Keywords

Introduction

In this paper, we elaborate on play orientation in mathematics teaching, an issue originating from curricula and policy guidelines stating that play should be an integrated part of the education for six-year-olds in Sweden (“preschool class”, a mandatory form of education the year before Grade 1). When the preschool class was introduced as a school form in 1998, it was said to integrate the best from the play-oriented preschool practice and the knowledge-oriented primary school (Govt Bill 1997/98:6, 2009/10:165). The preschool class has its own section in the national curriculum (Swedish National Agency for Education, 2018) with goals referred to as knowledge to strive for. As mentioned above, play should be an integrated part of the practice in preschool class at the same time as, for example, mathematical development is to be given a salient position. In an inquiry about ten-year compulsory school in Sweden (SOU, 2021:33), it is repeatedly emphasized that play is an important tool for exploring and understanding the surrounding world, which implies that a play-oriented and explorative approach should be an essential part of education for six-year-olds. However, what “play-oriented teaching” means in terms of teaching practices and the contextualization of learning content is not elaborated on in the preschool class education policy documents. This is also reflected in an evaluation of the preschool class practices, which revealed that child-initiated play, or “free play”, is the most common way of including play in the education. However, it is also pointed out that this kind of play most often lacks support from teachers and thereby includes no guided direction toward national curriculum goals (Swedish Schools Inspectorate, 2015).

Based on the above-described expectations and shortcomings in implementing play as an aspect of early childhood education for Swedish six-year-olds, there seems to be a need for a deeper inquiry into how play appears in the teaching practice. As a basis for intervention programs and further research on early mathematics education, here we present an observation study of mathematics teaching in preschool classes in Sweden, aiming to give an overview of how play, and in particular the notion playfulness, and mathematics as a content for learning are expressed in teaching activities. The specific research question to be answered is thereby how playfulness and mathematics as a content for learning appear in mathematics teaching about numbers with six-year-olds.

Making Sense and Making Meaning

Recent educational research (see Pramling et al., 2019) considers learning in the context of play to be instances in which the meaning of objects and actions may (and usually does) change in the creation of imaginary situations. It is conjectured that this is where new ways of understanding emerge, because the children move between the actual and imaginary experiences, “as if” something were in a certain way and “as is”, which means the way objects or occurrences are actually perceived in the situation. This is an iterative process that keeps the children’s play activity going and thus constitutes the motor for meaningful learning. Van Oers (2010) describes mathematical development as an emerging process in the context of children’s own activities in which actions and operations with numbers can be carried out, for example in the context of children’s play that makes sense to these children.

It is often stated that mathematics education should take its starting point in the lives, experiences, and needs of the learner (see Freudenthal, 1968), to which symbols may be introduced to shape and manipulate the experiences such that a problem is made possible to solve (thereby making mathematics “useful” to the learner). This process, known as “mathematization”, reduces the noise that the experiences in the real world induce. A precondition, however, is that the problem to be solved must be found in a context that is relevant to the learner and that mathematics is needed in order to solve the problem (Freudenthal, 1981). By this, one can draw the conclusion that the content to be learned should be considered meaningful in terms of practical use to the learner, as a tool for solving problems that the learner encounters. Following Leont’ev (1978), however, there are two dimensions of “meaning”. The first is the cultural meaning that is found in actions, objects, goals, or tools as well as the actions that are culturally attributed to them, a sort of standard way of understanding, which is mediated through, for example, books or by others in the same community. The second dimension is the personal meaning, or rather “sense”, that relates to the personal values that someone attributes to these objects, goals, tools, or actions. Thus, to aim for meaningful learning, teaching should include both a cultural and a personal dimension of meaning at the same time. That is, the cultural dimension relates to providing relevant cultural tools to the learner while the personal dimension relates to the involvement of the learner in practices that make sense to him/her.

Play Orientation in Early Childhood Education

Considering young learners, many scholars describe play as a context for putting abstract or academic concepts into a familiar setting in attempts to illustrate their culturally developed meaning, or how to use the concept in a way that is accepted by the main community. However, play orientation in educational settings could also be seen as “both culturally framed and unframed activities that are subsumed under the umbrella of ‘playfulness’” (Roopnarine, 2011, p. 20). In this way, play – or rather playfulness – is more of an approach that goes beyond certain activities. Whether a situation is regarded as “playful” in this sense depends on how those involved in an activity experience a joint focus of attention, goal, and boundaries (frames of the activity, in Roopnarine’s words) for what is possible to do in the situation. The playful approach thereby makes all participants agents in how the activity develops, regardless of whether the activity is scheduled, preplanned, or spontaneous.

Theories emanating from Vygotsky maintain that until the age of seven children learn to their best potential when the learning is embedded in playful activities (El’konin, 1999). Nevertheless, there seems to be a debate concerning how to conduct play-oriented teaching. Burghardt (2011) even experiences that the label “play” might better be avoided in trying to integrate playful activities into school curricula. On the other hand, there are research-based education programs aiming to further the idea of play as young children’s primary activity for learning. One such program is Developmental Education (Van Oers, 2014), which takes its starting point in the notion that any activity can be interpreted as more or less playfully formatted. This implies that play is not considered a separate kind of activity but rather part of a continuum, in line with how Roopnarine (2011) conceptualizes play (see quotation above). In accordance with this, the relationship between play and mathematics teaching can be seen both as “mathematics made playful”, for instance through games in which counting and mathematical operations are used, and as “mathematizing elements of play” in which the primary act is play and the teacher actively introduces mathematical concepts or operations to the playing child. Regardless of whether a play activity is framed in a context of manipulative or role play, it is usually characterized by children having a high degree of freedom in how they carry out the rule-governed activity. In such a context, the children can encounter tasks that are solved with tools that may look mathematics-like, but can also often be solved through intuition. It is then the actions of the teacher and how s/he articulates mathematical relationships that extend the children’s encounters with mathematics and that may induce learning. This means that children’s initiatives and explorations are important, but it is not enough that mathematical representations and concepts are present in a play activity – teachers must also provide new content and altering perspectives that extend the children’s experiences (Van Oers, 2010).

To support mathematical learning, it is not enough to merely confirm what the child him/herself initiates, as this will not contribute to extending the child’s knowledge or skills. Offering strategies for completing a task or a different perspective on how to interpret a problem is more a goal-oriented act that may direct the child’s attention toward skills or tools that help him/her complete a task in a more advanced way. The teacher’s way of responding to children’s initiatives and possible mathematical content in play activities thereby leads to different learning opportunities. Particularly, maintaining a shared focus and handling the balance between foregrounding play or the content for learning is a core issue and is not easily operationalized in educational settings (Björklund et al., 2018). In reality, teaching most likely moves across this continuum, sometimes starting from problems initiated by the children and sometimes from curricular goals determining what mathematics they are expected to learn about, but most often moving in between these. When this occurs, the two dimensions of meaning – the cultural meaning and the personal sense – are likely to be connected (Leont’ev, 1978).

The Study

In this paper, we present an analysis of teaching in Swedish preschool classes. It is not our ambition to classify teaching activities as play or not play; rather, we focus on the term playfulness as it comes through in the teaching of numbers. While playfulness does not have a clear-cut definition in the literature, we nevertheless understand the notion as shared attention and responsiveness to the other’s (the child’s) perspective and experiences, including imaginary creations, and particularly an openness in the direction of the activity whereby any participant may introduce alternatives and renegotiate the rules of a game, task, or play activity (Pramling et al., 2019; Roopnarine, 2011).

The data for analysis consists of fieldnotes and documentation in a protocol (originally developed by Venkat & Askew, 2018) focusing on teachers’ talk, gestures, use of artefacts, and notations in bringing forth numbers as the object of learning. Researchers (including the authors of this paper) made observations of teaching that the teachers themselves considered to be about numbers, the features of numbers, and how to make use of numbers in problem-solving. The data was collected during fall 2021. From the large data set, 81 episodes (from 46 individual teachers), observed and documented by the two authors of this paper, were used for analysis.

The object of analysis in this study is mathematics teaching about numbers, conducted in Swedish preschool class. Two features of this phenomenon are of specific interest: expressions of playfulness that come through in the mathematics teaching, and how mathematics appears as a content for learning. In this sense the analysis is phenomenological, aiming to reduce, describe, and search for essence in the observed phenomenon (Giorgi, 1997). Thus, the analysis started with identifying and describing the context in which the mathematics is being taught, as it appeared to the participants of the activity based on how the mathematics was framed as well as the teachers’ and students’ actions, freedom to pose suggestions and alternatives, and the nature of their engagement. Several qualitative differences emerged, in which similarities and differences in the intentions of the contexts shaped in the teaching activity came through. In this identification process, it was also possible to recognize a pattern in how the mathematical content appeared. In the following process, we condensed the contexts and connected meanings of the mathematical content into thick descriptions that bear a common meaning. Playfulness being the main object or phenomenon of inquiry is the guiding notion in these descriptions. The result of the analysis is thereby a condensed description of playfulness as it appeared in the observed lessons, and of how the content for learning was made discernable for the learners.

Results

In the Results section we present a synthesis of playfulness and the mathematical content for learning as it appeared during the teaching activities in the observed preschool classes. Three different appearances of how mathematics appears as content for learning in regard to the playfulness expressed in the teaching situation can be found in the data set. The observations are not exclusively classified to one or another as there are overlap** observations, but in this presentation, we describe the characteristic pattern of contextualization that was found in the data.

The Mathematical Content as the Primary Target

Firstly, a good many observations have the common characteristic of “completing a task”, in which the mathematical content appears as the primary target. The tasks are not contextualized based in the children’s experiences or lives. If manipulatives are used for visualizing number relations and operations, it seems irrelevant what kind of objects are used. The mathematical content constitutes the task, and is thereby the center of attention. There is rarely any playfulness observed, such as negotiating about rules or offering imaginary suggestions, and when there is some sense of playfulness this occurs when the teacher is building up tension in activities in which the outcome is not known and the children are involved in making guesses. The following example describes a teaching activity in which the mathematical content appears as the primary task:

The teacher hands out cards with numbers on them (1-20) and asks the children to arrange them in order from smallest to largest. The children take on the task and try to place the cards in the right order. During the activity the teacher supports the children, asking “Do you remember, 17, what number comes before?”. Child: “18”. Teacher: “What number is before?”. Child: “16”. Teacher: “Then this is your place” (points at a position on the number track). When all the numbers have been placed on the number track, teacher and children check together that the numbers have been correctly ordered by counting out loud together while the teacher points to one number at a time.

The goal for this group activity is clearly stated and does not allow for alternative solutions. The children are expected to participate in a certain way, and a specific content is in focus. There is no room for spontaneity and the focus is on completing the task, i.e., placing the number cards in the correct order.

Exploring the Mathematical Content in a Relevant Context

Secondly, playfulness is observed in cases in which teacher and children explore the mathematical content in a relevant context; that is, numbers are a central feature of the activity and exploring mathematical possibilities is necessary for completing the task. Here, the children are involved in activities in which they interact and solve tasks together with their peers and the teacher. Typically, the teacher directs the activity and asks questions like “Why…?” and “How come…?”. Mathematics is used as a tool for understanding the outcome of some investigation or suggestion, to help structure the children’s experiences. In these observations we can see that a common approach in the teaching takes its starting point in the children’s experiences and the teacher extends these by pointing out surprising results and hidden questions, and making them objects of inquiry. For instance, the teacher may introduce “conflicts” to highlight issues that can only be resolved through mathematical reasoning. In such cases, the goal for the activity is predetermined and known by the teacher, but the exact direction to take within the activity in order to arrive at the goal is not determined beforehand. This can be seen as a criterion for playfulness, as the approach allows for alternative routes and exploration. By structuring the mathematical content, the children make sense of their experiences and take part in the mediated cultural meaning in the process. Furthermore, this way of teaching mathematics, with playfulness and kee** content foregrounded, is based on either the children’s own lived experiences or a collectively created context shaped in the ongoing situation. Both seem to function as facilitators for engaging the children in the activity and connecting cultural and personal mathematical meaning in joint exploration. In the following example, the teacher and the children examine what fruits the children have brought to school as a starting point for a mathematically informed exploration:

Teacher: “Let’s try to find out which fruit is the most common one today!”. The teacher makes a horizontal axis on the whiteboard and asks the children what fruits they have brought with them today, writing their answers under the axis. The teacher then systematically asks for each fruit: “How many of you have apples with you today?”, and the children raise their hands. The teacher documents each answer with an X in separate stacks on the board for each fruit, and a stacked bar chart emerges. After this survey, the teacher uses the stacked bar chart to ask questions, hel** the children answer them by interpreting the data on the chart: “Which fruit is the most common one today?” “How can we know that without counting?” and “And if we compare pears and bananas?”. In the end, the teacher describes the use of diagrams in everyday life and says it is very useful to have the ability to interpret data presented in this way.

In this activity, the children’s experiences are taken as the starting point for mathematizing. The teacher helps reduce the “noise” and provides mathematical explanations for the children’s experiences. The engagement is strong among the children during the activity. The teacher points out surprising results on the chart, and challenges the children to figure out answers to different questions and make them objects of joint inquiry. In this activity, mathematics is necessary in structuring the mathematical content and reasoning about the results. Playfulness is also a central part of the activity, in terms of the explorative and curious approach that both teacher and students engage in.

Parallel Activities

Thirdly, the children are involved in activities framed in an imaginary narrative, usually participating by hel** a protagonist to complete a task. The mathematical task is then situated in a playful setting, but the narrative and the mathematical content are rather parallel activities. Children are invited to an imaginary setting, within which they are engaged in solving tasks through their own means and suggestions; that is, with a high degree of freedom. Mathematics may become part of the activity, but as the attention is not necessarily directed at exploring the mathematical content in a mathematically relevant context it is possible to participate without extending the children’s view on the present mathematical content. The teaching appears to constitute a combination of creating a playful setting in order to (re)gain interest among the children and of completing a task in which the setting does not support the mathematical inquiry. In the following example, the teacher invites the children into an activity that is shaped as an imaginary narrative:

The teacher tells the children about Findus (a well known cat in children’s literature) who wants to play a joke on Pettson (his owner) by hiding eggs in his boots. The teacher places two boots in front of the children and shows them five tennis balls to represent the eggs. The children are told that Findus needs their help to figure out how the five eggs can be divided between the two boots. The teacher captures each suggestion the children offer, and processes them together with the children by writing the solutions in triads on tablets. At the end of the activity, teacher and children state that they have found all the possible ways that the five eggs can be divided between the two boots.

This example is a teaching activity in which the mathematical task is situated in a playful setting. The task itself is well structured to facilitate an exploration of part-whole relationships in numbers, and its playfulness in playing a joke on a familiar fictional character seems to engage the children in participating in the activity. Nevertheless, the narrative and the mathematical content are parallel activities in the teaching.

Discussion

In this paper we set out to describe how playfulness and mathematics as a content for learning appear in preschool class teaching activities. The analysis is therefore focused on both the playfulness and the content for learning, as a contribution to the discussion on how play (or rather playfulness) may become an informed part of preschool class mathematics teaching. The appearance of these two features of preschool class teaching forms the phenomenon in our inquiry, and the analysis of the 81 observations reveals that the way in which these features emerge shapes the mathematical learning opportunities differently. Thus, we do not intend to make any quantitative comparisons of frequency of observations, as the data can best be described as “touchdowns in time” rather than necessarily being representative of classroom teaching from a broader perspective.

When we first identified expressions of playfulness, a continuum of the extent to which play was given space in the teaching activities appeared. At one end of this continuum, we find teaching characterized by an orientation toward completing a (mathematical) task with highly limited expressions of playful exploration or open-ended inquiry. At the other end, we find teaching framed in narratives and a use of props with the intention to engage the children in interaction whereby the teacher, through the playful narrative, guides their communication about some mathematical content. In between, we find examples of interaction that centers around exploring a specific content, characterized by active involvement (from both teachers and children), that induces an explorative approach with a high degree of freedom. These inquiries are often (but not exclusively) guided by the teacher’s open-ended questions that take their starting point in the children’s own experiences or a familiar setting and embrace alternative suggestions and imaginary proposals.

The common content in all our observations is numbers, but the mathematics appear in different ways in the observed teaching. In some observations the mathematics becomes the central task to engage in, often through the mediation of a standard solution to symbolically presented problems (e.g., numerals written on the whiteboard). However, the content can sometimes be presented very well in terms of visualizing mathematical structures and procedures but not connect to contexts outside the mathematical. The mathematical content is thereby heavily foregrounded. In these cases, there is also often a closed solution or expected way to complete the task at hand. Meanwhile, in other observations, there is a more open approach in which the children are invited to offer suggestions for how to complete a task. The teacher still has a clear emphasis on the mathematical content and goal of the activity, but encourages different solutions for reaching the goal.

Numbers are the central feature in the observed teaching acts, and do appear in all the observations. However, the analysis of how the mathematical content is foregrounded, seen through the lens of playfulness, reveals differences in the opportunities for learning mathematics as useful and relevant (and thus meaningful) to the children. The first and third categories exhibit teaching practices in which the mathematical task to be completed is central and the goal is clearly determined, as are the methods and tools to be used. In the first category, in which the mathematics constitute the context, the cultural meaning of mathematical tools and concepts comes through. However, if one focuses on merely the cultural meaning of tools and goals, the teaching risks being reduced to the training of specific operations (Leont’ev, 1978). This stands out in comparison with the second category, which exemplifies open-ended inquiries that to a greater extent involve the children’s experiences and suggestions as an outset for the teaching and are characterized by an openness in the direction of the activity, whereby any participant may introduce alternatives and renegotiate the rules of the game, task, or play activity. In other words, the children are invited to explore mathematical content in a relevant context, with their experiences taken as the starting point for mathematizing (see Freudenthal, 1981). The teacher helps reduce the “noise” and provides mathematical explanations for the children’s experiences. In this way, the interplay between mediating (cultural) meaning and (personal) sense (see Leont’ev, 1978; Van Oers, 2010, 2014) becomes operationalized.

There is no doubt as to the benefits of linking educational goals to play, as play can be seen as a motivating factor for children and make them perceive the learning as meaningful, enjoyable, satisfying, and thereby help to arouse their interest in further learning (Simeonsdotter Svensson, 2009). However, our study may contribute through problematizing how play and playfulness can be implemented in teaching in ways that facilitate the learning of a specific content, a task that is known to be challenging (Björklund et al., 2018). We claim that it is not enough to embed the content for learning in a playful context, as in the third category. Instead, we argue that there are greater opportunities for learning when the setting facilitates structuring the mathematical content to be comprehensible and useful to the learner. We observed this in the second category. Play then becomes more than simply having fun; it is a valuable educational tool that includes children’s experiences and meanings, creating opportunities for their deeper engagement.

The openness and high degree of freedom that playfulness offers can be characterized by a “what if” type of thinking (see Vaihinger, 1924/2001) that arises as children are challenged to go past their current level of understanding. This allows them to realize that changes to or variations in a specific task, as well as their consequences, can be anticipated and calculated. In this way, what-if thinking provides a kind of prospective thinking whereby the essential direction is forward (not looking back), which is in stark contrast to the commonly used notion of reflective thinking that instead presents a meta-perspective on an occurrence. This makes playfulness an asset in mathematics teaching, and our study has provided empirical observations of how this can take place in the Swedish preschool class. We suggest that this is an important insight to consider when develo** teaching practice and policy guidelines for how education for six-year-olds should be conducted; particularly in the Swedish context, where the role of the preschool class in the education system is under review. For mathematics to remain meaningful to learners, and for education to provide culturally mediated tools that support individuals’ development of mathematical knowledge and skills, it is essential to gain a deeper understanding of the significance of play and playfulness in regard to mathematics teaching and learning in early childhood education.