuous Cognition
Spoken languageHear and/or say concepts in spoken language
Bodily action (incl. gestures)See and/or express concepts in bodily actions
ArtifactSee and/or feel/make concepts in artifacts
SymbolSee and/or write concepts with symbols
ImageSee and/or draw concepts with images
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Lundvin, M.; Palmér, H. A Play-Responsive Approach to Teaching Mathematics in Preschool, with a Focus on Representations. Educ. Sci. 2025, 15, 999. https://doi.org/10.3390/educsci15080999

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Lundvin M, Palmér H. A Play-Responsive Approach to Teaching Mathematics in Preschool, with a Focus on Representations. Education Sciences. 2025; 15(8):999. https://doi.org/10.3390/educsci15080999

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Lundvin, Maria, and Hanna Palmér. 2025. "A Play-Responsive Approach to Teaching Mathematics in Preschool, with a Focus on Representations" Education Sciences 15, no. 8: 999. https://doi.org/10.3390/educsci15080999

APA Style

Lundvin, M., & Palmér, H. (2025). A Play-Responsive Approach to Teaching Mathematics in Preschool, with a Focus on Representations. Education Sciences, 15(8), 999. https://doi.org/10.3390/educsci15080999

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