Games in elementary mathematics programmes

Valerie Walkerdine’s (1988) analysis of mathematics learning contexts described games as signifiers of fun. Until recently, the taken-for-granted need to improve mathematics programmes by adding a fun element has gone largely uncontested and has become what Wilfred Carr and Stephen Kemmis (1986) describe as “folk wisdom” (p. 42). An implication of the perceived need to include games in mathematics is the belief that learning mathematics cannot itself be a pleasurable endeavour, thus reinforcing the existing negative stereotype of the subject as hard and abstract, as noted by a mathematics education commentator, Paul Ernest (2019). The effect of this, Marta Kawka and Kevin Larkin (2018) suggest, can be that students neither see learning as enjoyable in its own right, nor experience the deep satisfaction that can come from working hard over time to learn key mathematical concepts. The need to include a fun element in mathematics can also reinforce the separation of mathematics as a cognitive endeavour from the role of emotion in learning mathematics (Walkerdine 1988).

More than 30 years ago, Ernest (1986) posed the question, “Are games just an enjoyable interlude or can games be used to actually teach mathematics?” (p. 2). Ernest cited the findings of George Bright and colleagues’ (1985) programme of research comprising 11 studies from 1979 to 1982, drawing on their findings to argue that including games in the mathematics programme can have a positive influence on students’ achievement. In more recent studies where mathematics achievement data have been included, the findings relating to the effects of games are mixed. For example, Jennifer Young-Loveridge (2004) included games in a long-term intervention that successfully raised 5-year-olds’ numeracy achievement. On the other hand, in a more recent study by Leicha Bragg (2012), the 10- to 12-year-olds in the game-playing classes made smaller achievement gains than those who engaged in non-game activities.

Simply playing a mathematics game is not sufficient for learning the mathematical concepts and reasoning that are important for scientific endeavour. A game often involves students practising skills or developing conceptual knowledge and problem-solving strategies (Ernest 1986) and can be based on skill, strategic thinking, or probability. Lynn Burnett (1993) reported that discussion with students about the mathematics in games that they played, and the social context of game playing, can support conceptual learning. Building on this work, Bragg (2003) suggested that it was actually students discussing the strategies they applied during a game that may have the potential for helping them learn mathematics, rather than the actual playing of a game. However, in her more recent research, Bragg (2012) found no evidence to “support the claim that games are a very useful vehicle for promoting mathematical learning, even when game playing is supported by teacher-led discussion” (p. 1463). Moreover, playing mathematics games has the potential to distract from learning the actual mathematics content. John Gough (1999) suggested a disadvantage of games is that students’ focus on winning can distract them from thinking about the mathematics concept involved.

Since 2000, the research focus appears to have shifted away from mathematical games involving equipment such as cards and dice, and towards digital (computer) games (e.g., Kebritchi, Hirumi and Bai 2010) that are typically played by individual students or small groups. The use of digital games is not examined in this paper and nor are games examined as a mathematical task. In the research literature related to mathematics games that we located, we were struck by the predominance of research that involved games played by either small groups, pairs, or individual students. In fact, we found no research relating to whole-class mathematics games in primary classrooms. Additionally, the assumption that whole-class mathematics games are inclusive of all learners is largely unexamined. When games are played with the whole class—representing a range of understandings of any mathematics concept—it is difficult to see how any one game could be associated with mathematics learning for all students. Moreover, for a student who does not excel at mathematics, what might otherwise be the fun element of a game can be overshadowed by the student’s fear of being expected to publicly respond to problems they might be unable to solve. During our years of teaching, we have observed that for some students, their anxiety seems to prevent them engaging with any cognitive element of a whole-class mathematics game.

In this paper, we are not looking at the efficacy of games to promote mathematical learning, but rather the relationship between fluctuations in classroom emotional climate and playing a game. The mathematics games that we focus on are non-digital games and are played together by the whole class, sometimes including the teacher. Initially, we considered key elements of Bright, Harvey and Wheeler’s (1985) definition of a game as being “a challenge against a task or opponent” and “governed by a definite set of rules” (p. 5). Typically, a game’s rules include its aim or what constitutes winning, and the understanding that once a turn has been taken, a player cannot change their move or response. More recently, an alternative perspective on rules in games posited by Kenneth Liberman (2013) was that “One of the reasons for the impossibility of the folkloric notion that rules provide for an order in advance is that rules cannot ever be complete. They cannot account for every unforeseen game contingency” (p. 86). Following Liberman, we observed enacted game rules (as distinct from formal rules) and how these were evident in the unfolding experiences of orderliness during game playing.

Emotional climate and games

The notion of emotional climate is key to our investigation of the playing of whole-class mathematics games. In this paper, we draw on Alberto Bellocchi, Stephen Ritchie, Kenneth Tobin, Maryam Sandhu and Satwant Sandhu’s (2013) definition of emotional climate as “the collective state of emotional arousal produced by a class” (p. 530). Social theories that illuminate constructs associated with emotional climate focus on social interactions which produce emotional energy. In interaction ritual theory, Randall Collins (2004) used the term emotional energy to describe the special kind of energy arising from participation in a ritual.

To examine the flow of emotional energy during game playing, we attended to interaction structures that Collins refers to as interaction ritual chains. It is useful to our analysis of game playing to draw on Collins’s (2004) four conditions for interaction rituals characterised as follows: (1) two or more people experience one another’s bodily presence; (2) boundaries show which participants are included and excluded from interactions; (3) a mutual focus of attention is on common objects or activities; and (4) there is a shared mood or emotion. When these four conditions are met, interaction rituals generate successful interactions which, when repeated, produce interaction ritual chains. As Bellocchi, Ritchie, Tobin, Sandhu, M., and Sandhu, S (2013) observe, “Individuals will then seek to reproduce successful interaction rituals in future interactions in order to reproduce positive emotional energy” (p. 531). They identified a relationship between an interaction ritual’s success and the intensity of participants’ shared emotional experience. The intensity of social interactions varies from high to low emotional energy, which can be associated with positive, negative or neutral emotional climates (Bellocchi, Ritchie, Tobin, King, Sandhu, Henderson 2014). Various terms are used to describe high intensity emotions including “euphoric interaction” (Goffman 1961), “collective effervescence” (Collins 2004), and “high positive emotional exchanges” (Davis and Bellocchi 2018a). In this paper, we have used all three descriptors. On the other hand, low intensity emotions are less likely to result in the reproduction of interaction ritual chains or feelings of solidarity.

Game playing can be construed as a ritual activity generated through rules and patterned interaction; that is, social interactions are structured in games. The work of Erving Goffman (1961) on “Fun in Games” illuminated some of the contradictions that caught our attention in our inquiry into game playing in mathematics. We agree with Goffman’s observation that “Games can be fun to play, and fun alone is the approved reason for playing them” (p. 17). He established a theoretical notion of a “gaming encounter” to explain how games impose an entirely different structure on interactions between participants that is distinct from other classroom activity, such as instructional lessons. Furthermore, Goffman suggested that a gaming encounter has the potential to totally absorb or in his words “hold the participants entranced” (p. 67) and generate group solidarity. To unpack the relationship between games and emotions, we use the work of Goffman (1961), Collins (2004), and Ritchie and Tobin (2018), as well as James Davis and Bellocchi (2018b).

Important in game playing is the physical positioning of players, drawing on the point by Collins (2004) that one of the ingredients of rituals is the physical assembly of the group associated with emotional energy. When people come together in the same place for a shared focus, such as to play a whole-class game, there is what Collins termed “physical attunement, currents of feeling” and “bodily inter-orientation” (p. 34). Similar to the work of Bellocchi, Davis and Donna King (2018) in science classrooms, in this paper, we are interested in how game playing as a ritual activity in mathematics classrooms is “a site for reproduction of emotions such as enjoyment for individual students, the class and [the teacher]” (p. 58).

We consider the collective emotional climate alongside individuals’ expressed emotions. Essentially, we are trying to understand the everyday lived experiences of classroom game playing as part of a classroom mathematics programme. Having encountered games in mathematics programmes as both classroom teachers and researchers, we are intrigued by the association of games and fun. In this paper, we address the question: How are fluctuations in the classroom emotional climate associated with playing whole-class mathematics games?

How we went about our study

Our interpretive study used event-oriented social inquiry and included multiple data collection methods (Tobin and Ritchie 2012), as we outline below. The lead researcher and one other researcher (the second and first authors, respectively) were both experienced elementary school teachers who also have considerable experience in mathematics education. The third and fourth researchers in the team were postgraduate students working in educational psychology (noted in the acknowledgements). The data reported here were from an initial exploratory study that was built on in subsequent research into the use of cogenerative dialogues and student wellbeing and to an ongoing interest in the association of affect and emotion with learning in mathematics.

Study context

The primary school was located in a moderately high socio-economic community and was selected because of its proximity to the university. There were 40 participants in all: two teachers who shared a class of 34 Year 7 and 8 students (ranging from 10- to 13-years-old); and four teacher aides. Marama was the teacher responsible for the class and was released by another teacher, Trish, for two days each week for other duties (we use pseudonyms for the teachers and students throughout this paper). Marama and Trish each had more than 10 years’ teaching experience and both had expressed to the school principal their interest in participating in this study. The principal was part of the lead researcher’s professional network.

Emotional climate

An interactive response card system, TurningPoint™ (henceforth referred to as ‘clickers’), was used during 8 lessons over a 2-week period to collect data relating to the emotional climate. Teachers and students used clickers that transmitted responses to a receiver, which collected the data. Software on a laptop then collated the clicker data in real time.

The question to which participants were asked to respond with the clickers was, “How does the classroom climate feel?” Classroom climate was described to students as being how the classroom feels, just as the climate outdoors is how the day feels. The Likert-type scale relating to participants’ clicker responses was: 1 = very negative; 2 = negative; 3 = neutral; 4 = positive; 5 = very positive. For the first few days of using the clickers, the scale was written on the whiteboard to provide a visual reference point. Participants had up to 30 s to give their responses. Each participant was assigned a clicker, which was labelled with their name. Clickers were attached to lanyards, so they could be worn around the neck. This allowed students and teachers to go about their usual business during class time with a minimum of inconvenience from the presence of the clickers. A cell phone chime every five minutes was initially used to indicate when participants should use their clickers. During the third observed lesson, when the noise level in the class was higher than on previous occasions, there were more missing clicker data due to people not being able to hear the cell-phone chime. Trish suggested that the researchers instead use an actual chime-bar to signal that it was time to use the clickers. This was louder and was introduced the next day, resulting in fewer missing data.

Especially over the first few days of using the clickers, individual students’ responses were monitored to identify, for example, if any student gave consistently extreme responses. One such student consistently responded “very negative” and the researchers informed Marama, who was already aware that the student was experiencing high levels of anxiety. This served as a reminder that the mean collective emotional climate obscured the experienced emotions of individuals.

Classroom observations

Over the two-week period, eight mathematics lessons were videoed by the research team. Each lesson was between 50 and 60 min in duration. Three members of the research team were present in the classroom during each lesson and observed a mixture of whole-class learning experiences, small-group instruction, and practice activities in various forms—including whole-class games. Three video cameras were positioned around the classroom to capture clear images of what was happening in various locations within the large space, and also to capture sound where possible. One camera was stationary, near the researcher who was responsible for monitoring the clicker responses. The other two researchers re-positioned their cameras during the lessons, as they thought necessary. As much as practicable, cameras were positioned unobtrusively. Still shots from the video are included with participants’ informed consent.

Cogenerative dialogues

In addition to data collected during the mathematics lessons, three cogenerative dialogues (cogens) were held with groups of participants in a small room near the classroom. Cogens involved one of the researchers and a group of six or seven students. Students were included either at the teachers’ suggestion, or because their clicker responses had been of particular interest. Two students were included in all three cogens. The class teacher also attended the second cogen session. Each session ran for between 20 and 40 min.

The researcher set a few ground rules at the start of each session, then asked students to talk with each other about “What helps you to learn in maths?” Occasionally the researcher probed a student’s comment, but the students largely steered the conversation themselves. These sessions were recorded by video cameras, monitored by a second researcher who did not participate in the cogen. In this smaller room, making a clear audio recording of the dialogue (on the video recordings) was easier than it was in the classroom setting.

The clicker data and videoed mathematics lessons are the main focus in this paper, with excerpts from cogens included where relevant. Additional data collected during this project included field notes made by team members during each lesson and subsequent joint reflective notes.

Data analysis

Following a hermeneutic phenomenological framing (Tobin and Alexakos 2021), in our analysis of the classroom data, we responded to the questions, what is happening during the game playing and why, and what more is there? Within that framing, our specific focus was on the association of the classroom emotional climate and whole-class mathematics games.

Data analysis was conducted in two stages. First, the ordinal clicker data were summarised by producing line graphs of the mean of all students’ responses at each clicker point. The resultant ‘landscape’ of the classroom emotional climate during each lesson allowed us to identify peaks and valleys that stood out as fluctuations in emotional climate.

The second stage was to locate these apparent fluctuations in the video data of the mathematics lessons, to identify likely events for closer analysis. Reviewing the sections of video that included the three peaks and valleys revealed that each one centred around playing a whole-class game. Our focus in this paper on mathematics games emerged from our analysis of the clicker data and was not something we had set out to investigate.

We describe and analyse the features of emotional energy that we observed in the videos of mathematics lessons which coincided with the fluctuations in the classroom emotional climate, to build an understanding of “taken-for-granted experiences of emotional energy that were typically unnoticed by students” (Davis and Bellocchi 2020, p. 362). Close readings of transcripts of the cogens enabled us to identify more about the students’ understandings of emotional climate and their lived experiences of playing whole-class mathematics games.

Fluctuations in emotional climate

Line graphs of the clicker data for each lesson supported the identification of three events for closer analysis and are presented as part of each event described below. The graphs for Events 1, 2, and 3 included noticeable changes in mean ratings of the emotional climate (i.e., a difference of more than 0.5 of an interval between two mean responses), and the events around these particular points were investigated more closely. All three instances of noticeable changes in the emotional climate occurred during whole-class mathematics games. These were the only three occasions on which games were played over the eight mathematics lessons. On two of these occasions, the clicker data showed peaks in the mean positivity of the emotional climate in the classroom and on the third occasion a valley was evident. Drawing on Collins’s (2004) four conditions for interaction rituals, the three cases show: a successful interaction with dramatic emotional energy in Event 1 associated with a positive emotional climate; a successful interaction with undramatic emotional energy in Event 2 associated with positive emotional climate; and an unsuccessful interaction in Event 3 associated with negative emotional climate and interactional repair.

Event 1: ‘Round-the-World’

The first noticeable fluctuation in the clicker data was at the end of a lesson, immediately after the class finished playing a whole-class mathematics game called ‘Round-the-World,’ commonly used during mathematics lessons in New Zealand classrooms. Figure 1 shows the peak in the clicker responses towards the end of the lesson.

Fig. 1
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Clicker responses that include Event 1 (indicated by the rectangle)

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The mathematics in the game typically focuses on number knowledge and concepts, and often basic facts. Following the metaphor, round-the-world, the class sat on the carpet in a large circle. This whole-class game imposes a circular interaction structure where the physical positioning enables all players to see one another’s reactions and gestures as interactions unfold generating a sharing of emotions.

In the game we observed, Marama posed a number problem at a difficulty level suitable for the pair of students involved. The first of the two students to answer correctly moved round to the next student in the circle to answer the next question. When everyone had taken a turn, the remaining students challenged one another until one student was left. This student then challenged the teacher, with the runner-up student giving the problem. Students who were not having their turns were expected to encourage those who were, thus involving everyone throughout the event of the game. The moment when the student, Jack, correctly answered the problem before the teacher, and realised he had won the game, is shown in Fig. 2.

Fig. 2
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‘Round-the-World’ event

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Jack’s victorious cry and fist raised above his head were echoed in the behaviour of the other students who cheered, laughed, clapped, and surged forward towards the student and teacher, several getting up on their knees. The teacher (somewhat blurred, directly to the left of the winning student) is throwing her head forward, feigning shame at losing the game.

Students talked about this moment in the cogen sessions when we asked in what situation they might press a five on the clickers. Hope’s response was “Round-the-World winning,” and the other students voiced their agreement. Paul added that he would give a rating of five when “like say the atmosphere gets positive because someone makes a joke or something, like if you are talking and having a good conversation”. When asked what would induce them to click a one, Wayne responded: “We get beaten, Whaea (the teacher) beats us in Around-the-World.” In this study, this was the only instance of a whole-class mathematics game that involved a student competing directly with the teacher. In a separate paper (Higgins and Bonne 2014), we have explored student–teacher alignments in more detail to illuminate changing power dynamics in different classroom situations.

Commentary. The structure of the game simultaneously enabled the teacher to retain the symbolic capital of ‘teacher’ while also enjoying the role of player. More than all players just being on an even footing, Goffman (1961) explained that:

shared spontaneous involvement in a mutual activity often brings the sharers into some kind of exclusive solidarity and permits them to express relatedness, psychic closeness, and mutual respect; failure to participate with good heart can therefore express rejection of those present or the setting. (p. 40)

The students regarded Marama as a player, and this transformed the rules relating to how students (other players) could interact with her. Goffman described how in encounters such as in this game, a player’s role beyond the context of the game is irrelevant. However, Wayne’s comment (above) about the likely effect of being beaten by the teacher suggests this is not always the case.

Goffman’s (1961) idea of “the membrane of the encounter” (p. 66) helps us think about ‘Round-The-World’ as having a metaphorical membrane around it that supports collective engagement in and focus on the game. The inward-facing circle enabled a common gaze on the unfolding sequence of player interactions. The reproduction of emotions drew on the class’s collective knowledge of which people were likely to win (due to their mathematics skills), which afforded shared expectations and generated tension and built excitement.

As play progressed, a growing sense of excitement was evident in gestures and laughter, which Davis and Bellocchi (2018a) term “high positive emotional exchanges” (p. 24) and Goffman (1961) describes as “euphoric interaction” (p. 43). The game culminated in what Collins (2004) describes as ‘collective effervescence’ as part of an interaction ritual chain marked by synchronous laughter and cheering providing the conditions for successful interactions. We note that up to that point the emotional energy had been what Davis and Bellocchi describe as mundane, but then at the end of this game a dramatic peak was “experienced contemporaneously at individual-collective levels” (Davis and Bellocchi 2018a, p. 16). We see this event as characterising successful interactions associated with positive emotional climate.

Event 2: ‘Skunk’

The second event we identified in the clicker data was early in a lesson when Marama led the whole class for a game called ‘Skunk,’ a well-known probability game. In preparation for playing this game, students wrote the letters S K U N K across their page, with each letter representing a different round of the game; play begins with the ‘S’ column and continues through to the second ‘K’ column. The object of ‘Skunk’ is to accumulate the greatest possible points total over five rounds. Players begin the game standing up and sit down when they no longer want to risk losing points. Figure 3 shows the peak and subsequent valley in the clicker responses during the first 15 min when ‘Skunk’ was played.

Fig. 3
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Clicker responses that include Event 2 (indicated by the rectangle)

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At Marama’s request, a student explained how the game is played—ostensibly for the benefit of a student who had recently joined the class, but probably as a useful reminder for all students, as they may not have played the game for some time—and then the game got underway. Students began the game standing beside their tables, with a pencil and paper to record their points. Figure 4 shows Marama preparing to roll the two large dice. Behind Marama, we can see she has shown students how to set up their page to play the game.

Fig. 4
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Marama preparing to start a game of ‘Skunk’

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The image in Fig. 5 shows some of the students reacting to achieving high scores. Some students responded to getting a high score during the game by jumping up from their chairs and clapping or punching the air exuberantly, while other students remained seated.

Fig. 5
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‘Skunk’ event

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When students talked about playing ‘Skunk’ in one of the cogen sessions, their remarks suggested that perhaps not all students had fully grasped the concepts that were the mathematics focus of the game, with one student describing probability as being “luck.” Another student disagreed and explained that:

It’s kinda probability because it’s like you‘re thinking, is it gonna be bust? So should I stand up or sit down? Should I take the chance? You’re kinda thinking, oh, you’re thinking in the future. Thinking about prob- … what might happen.

Commentary. This event differed from Event 1 in three salient ways. First, this game did not involve the whole class sitting in a circular arrangement as they had for ‘Round-the-World’; instead, the players sat at their desks that were arranged in groups, affording them a view of and proximity to a group of their peers, rather than the whole class. While students added their scores during game playing, the classroom was largely quiet with individuals being introspective and not interacting with others. Bellocchi Ritchie, Tobin, King, Sandhu, Henderson (2014) point out that when students work in smaller groups without a collective focus, “the class cannot develop a sense of bodily co-presence of other groups” (p. 1313). We suggest that the nature of individual students’ introspective emotional energy was associated with a collective energy that could be described as undramatic (Bellocchi, Davis and King 2018). In contrast, dramatic moments occurred at other instances in this game playing sequence with audible exclamations whenever the number one was rolled because it affected everyone’s scores, thus changing the emotional climate of the classroom.

Secondly, the teacher was leading the ‘Skunk’ activity (as shown in Fig. 4) rather than participating alongside the students as another player. Here she was drawing on the symbolic capital of ‘teacher.’ It was unclear why the teacher opted not to play, but this might have been related to wanting to supervise the students, who were located around the classroom space. In contrast to ‘Round-The-World,’ in ‘Skunk’ the teacher did not control the numbers involved; instead, they were generated by rolling two dice with the numbers 1 to 6 limiting the possibilities. The risk of losing their points at each roll of the dice had the potential to build collective tension amongst the players. However, although some students showed their enjoyment of the game in Fig. 5, the euphoria that was evident at the climax of ‘Round-the-World’ did not appear in ‘Skunk’ and no collective effervescence was apparent.

Thirdly, the nature of the peak in the emotional climate data for this event was modest when compared with the peak for Event 1, with a maximum mean of 3.6 compared with 4.7, respectively. As we said at the start of this commentary, we think the students being seated at groups of desks probably impacted the collective emotional climate which was relatively calm during this lesson.

We consider ‘Skunk’—as we do for the other games observed—as a site for reproduction of emotions. In ‘Skunk’, while we observed successful interactions that maintained the orderliness of the game, there appeared to be fewer opportunities for a collective sense of tension to be generated during the game. We see this event as characterising successful interactions associated with positive emotional climate.

Event 3: ‘Greedy Pig’

The third noticeable fluctuation in the emotional climate was associated with a whole-class mathematics game at the end of a lesson. The game ‘Greedy Pig’ is a longer version of ‘Skunk’ discussed in Event 2. Figure 6 shows the valley in the clicker responses a short time before the lesson ended. The emotional climate at click 9 (mean 1.8) was the lowest mean rating for any lesson.

Fig. 6
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Clicker responses that include Event 3 (indicated by the rectangle)

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Towards the end of this lesson, when Trish introduced the game ‘Greedy Pig’ using rules that differed from those with which students were familiar, the students and teacher became confused. Trish’s response was to step back and invite one of the students to lead the game. However, the student’s explanation failed to get the game back on track and students started talking amongst themselves and looked confused. Trish realised she needed to intervene: “Ok, we’re going to stop there. I think it’s what’s called a bit of a disaster.”

Figure 7 shows Trish’s reaction to the students informing her that they usually begin the game standing up, not sitting down as she had instructed them. When Trish (right of centre at the back of the photo) realised that the instructions she gave for the game of ‘Greedy Pig’ seemed to have confused the students, she hit the palm of her hand against her forehead. At this point, Trish instructed students to abandon the game and brought them all down to sit on the carpet to talk to them about what had happened. Trish said: “I’ve no idea how you’re feeling, but I think that’s one of the worst maths lessons I’ve ever ta-” and was then interrupted by the sound of the clicker alarm. Click 9 in Fig. 6 occurred just as the game had been abandoned and Trish had brought the students to sit on the carpet.

Fig. 7
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‘Greedy Pig’ event

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The abandoned game was not specifically mentioned by the students who participated in a cogen that was held directly after this mathematics lesson observation. Their conversation (see below) did, however, touch on the nature of their relationship with Trish compared to the one they had with Marama. The students suggested that one important aspect that supports their learning is having a teacher whom they respect, someone to whom they can relate as “a big student”.

Paul:

You kinda need a teacher that you respect, like, as a person because otherwise you don’t really listen to them (students laugh)

Jo:

So what does: does anyone want to expand? What does having a teacher you respect mean?

Paul:

Well, it has to be like, a teacher that like, tries to extend the games for as long as you can and she makes jokes in class and she’s a really funny teacher and stuff

Jamie:

She knows what kids really like.

Rob:

She’s like one of the students (began speaking at the same time as Jamie).

Jamie:

Yeah she’s also she also knows how to kinda help and work with the kids who are naughty in class but she, and she does in a way that’s really good but you don’t always feel (really dumb?)

Paul:

Yeah she’s a big student (began speaking at the same time as Jamie).

Rob:

And she knows where the boundaries are set.

Jamie:

Yeah, so she has boundaries but um:(looks down) it’s my stomach! (students laugh) She has boundaries but she also knows how to have a lot of fun.

Paul:

Yup, like she’s fun but like if you get out of line like she can (let them know ?) sort of thing. … But then at the start of the year, with [Trish], we kinda didn’t respect her because um:, well, I guess we didn’t really know her that much and we kinda missed [another teacher].

Commentary. In this event, the teacher’s interpretation of the rules differed from the students’ interpretation of the participation structures when they played this game in their class. It is conceivable that the teacher was drawing on experiences of playing the game differently in other classes. As for ‘Skunk,’ the arrangement of players was dispersed in groups around the classroom, so players were immersed amongst their other seat companions and interactions shaped accordingly. However, from the outset, there was confusion about whether the students should sit or stand to begin the game, disrupting the collective energy flows and mutual focus, hence compromising the conditions for generating successful interactions (Collins 2004).

In the first two game events, the students and teacher demonstrated a shared understanding of the games’ governing rules. In the case of Greedy Pig, however, the discrepancy in understanding the rules generated what Bellocchi (2018) terms “a rupture of high intensity” (p. 88) and collective emotions of confusion and disappointment were evidenced by gestures and comments of despair. Through the confusion about the rules, the game’s structure was impacted causing a breakdown of orderly play (Liberman 2013). To salvage the game and to satisfy students’ expectations of participating in the game, Trish invited students to step into a teacher-like role. When this attempt at interactional repair did not lead to successful interactions, she signalled her withdrawal from leading the game by pushing her palm to her forehead. We see this event as characterising unsuccessful interactions associated with interactional repair.

Previous to playing this game with Trish, the students in our study had played a version of the game with Marama, for whom they expressed a shared feeling of respect during the cogen, presented above. We suggest that the high regard they had for Marama as a “big student” was also associated with their respect for her enactment of the game of Skunk. In this case, the respect is associated with the game playing, not the teacher. This is in line with Davis and Bellocchi (2018b) who explain that “It is the idea associated with the person or the artefact, which is treated with respect” (p. 1426). The students respected the game enacted by the collective that included Marama, underscoring that although Trish was familiar with the principles of the game she did not know their game. Liberman (2013) reminds us “the rules are used as a fabric for collecting procedures of orderly play; and the procedures they collect become just what the rules mean” (p. 85).

Reflecting on emotional climate and game playing in mathematics

We have generated insights into fluctuations in classroom emotional climate, which occurred during the playing of whole-class mathematics games. Our evidence suggests that when positive emotional energy is generated from players’ interactions, it is likely that the individual and the class will want to experience the game again, in the expectation that each occurrence of the game will reproduce similar emotional energy (Bellocchi, Ritchie, Tobin, Sandhu, M., and Sandhu, S 2013). This is perhaps one reason that games are seen by teachers and students as a mechanism to restore a positive classroom climate, and as signifiers of fun (Walkerdine 1988).

In the context of game playing, we sought to interpret: “What is happening in social contexts, why is it happening from the perspectives of participants, and what more can be learned when disharmonies occur in social settings” (Tobin and Alexakos 2021, p. 17). Each context of the game playing provided different manifestations of the rules. As Liberman (2013) explains, “the rules and the context each provide the other with whatever sense and coherency they are going to have” (p. 85). Specifically, we explored the question, how are fluctuations in the classroom emotional climate associated with the playing of whole-class mathematics games? While the first two events provide evidence that introducing a whole-class mathematics game can indeed lift the emotional climate and involve successful interactions, in the third event disharmony occurred when collective confusion prompted interactional repair. Collins (2004) described four conditions for interaction rituals: (1) two or more people experience one another’s bodily presence; (2) boundaries show which participants are included and excluded from interactions; (3) a mutual focus of attention is on common objects or activities; and (4) there is a shared mood or emotion. The first two events met all four conditions that characterise successful interaction rituals. In the third event, however, the orderliness of the play was compromised and the condition of mutual focus of attention was disrupted, constraining successful interactions.

We consider Goffman’s (1961) suggestion that: “… we could learn about the structure of focused gatherings by examining what happens when their orderliness breaks down” (p.19). We are thinking of “orderliness” as collectively understood rules or procedures of play. We saw evidence of orderliness in the first two events, and we observed it break down when the rules or procedures by which the game was played were interpreted differently by the teacher and students. Although Bright, Harvey and Wheeler’s (1985) definition of a game included being “governed by a definite set of rules” (p. 5), more recently a different view was expressed by Liberman (2013) as rules being constituted by a shared belief that they "bear a coherent order prior to the moment when they are applied" (p. 83). Our study showed that the emotional climate fluctuated positively when the students’ and their teacher’s shared beliefs maintained orderliness. This illustrates Liberman’s point about “how rules are made to serve as a means for stabilising the particular orderliness that game players develop in the course of their play” (p. 84). When this orderliness was interrupted because the teacher and students did not share the same beliefs about the game they were playing, the fluctuation in emotional climate was negative.

In summary, in the spirit of interrogating folk wisdom, we have used evidence to generate locally relevant theory of classroom life that explains why games can be a useful part of a teacher’s repertoire. The taken-for-granted wisdom that games can enhance emotional aspects of a classroom learning environment is supported by some of our evidence. However, the incomplete nature of a game's rules provided an opportunity for a game in the third event to reduce the emotional climate across a class. Following Liberman (2013), we observed enacted game rules (as distinct from formal rules) and how these were evident in the unfolding experiences of orderliness or disorderliness during game playing and the associated fluctuations in emotional climate.

Conclusions, limitations, and future research

As teachers and teacher educators, we continue to be curious about the folk-wisdom that game playing adds a necessary fun element to a mathematics classroom. In our study, games were brought to our attention because they aligned with fluctuations in the emotional climate of the classroom and we wondered about the connection. Although our study was limited to one classroom, we used multiple methods to help understand these three fluctuations. We sought to expand the conversation about the complexities of classroom emotional climate and to make an initial contribution to the literature relating to whole-class mathematics games in primary classrooms.

Collins’s (2004) theory of interaction rituals stresses the point that social interactions produce emotional energy. We suggest a renewed focus on the relationship between interactions and emotions in mathematics classrooms in future research. From our findings, we cannot comment on any association between emotional climate and the learning of mathematics; this remains to be investigated. Nor do we assume that a peak in emotional climate is the only indication of enjoyment during the learning of mathematics. For example, future investigations might explore the extent to which teachers highlight for students the enjoyment and satisfaction that can accompany persevering with a challenging problem, which could be associated with sustained periods of non-dramatic emotional energy. Although in our study we were examining the collective emotional climate, we also noticed an individual who was not swept up in the euphoric moments during game playing. This signals a need for future research to examine the assumption that whole-class mathematics games are fun for and inclusive of all learners.