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Group flow in small groups of middle school mathematics students Armstrong, Alayne Cheryl 2005

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GROUP FLOW IN SMALL GROUPS OF MIDDLE SCHOOL MATHEMATICS STUDENTS by ALAYNE CHERYL ARMSTRONG B.A.H., Queen's University, 1988 M.A., University of Manitoba, 1995 B.Ed., University of British Columbia, 1997  SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in FACULTY OF GRADUATE STUDENTS (MATHEMATICS EDUCATION)  THE UNIVERSITY OF BRITISH COLUMBIA December 2005 © Alayne Cheryl Armstrong, 2005  11  ABSTRACT This paper investigates situations in middle school mathematics classroom settlings where group flow seemed to be occurring, identifies observable characteristics that distinguished them as being group flow experiences, and suggests conditions which may encourage the occurrence of group flow. The focus was on the observation of small groups engaged in mathematics tasks in a regular classroom setting by the teacher-researcher through video and audio taping. Theoretical frameworks used to analyse the recordings were Sawyer's (2003c) model of group flow (following from M . Csikszentimihalyi's theory of flow), and the conditions required for the emergence of collective learning systems proposed by Davis and Simmt (2003). This paper suggests if the conditions for the emergence of a collective learning system are present, a group is more likely to develop and maintain a collective zone of proximal development where ideas can be negotiated. As a collective understanding emerges for the group and group flow becomes a possibility, certain behaviours that are both physical (posture, positioning, gestures, facial expression) and verbal (tone of voice, echoing and repeating words/phrases, rate of speech, fragmentation of speech) are observed. Group flow appears to manifest itself in a quicker form of these physical and verbal behaviours, a synchronization of action - for instance a physical closeness, an echoing of gestures and phrases, a quick fragmented way of speaking where members seemed to be finishing off each other's sentences - that suggests a parallel synchronization of thought. The more that group members appear to be "of one mind," the more likely it is that group flow may be observed.  iii  TABLE OF CONTENTS Abstract  ii  Table of Contents  iii  Acknowledgement  iv  Chapter 1: Introduction  5  Chapter 2: Literature Review  12  Chapter 3: Methodology  38  Chapter 4: Results  50  Section 1: Structured Mathematical Task  50  Section 2: A More Open-Ended Mathematical Task  72  Section 3: Open-Ended Questions with Fewer Pre-existing Structures  105  Chapter 5: Conclusions  131  Bibliography  143  Accompanying Materials: C D of thesis paper with videoclips  ACKNOWLEDGEMENT Many thanks go to the following: at U B C , Dr. Ann Anderson, Dr. Cynthia Nichols, Dr. Lynn Fels, Dr. Lyndon Martin, Katharine Borgen, and the members of the Mathematical Understanding reading group; Ann Shaw, Don Plummer, my school's staff, and the students who agreed to participate in this study; Glenn Davies.  5 CHAPTER 1: INTRODUCTION What is needed is a dialogue in the real sense of the word 'dialogue,' which means'flowingthrough,' amongst people, rather than an exchange like a game ofping-pong. The word 'discussion' really means 'to break up everything,' to analyze and have an exchange, like a game. Therefore we need this dialogue; the spirit of the dialogue is not a competition, but it means that if wefindsomething new, then everybody wins. - David Bohm (1998) Statement of Problem M y situation was what Schoenfeld (1992) might term a "problem-driven" one - simply put, I wanted to know what was going on in my classroom. I'd first noticed it while teaching drama, when I'd assemble the students randomly into small groups and assign them a situation around which to build a skit. Generally, the kids would enjoy themselves and produce something that was at least mildly entertaining, but once in awhile a group would catch on fire and produce something that observers and participants alike recognized as being phenomenally good. Then I started noticing it occasionally in the other subjects I taught, math and science. Groups of students who, individually, were rather lacklustre about math, together would sometimes rip excitedly through Problems of the Week like nobody's business. After they'd present their ideas to the class, inevitably they'd be told rather bluntly by one of their classmates, "I didn't know you were smart." Meanwhile, during some science lab activities, students would be noisily investigating something and, when told that it was time to clean up, would comment, "It's lunchtime already? We just got started!" What was going on? As a teacher with classes comprised of as many as 34 students, I'm physically unable to spend much time with each student individually. So I teach them as a group, a shifting nebulous entity that is both singular and plural at the same time. I'm  6 fascinated by the different cultures that develop in my classes over the course of a year; the classes themselves too seem to have different personalities (Bowers and Nickerson, 2003). This has led me to wonder, what are the possibilities of working with a class as a collective, and what's at play when a collective, or a smaller group within that collective, really gels and begins to perform the task at hand at the peak of the group's abilities?  Need for Study In his theory of optimal experience, Csikszentmihalyi has described flow as "the state in which people are so involved in an activity that nothing else seems to matter; the experience itself is so enjoyable that people will do it even at great cost, for the sheer sake of doing it" (1990, p. 4). He has suggested that flow may occur when there is an inherent interest in the activity, as well as a match between the level of challenge that the activity offers, and the level of skills that the participant possesses. The concept of flow fits in nicely with the idea of learning and it is not surprising that it has been studied in educational settings such as foreign language classrooms in secondary schools (Egbert, 2003), school physical education programs (Mandigo & Thompson, 1998), Grade Five creative writing programs (Abbott, 2000), school music programs (Custodero, 1999, 2002), Grade Five and Six math classrooms (Schweinle, Turner and Meyer, 2002), and high schools in general (Shernoff and Csikszentimihalyi, 2003). In terms of peak performance, flow theory has traditionally focused on individuals, although there have been references to group flow by Csikszentmihalyi (1988a&b, 1990, 1997), Emile Durkheim (1967), and Sato (1988). In recent years, in his studies on group creativity, R. Keith Sawyer has expanded on Csikszentmihalyi's  7 theory to develop a model of "group flow," which he describes as "a collective state that occurs when a group is performing at the peak of its abilities" (2003c, p. 167). He argues that while group flow and individual flow may both be considered emergent, arising from the activities of individual agents, they are not the same thing: individual flow represents the individual performer's state of consciousness while group flow is a property of the entire group and "cannot be reduced to psychological studies of the mental states or the subjective experiences of the individual members of the group" (2003c, p. 46). He adds, "It depends on interaction among performers, and it emerges from this process. The group can be in flow even when the members are not; or the group might not be in flow even when the members are" (2003c, p. 47). His studies are largely based on improvisational theatre and jazz music as examples of the occurrence of group flow, but he has suggested that it can apply to the classroom as well. The occurrence of group flow suggests that those groups experiencing it are at a peak level of performance, working in a collective zone of proximal development that offers a satisfying balance between skill level and challenge - in other words, the group is learning. The implications for educators of group flow being an intrinsically motivating phenomenon are powerful. As teachers in the classroom are almost inevitably working with groups, both large and small, a better understanding of what leads a group to reach flow would be beneficial to both teachers and students. It would be helpful for teachers to understand what conditions may encourage groups to perform at their peak and to recognize signs of the occurrence of peak performance. Understanding these conditions could help increase teacher awareness of how classroom routines and conventions, task structure, and subject content may better facilitate students in their  8 learning. Recognizing signs of peak performance could help teachers determine which situations and tasks promote learning situations in their classrooms. As math is viewed anxiously by some students, a better understanding of how to help students support each other during group work may lead more students to success in math and thus may promote a more positive attitude towards the subject. If traditional flow theory suggests that the individual can organize the normally scattered self to produce a momentarily well ordered consciousness, it seems possible that a group might be able to experience the same kind of self organization on a larger scale. M y study applies Sawyer's model in observing the behaviour of small groups in middle school mathematics classes. Although in most studies flow has been treated as a self-reported phenomenon, works by Sawyer (2003c) and Custodero (1999, 2002) suggest that it is also observable by outsiders. Engstrom (1994) argues that group work has its benefits to researchers in that group members essentially have to "think" aloud in order to communicate their ideas with others, thus giving researchers/observers access not only to their physical gestures and facial expressions, but to their thinking. Davis and Simmt consider math classrooms to be "adaptive and selforganizing complex systems" (2003, p. 138) and note the "expanded possibility that comes about when differentiated agents, who operate at a local level with local rules, come together in manners that complement and amplify existent possibilities while opening up others in the space of joint action" (2003, p. 147). This echoes Sawyer's argument that in some situations the whole (group) can be more than the sum of its parts (individual members) - i.e. group flow - while also foregrounding the nature of a classroom group as a learning system. I use conditions for complex systems proposed by Davis and Simmt as a way to flesh out Sawyer's model when  9 considering how group flow may occur in small groups tackling less structured tasks. The less structured the task, the more negotiating the group must do to establish a space of joint action (Davis & Simmt, 2003), or a collaborative zone of proximal development (Goos, Galbraith and Renshaw, 2002), in which they can develop their ideas as a collective. Once this zone is established, the possibility of group flow is present although not predictable.  Research Questions The purpose of the study was to observe situations where group flow seemed to be occurring, and to identify observable characteristicsthat distinguished it as being a group flow experience. The research questions that guided this study were: 1) what are the observable characteristics of group flow, and 2) what conditions may help to promote the experience of group flow in small groups in a regular mathematics classroom setting? The focus was on the observation of small groups engaged in mathematics tasks in a regular classroom setting by the teacher-researcher through video and audio taping.  Significance of Study At its heart, my study probes the nature and growth of collective mathematical understanding. Although there has been research about groups in classroom, particularly on cooperative learning, the focus has largely been on the dynamics of groups as a collection of individuals, and on how teachers should best structure tasks for group work. There is little so far in educational literature describing how classroom groups function as learning systems and how the collective understanding of groups works as an emergent process. M y study aims to offer a contribution to that  10 view. As well, my study offers a contribution to flow theory by describing group flow behaviour in the setting of a middle school mathematics classroom.  Thesis Outline In this study, I examine group flow in three different contexts using different frameworks for the analysis. •  In the first session, a pair of students who had an established working relationship worked on a highly structured task involving fraction worksheets. Using Sawyer's model to analyse the conditions of the situation, since his observations of behaviour in creative group situations such as jazz groups and improvised theatre troupes have parallels with classroom situations, as he himself has briefly noted (2003c), I suggest that the balance between the structure of the activity and the nature of the goal provide conditions that, encourage group flow. I then characterize the students' behaviour as being observable evidence of group flow.  •  In the second session, a group of four students worked on a less-structured task, an open-ended mathematics problem. After briefly positioning the situation using Sawyer's model, I then use the conditions for emergent learning systems proposed by Davis and Simmt to further suggest how a collaborative zone of proximal development might develop in a less-structured situation. I then characterize the behaviour of a pair of students within the group as being evidence of group flow.  •  In the third session, I compare the behaviour of two groups of students who are working on the same math problem. One group, although finding the "right" answer quite quickly, did not display evidence of achieving group  11 flow. In contrast, within the second group, which itself was unable to reach a solution before the end of the session, a pair of girls appeared to reach group flow while working through a particular aspect of the problem. Using both Sawyer's model and the conditions of Davis and Simmt, I suggest why group flow was achievable for one group but not the other. M y study suggests if the conditions for the emergence of a collective learning system are present, a group is more likely to develop and maintain a collective zone of proximal development where ideas can emerge. As a collective understanding emerges for the group and group flow becomes a possibility, certain behaviours that are both physical (posture, positioning, gestures, facial expression) and verbal (tone of voice, echoing and repeating words/phrases, rate of speech, fragmentation of speech) are observed. Group flow appears to manifest itself in a quicker form of these physical and verbal behaviours, a synchronization of action - for instance a physical closeness, an echoing of gestures and phrases, a quick fragmented way of speaking where members seemed to be finishing off each other's sentences - that suggests a parallel synchronization of thought. The more that group members appear to be "of one mind," the more likely it is that group flow may be observed.  12  CHAPTER 2: LITERATURE REVIEW In this chapter, I will review literature related to my study, beginning with how researchers in mathematics education have taken an interest in the social dynamics of mathematics students, developing theories to bridge the gap between the individual and the group of which he/she is a member. Next, I will consider studies of group dynamics in the classroom, considering those that focus on the individuals as part of a group, and those that focus on the group as an entity in and of itself. I will then discuss research about the concept of flow as experienced by the individual, and its application in different types of classrooms and subject areas. Finally, I will discuss recent research on group flow, learning systems, and the nature of collective understanding. It is my contention that by using a learning systems lens to observe the behaviour of groups in middle school mathematics classrooms, my findings will contribute to the newly developing field of group flow and also to studies exploring classroom groups as learning systems.  Social <-  Individual: A Reflexive Relationship  In recent years, some researchers in mathematics classrooms have shifted their focus from the practices and cognition of the individual to that of the classroom community, following Vygotskian theories that view learning as a social process that a child's inherent mental functioning is further developed through social interactions with those in his/her culture who have more experience (Wertsch, 1985). For instance, in the mid-1990's, Paul Cobb's research group came to view the construct of communal mathematical practices in the classroom rather than those of individual students as being particularly useful to observe, especially the development of socio-mathematical norms, arguing that "learning involves the interactive  13 constitution of mathematical meanings in a (classroom) culture" (1995, p. 1). In moving from a constructivist to what he terms an "interactional" perspective, Cobb writes: individual students are seen as actively contributing to the development of both classroom mathematical practices and the encompassing microculture, and these both enable and constrain their individual mathematical abilities. This notion of reflexivity...implies that neither an individual student's mathematics activity nor the classroom microculture can be adequately accounted for without considering the other (1995, p. 9). The interactive nature of the relationship between the individual and the collective is an important one. Bohm's theory of implicate order diminishes the idea of there even being a separation between individual and group: "The implicate order implies mutual' participation of everything, with everything. N o thing is complete in itself, and its full being is realized only in that participation" (1998, p. 106). Cobb echoes this when he argues for a distributed theory of intelligence: "it does not merely mean that individual activity and communal practices are interdependent. Instead it implies that one literally does not exist without the other" (1998, p. 197). Studying mathematics students' learning means taking into account the interactive behaviour of the collective of which they are a part, and this belief provides a foundation for my study.  Taken-as-shared -> Joint space: Sharing Mathematical Thinking Working, then, from the premise that the relationship between the individual and his/her collective is a reflexive one, and contributions to the development of  14 mathematical ideas travel both ways, one might next consider how these contributions come to be exchanged. Saxe (2002) suggests that the spread of forms of knowledge within collective practices is a social one, which he terms  sociogenesis, that happens  as individuals within the group appropriate each other's ideas. However, in the idea of each individual constructing his/her own learning in isolation, the sharing of knowledge with others becomes problematic even i f some kind of appropriation is involved. For instance, Forman and Larreamendy-Jones argue in a vein similar to Saxe's that participants in a collaborative effort constantly regenerate the context of their learning situation, but they note that the individuals "may or may not learn the same things about a common task" (1995, p. 550). How then can ideas ever really be shared? The concept of  taken-as-shared helps to bridge this apparent gap between  individuals. Cobb, Yackel and Wood have described taken-as-shared as being developed between individuals through their social interactions, and as being something that evolves as students make adaptations "which [eliminate] perceived discrepancies between their own and others' mathematical activity while pursuing their goals" (1992, p.l 18). The relationship between the individual and the group is interdependent and reflexive. Learning is a circular process as individuals mutually adjust to each other and their environment, while resulting in a group consensus which is interactively constituted. Forman and Larreamendy-Jones expand this idea, suggesting how the changing relationship between individuals results in the evolution of what is taken-asshared. Their premise is that math itself is culture, and thus taken-as-shared is a starting point for mathematical conversations: "because explainers need to go beyond what they can safely take as shared they must provide further information... [OJnce  15 explanations are out in the public domain, they too become shared knowledge reference points for further conversational contributions that will expand even further what the conversants can take as common information" (1998, pp. 107-108). In that way, the taken-as-shared changes as the individuals negotiate. However, this argument seems to suggest that while taken-as-shared is a common agreement that develops through interaction, it somehow has an independent existence outside of the individual and the group. Voigt is able to work around this difficulty. He writes that the concept of taken-as-shared goes beyond suggesting that individuals can come to agree that they have ascribed the same meaning to an idea: "from the observer's point of view, the meaning of taken-as-shared is not a partial match of the individual's constructions, nor is it a cognitive element. Instead, it exists in the process of interaction" (1995, p. 174), and not beyond it. He calls the relationship between mathematical meanings that are shared a theme and explains it using a simile: "the theme can be described as a river that finds its own bed" (1995, p. 175). For Voigt, the theme lies both inside and outside the individual and the group of individuals - it is present neither in one, nor the other, but in the moments in which the individuals are negotiating and that the group itself is acting as one. To consider how these kind of intellectual negotiations might occur within a group, it helps to think of a site of interaction - not a physical one but more of a cognitive one. Vygotsky describes the  zone ofproximal development as "the distance  between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers" (1978, pp. 85-86). In their work on collaborative problem-solving in a Grade Five mathematics  16 classroom, Zacks and Graves argue for a reconceptualization of the zone of proximal development "as an intellectual space, created in the moment as the result of the interaction of specific participants engaged with each other at that specific point in time" (2002, p. 233). John-Steiner works with a similar concept,  mutual zone of  proximal development (1996), as do Goos, Galbraith and Renshaw (2002) with  collaborative zone ofproximal development, both of these terms being reserved for situations in which the collaborators who are interacting share equal levels of competence in that specific domain.' Davis and Simmt, meanwhile, use the phrase space of joint action (2003). In my study, I will suggest that a collaborative zone of proximal development is an intellectual site where students are able first to negotiate shared meaning within their group (or part of their group), possibly reaching a state of group flow in the process.  The Nature of Classroom Groups Assuming then, that individuals and groups learn through the reflexive negotiation of shared meaning, it's helpful to explore the nature of groups in the mathematics classroom, and the conditions that encourage them to work effectively. In considering groups, there are different approaches: one might focus attention on the individuals that comprise the group and the relationships between them; or one might view the group as an entity in itself, a learning system. In the discussion that follows, I will first consider the perspective that is most common in the literature, that of the group as a collection of individuals.  I will use collaborative zone of proximal development in this paper, as I feel this term effectively foregrounds the interactive nature of the emerging zone of proximal development 1  17 Perspective I: A Group as Collection of Individuals One of the popular education movements in the last decade has been Cooperative Learning, which Elizabeth G . Cohen defines as "students working together in a group small enough that everyone can participate on a collective task that has been clearly assigned. Moreover, students are expected to carry out their task without direct and immediate supervision of the teacher" (1994, p. 3). In her discussion of studies of cooperative learning, Cohen notes that certain conditions are necessary in order for the theoretical advantages normally attributed to cooperative learning to actually be obtained (1994, p. 2). She argues that "students do not naturally construct knowledge when you put them in groups" (1996, p. 2), and furthermore, "when students work together, sharp inequalities appear in the rates of participation and influence of the different members of the group" (1996, p. 3), a pattern that is evident regardless of the subject matter being studied or the age of the student. As a result, teachers must actively teach students interaction skills (1994, p.7) and choose the task carefully to ensure that it favours the type(s) of interaction for which the teacher is aiming. For example, if the main type of interaction desired is for students to offer each other assistance, then the motivation of students to do so as well as the preparation for constructive assistance of one another become important factors for predicting the relative success of the groups. If, on the other hand, an extensive mutual exchange of ideas and strategies is desired, then too sharp a division of labor or limited participation of low-status students may impede the very  18 interaction necessary for the achievement of conceptual learning." (Cohen, 1994, p. 4) She argues that in true group tasks, particularly ones involving loosely structured problems, members need to be reciprocally interdependent so that task completion is only possible i f members exchange resources with one another (Cohen, 1994, p. 8). Another aspect of group work is peer relations. "When students work together, sharp inequalities appear in the rates of participation and influence of the different members of the group. Some members do too much of the talking and too much of the final decision-making. Others participate very little and what they do say appears to fall on deaf ears" (Cohen, 1996, p. 3). Although Cohen notes that this behaviour can be observed in classes dealing with all subject matters, and with groups of students ranging in age from second graders to adults, it is not clear from her discussion i f "talking" is the only form of participation that is deemed as legitimate by the observer, or how "final decision-making" is defined. The nature of the acts of participation and decision-making (negotiation) are far more complex than Cohen describes, and in my study I will argue that the observation of body language and gesturing is important in determining the level of participation of individuals within the collective. Lloyd and Cohen have found that each student's status within the class will affect how much he/she contributes (1999). Peer status (one's popularity within the class or group), and academic status (how "smart" one is perceived as being in a certain subject matter) are  local status characteristics - there are expectations for  competence based on high and low status, however, they are held locally, depending on a particular school's, class's, or group's culture (Lloyd & Cohen, 1999).  Status  19  generalization "describes the process whereby initial status differences in a group given a collective task organize expectations for competence at that task. As a result of the process of status generalization, high-status members are more active and influential on the task, even though the task may have no relevance to initial differences in status" (Lloyd & Cohen, 1999, p. 194). Lloyd and Cohen's research notes that status generalization is based on situation, depending on where the individual stands with respect to the group, and that teachers can work to help lower status students through the creation of group tasks with multiple intellectual aspects so that there are more entry points for meaningful contributions by all members. For my study, I wanted the students to feel comfortable in sharing their thoughts with their peers, so I kept Lloyd and Cohen's findings in mind when setting up the groups whose collaboration would be videotaped. In terms of classroom groups and the study of mathematics, a number of researchers have looked at the interactions that take place between the individuals within groups and the mathematical norms that emerge, and what follows is a brief description of some of their work. Goos et al. (2002) study how peer collaboration between senior secondary students does not guarantee success in a math task, particularly i f students do not challenge each other's ideas in order to promote the more useful strategies and abandon the weak ones. Kieran (2001) looks at the problems that arise in pairs of 13 year olds when one partner has difficulty expressing his/her emergent thinking to another (2001). Watson and Chick (2001) focus on the cognitive, social and external factors influencing different outcomes of math collaborations of small groups in Grades Three, Six, and Nine. Others explore the role of explanation, justification and argumentation in math (Yackel, 2001), cyclical patterns of interaction in college level math classes for preservice secondary math  20 teachers (Bowers & Nickerson, 2001), and how ideas spread in classrooms (Anderson & Nguyen-Jahiel, 2001). Forman and Larreamendy-Joerns (1995) investigate how a variety of goals, interests, and expertise emerge during the social interaction of samegrade pairs of Grade Four and Grade Seven math students as they work on transformational reasoning tasks, and develop explanations and peer collaborations. In another study by Forman and Larreamendy-Joerns (1998), they apply Grice's model of the informal rules that underlie different kinds of discussions to compare mathematical and everyday conversations in a Grade Two mathematics classroom, and determine that mathematical conversations have particular requirements that are reinforced by the teacher's requesting and revoicing strategies during the lesson. In a study I found to be very helpful in analyzing how the students in my study were speaking, Forman and Ansell (2002) examine how inscriptions are used by 2  students in developing their ideas in order to support or oppose arguments while problem solving in a mathematics classroom, and suggest how teachers can support student contributions through revoicing their arguments. In various studies (Forman et al, 1998; Forman & Ansell, 2001), revoicing has been framed as a way in which teachers can orchestrate discussions within their classes. Through repeating, rephrasing, summarizing, or elaborating upon student ideas, teachers privilege student voices and help structure their talk. As we'll, by revoicing certain ideas, a teacher legitimates them with the weight of the teacher's authority and thus increases the likelihood of the rest of the class accepting these ideas. In their later work, through their generalization of the process of revoicing as one involving "listeners" and "speakers," Forman and Ansell (2002) suggest that revoicing is a strategy that may be  Inscriptions are symbolic and material objects that serve to both preserve and represent aspects of the world and make them available to be discussed and argued (Forman & Ansell, 2002). 2  21 used by all listeners, not just teachers, although the focus of their study continues to be on ways in which teachers can influence class discussions. In its more elaborate forms (rephrasing, summarizing, elaborating, and translating), [revoicing] allows the listener to reframe the speaker's utterance in a way that can be evaluated by the original speaker as well as by other listeners. In this way, listeners can try to clarify a speaker's utterance by articulating presupposed information, by substituting technical vocabulary for less, precise linguistic terms, or by further explicating the speaker's intentions. (Forman & Ansell, 2002, . p. 258-259)  I would like to argue that in a classroom context where control is decentralized, such as in a small group task where the teacher is not present, revoicing can be used by students to orchestrate their own discussions, and thus potentially develop and maintain a collaborative zone of proximal development. Finally, Gordon-Calvert (2001) examines the conversation between two university students as they worked through a math activity, and notes how their social and mathematical relationships are influenced and maintained, observing the pair's physical proximity and positions, gestures, and recursive conversational patterns. While these studies are useful for considering what helps, hinders, and may emerge from group work within a mathematics classroom, none of them address what occurs when a collective works so well together that it achieves group flow. To consider that, it seems more useful to consider a group from the perspective of it being a single entity.  22 Perspective II: The Collective as a Single Entity Groups vary in terms of the cohesiveness of their members. A group may merely be a number of items that happen to be located close together - sit a bunch of strangers at a table and before they interact at all they could still be classified as a group. There can be more specialized groups, for instance collections (according to the dictionary, "any group of things systematically assembled") - the "smart" kids, the "average" kids, the "low end" kids, to cite a traditional form of classroom grouping. To consider group flow, one needs to focus on a group that is more than a mere collection, where there is collaboration resulting in some form of unity. Collaboration is not the same as a group of people brainstorming (John-Steiner and Mahn, 1996) - i.e. generating a bunch of ideas. When the members of groups collaborate, something special can happen on the wavelength of the implicate order Bohm describes (1998). As Davis and Simmt write, "For reasons that are not fully understood, under certain circumstances agents can spontaneously cohere into functional collectives - that is, they can come together into unities that have integrities and potentialities that are not represented by the individual agents themselves" (2003, p. 141). However tempting it may be to try to do so, this collaborative process is difficult to analyse into individual contributions as the phenomenon is not deterministic (Davis & Simmt, 2003); one might even argue that the individual pathways of growth of understanding within the collaboration do not exist (Martin & Towers, 2003), as the growth is occurring on a collective level. Viewing a group as a collection of individuals is not helpful in trying to explain why it is that, for some groups on some occasions, their whole is greater than the sum of their parts.  23 [T}he suggestion that a new transcendent unity can emerge from a group of previously disorganized agents is not a claim about a superorganism, a superior consciousness, or a metaphysical event. It is, rather a statement about expanded possibility that comes about when differentiated agents, who operate at a local level with local rules, come together in manners that complement and amplify existent possibilities while opening up others in the space of joint action. (Davis & Simmt, 2003, p. 147) In this study, I will argue that this kind of joint collaboration, when it results in the peak performance of the collective, may result in group flow. Before considering that, however, a discussion of individual flow is in order.  Review of Flow Theory What is Flow? In his theory of optimal experience, Csikszentmihalyi describes flow as "the state in which people are so involved in an activity that nothing else seems to matter; the experience itself is so enjoyable that people will do it even at great cost, for the sheer sake of doing it" (p. 4, 1990). It is a state of experience in which an individual is intensely focused on and absorbed by an activity, so much so that the experience is inherently self-motivating, what Csikszentmihalyi initially termed as autotelic. In his initial studies, Csikszentmihalyi (1988a) and his students interviewed a number of "experts" - musicians, artists, scholars, amateur athletes, rock climbers among others - about their experiences during their preferred activities, and this was later followed by interviews conducted from the late 1970's to the mid 1980's by his colleagues  24 internationally with people from a variety of professions and stations in life. Even those individuals who had never heard of the term "flow" had experienced the feeling and had their own phrases for it: "in the zone," "getting into it," "jamming," "in the groove," "everything is just clicking," "peak performance." Csikszentmihalyi proposes that this state of flow occurs when the individual's level of skill is matched by the level of challenge involved in doing the activity so that he/she is motivated to continue: "every flow activity, whether it involved competition, chance, or any other dimension of experience, had this in common: It provided a sense of discovery, a creative feeling transporting the person into a new reality. It pushed the person into higher levels of performance, and led to previously undreamed-of states of consciousness" (1990, p. 74). He notes that participating in certain activities does not guarantee flow will occur: "It is not only the 'real' challenges presented by the situation that count, but those that the person is aware o f (1990, p. 74). In short, flow seems to occur when the person is willing and able to make the most of the activity in which he/she is involved, whether it is experienced in situations such as rock climbing or in the everyday peasant lifestyle of traditional cultures such as the Occitans in northern Italy (Csikszentmihalyi, 1988).  3  Starting in the late 1970's, Csikszentmihalyi and his colleagues began using a different method to collect data, to address the drawbacks of using interviews and questionnaires (for instance, limits of memories, difficulties for subjects not used to reflecting) and to look at flow as an everyday experience. He started using the Experience Sampling Method (ESM), in which subjects are randomly prompted by an electronic pager several times a day (depending on the study) to record information in  Richard Mitchell argues that flow is not reported as a common phenomenon in modern western society because our discursive language, more suited for analysis, is not structured to adequately describe such ephemeral experiences (1988).  3  25 an Experience Sampling Form (ESF). The form contained both numerical scales with which to rate the intensity of experiences as well as open-ended items. The increased richness of data resulted in an elaboration of the flow model by both Csikzentmihalyi and other flow researchers. 1)  There is an inherent interest in the activity which prompts the individual to pursue it in the first place.  2)  The individual perceives that a balance exists between his/her skill level and the level of challenge involved in the activity. This balance is fragile (Csikszentmihalyi, 1997): if the challenges and the skills are both too low, apathy results; if the challenge is too high and the skills too low, the individual becomes anxious; finally, if the challenge is low and the skill too high, there is relaxation and little effort  3)  The activity should be one that enables intense concentration: "the ability to harness concentration for more complex mental tasks may be one of the hallmarks of achievement and talent development" (Shernoff, Csikszentmihalyi, Schneider & Shernoff, 2003, p. 161).  4)  There is a clearly defined goal in performing the activity.  5)  The individual has a sense of control of the activity and the level of his/her performance during it. Throughout the activity, the individual receives regular feedback about his/her performance which enables him/her to monitor his/her success and provides motivation to continue.  26 6)  The individual becomes less conscious of the self and of the passage of time.  7)  4  And finally, there is a sense of satisfaction and perhaps enjoyment (although not always) in participating in the activity, depending on the nature of the activity. Csikszentmihalyi notes that this feeling may occur in retrospect since during the activity all concentration is focused on the performance (Csikszentmihalyi, 1990).  Flow in Educational Settings. This last aspect of flow, the self-perpetuating motivation, has caught many educational researchers' interest. Ideally a classroom setting would promote engagement in and excitement about the learning process. Students who are engaged in their learning are more likely to push themselves to meet challenges, and to persist in the face of difficulties, resulting in an improved performance. Shernoff et al. note that optimal engagement, which "appear[s] to be promoted by a moderate difference between the challenge of a task and an individual's skills" (2003, p. 172), is similar to working within a student's zone of proximal development. " A s individuals seek to master new challenges, they develop greater levels of skill. Once mastered, they must identify progressively more complex challenges to create an ideal match for their skills. Flow thereby invokes a growth principle, in which a more complex set of capabilities is sought after and developed" (Shernoff et al., 2003, p. 161). The feeling of flow encourages someone to develop the scaffolding required for his/her own private zone of proximal development, with little or no external scaffolding required, and this facilitates further learning. In her summary of the variables that Csikszentmihalyi and flow researchers have found to occur during flow experiences, Joy Egbert (2003) notes that Jackson and Marsh (1996) argue two conditions - lack of self-consciousness and the perception of time passing more quickly - are not always present. 4  27 The phenomenon of flow has been studied in different school settings, generally focusing on measuring the flow experiences of individual students, and on providing a recommendation of types of school activities that may foster the more frequent occurrence of flow. Egbert (2003) used surveys, post-task interviews, and observational data to study the presence of flow in secondary school Spanish classrooms. She found that that flow does exist in these classrooms, and argued that flow theory provides a useful framework for the creation and evaluation of language learning activities. Flow Theory was used as a model by Mandigo and Thompson (1998) to consider how children and youth might be motivated in school physical education programs and encouraged to remain physically active in adult life, and to make recommendations for teachers about structuring physical education activities that would lead to more optimal experiences for students. Using interviews and field observations, Abbott (2000) studied the intrinsic motivation of two Grade Five students in elementary school writing classrooms and found that these boys selfreported flow experiences similar to what adolescents and adults have reported in other studies, suggesting that flow can occur at a variety of ages, and that its presence is more likely when subjects have more control over the creation of their products. In her survey of elementary school music programs, Custodero (2002) noted the significance of peers acting as "others" in facilitating individual students' flow experiences. For instance, if the level of challenge of a musical activity is too difficult for a student, the peer can provide a model to imitate and thus help the student meet the challenge, providing a more optimal experience. Finally, in studying student engagement in high school settings using the Experience Sampling Method, Shernoff et al. (2003) concluded that students reported feeling engaged when their skills and the challenge of the task were both high and in balance (i.e. flow conditions), and this  28 included participating in both individual and group work, rather than more passive activities such as listening to lectures, writing tests, or watching videos. The only researchers to find Flow Theory inadequate for their purposes are Schweinle, Turner and Meyer (2002) who studied the reciprocal-relationship of cognition, motivation and affect in Grade Five and Six mathematics classrooms. Classes were classified by researchers as being in flow or not based on the ratio of challenge to skills, and students participating in the study wrote a self-report in the last five minutes of class describing their current emotional state and the feelings about the.math class they had just experienced (Experience Sampling Method). Although the researchers found that positive emotional experiences and motivation are connected, they also found that the most positive experiences occurred when skill level was much higher than the level of challenge. They conclude that flow theory, while an interesting construct, is not adequate to describe the experiences of younger children, who might not have enough experience with the positive aspects of challenges to fully appreciate their benefits, or the complexities of actual classrooms where students do not have much freedom to choose tasks that match their interests and/or abilities. From the above studies, it is apparent that Flow Theory can play at least some role in guiding teachers to structure activities that promote a more optimum classroom. experience. Students need to be given choice whenever possible both in terms of the type of activity (to best suit their interests) and the level of activity (to create a balance between skill level and challenge). They need to be active in their learning and have some sense of control over the task and the outcomes. A s Schweinle, et al (2002) point out, the classroom is a complex environment, with many factors to consider such as individual interests and abilities, as well as factors the researchers do  29 not mention, such as social dynamics. Flow Theory based on individuals may not be suitable to address the complexity of a classroom situation. On the other hand, as Custerado (2002) points out, the help of peers can provide the scaffolding required for some students to reach flow - perhaps the complexity of a situation where individuals interact can actually promote flow.  Flow as a Collective Experience Csikszentmihalyi considers flow to be an experience of complexity, and points out that complexity involves the integration of autonomous parts. A complex engine, for instance, not only has many separate components, each performing a different function, but also demonstrates a high sensitivity because each of the components is in touch with all the others. Without integration, a differentiated system would be a confusing mess. Flow helps to integrate the self because in that state of deep concentration consciousness is unusually well ordered. Thoughts, intentions, feelings, and all the senses are focused on the same goal. Experience is in harmony. And when the flow episode is over, one feels more "together" than before, not only internally but also with respect to other people and to the world in general... [he quotes a climber] '...Your comrades are there, but you all feel the same way anyway, you're all in it together.' (1990, p. 41)  30 It stands to reason that, if the individual can order the normally scattered self to produce a momentarily well ordered consciousness, theoretically the same process could happen on a larger scale with a group of people. Can a group experience flow in the same way an individual can? Although group cohesion is not something Csikszentmihalyi focuses on, he does refer to group flow and describes a few situations where it may occur. One of the sites of group flow that Csikszentmihalyi discusses is within the family. For instance, he describes the late Canadian author Robertson Davies' happy marriage as possessing "joint-flow" (1997, p. 112) and cites the couple's common interests as being what made this flow possible. As for achieving flow in family relationships, Csikszentmihalyi notes that the following are required: a common group goal, open channels of communication in order to provide clear feedback and to address the differing needs and desires of the various family members, balancing challenges and skills, and a setting of trust. When all this is in place "then life in [the family] becomes an enjoyable flow activity. Its members will spontaneously focus their attention on the group relationship, and to a certain extent forget their individual selves, their divergent goals, for the sake of belonging to a more complex system that joins separate consciousnesses in a unified goal" (Csikszentmihalyi, 1990, p. 185). Another occasion of collective flow that Csikszentmihalyi discusses is when adolescents are with their friends and, for instance, they're trying to "gross each other out": "The goal of such sessions emerges by trial and error, and is rarely made explicit; often it remains at the participants' level of awareness. Yet it is clear that these activities develop their own rules and those who take part have a clear idea of what constitutes a successful 'move', and of who is doing well." (1990, p. 56) He further notes that theatre improvisational groups and good jazz combos work in a  31 similar fashion, as do groups of scholars or debaters, when their "moves" mesh together well. Even a good conversation in a casual setting has flow to it. Finally, Csikszentmihalyi cites the live concert crowd as another example of group flow: "There are few other occasions at which large numbers of people witness the same event together, think and feel the same things, and process the same information. Such joint participation produces in an audience the condition Emile Durkheim called 'collective effervescence' or the sense that one belongs to a group with a concrete, real existence" (1990, p. 110). Victor Turner calls the same experience "communitas" (1974).  What is Group Flow? R. Keith Sawyer is an educational researcher who has worked with Csikszentmihalyi and who has taken a particular interest in complexity, improvisation, and group creativity, and he has further developed the concept of group flow. O f flow as a collective phenomenon, Sawyer argues that it is related to but is not the same as the psychological concept of flow: "Csikszentmihalyi intended flow to represent a state of consciousness within the individual performer, whereas group flow is a property of the entire group as a collective unit" (Sawyer, 2003c, p. 43). Unlike Csikszentmihalyi who, when he considered the possibility of group . flow, regarded a group as a collection of individuals, Sawyer regards a group as an entity in and of itself and he defines group flow as "a collective state that occurs when a group is performing at the peak of its abilities" (2003c, p. 167). He argues that while group flow and individual flow may both be considered emergent, arising from the activities of individual agents, they are not the same thing: individual flow  32 represents the individual performer's state of consciousness while group flow is a property of the entire group and "cannot be reduced to psychological studies of the mental states or the subjective experiences of the individual members of the group" (2003c, p.46). For instance, an individual in a group may experience flow, while the group itself doesn't, and a group may be in flow although its members aren't. Sawyer relates the following from his interviews with improvisational performers in both theatre and music: Improvisational musicians and actors alike often describe the experience of walking off of the stage at the end of the night, feeling that the performance had been really bad, and then hearing later that the audience had found it to be a stellar performance. This is not only an issue of expertise; even. regulars and aficionados in the audience sometimes have different opinions of a performance than the performers themselves.... Inversely, most group performers can tell a story of at least one night's performance that they thought was particularly good, but later as they were discussing the performance with knowledgeable, trusted colleagues who had been in the audience, they discovered that it was not one of their best (2003c, pp. 46-47). It would seem, then, that a way to recognize group flow, its existence and its quality, would be through someone outside the experience - either an observing on-looker (i.e. audience), or a performer him/herself, looking back using the distancing effect of time to recognize whether or not an occasion was a flow experience  33 Noting that some groups are more process-oriented (improvisational theatre groups, and jazz bands for example) and others are more task or product-oriented (such as a engineering team fixing a design flaw in a car), Sawyer proposes a model in which "group flow is more likely to occur when the degree to which the group must attain an  extrinsic collective goal is matched by the number of pre-existing structures  5  shared and used by the performers"[italics are Sawyer's] (2003c, p. 167). According to Sawyer, the nature of the extrinsic goal may range from being an unknown goal to being a known one. For example, an improv group, which does not know ahead of time about what its skit will be, has an unknown goal, although it still has the intrinsic goal of being entertaining as it works through the process of inventing the skit - this group is, says Sawyer, problemfinding.At the other end of the range, an engineering team fixing a specific design flaw has a known (extrinsic)  goal and is thus problem solving. As for pre-existing structures, Sawyer proposes that these structures come in at least four types: 1. A n overall outline of the performance that all participants know in advance; 2. A shared repertory of ready-mades, with a knowledge of how they are usually sequenced; 3. Defined roles for each of the performers (the flexibility of these roles will depend on the nature of the group); 4. Common agreement on the conventions - the set of tacit practices governing interaction in the group.  Sawyer defines pre-existing structures as "the performance elements that are associated with a ritualized performance" (2003c, p. 168). 5  34 The greater the number of shared structures there are, the more restrained and predictable the performance of the group will be. If there are too few shared structures, there will not be enough to direct group members and the results will be uncontrolled, random, and ineffective. Achieving group flow involves finding a balance in the relationship between the number of shared structures and the nature of the collective goal (extrinsic or intrinsic). The more extrinsic the goal, the more shared structures are required to ensure that the group is able to achieve this goal in an effective manner. When there is no extrinsic goal - in other words, when the group is problem-finding - then fewer shared structures are required, and the group's performance can be more exploratory and improvisational. Sawyer's work on group creativity focuses mainly on music and theatre, but he does discuss the literature about collaboration and creativity in other team situations, and briefly touches on group interactions in the classroom, mainly drawing on the work of Azmitia (1996), Cohen (1994), and Webb and Palincsar (1996) who all suggest that small groups function most effectively when there is an implemented structure that is of a kind suitable to the task. His model predicts that most classroom collaborations have an extrinsic goal and therefore "would need more scaffolding and structuring from the teacher" (Sawyer 2003c, p. 185). His work also touches on the simile of "teaching as performance," arguing that viewing teaching as a form of improvisation foregrounds "the collaborative and emergent," and not to mention, creative nature of classroom practice (Sawyer 2004a), and he further suggests that  35 improvisational theatre exercises would be effective in helping to train pre-service teachers (Sawyer 2004b).  6  Group Flow in the Classroom According to Sawyer, "the study of group flow thus requires a fundamentally social psychology, and must proceed by examining the interactional dynamics among members during performance" (2003c, p. 47). Returning to his model for group flow, it would seem that in order for group flow to occur there must be: •  an agreement between the members about the task - what its purpose is, what its overall structure might be like; the teacher may set up many structures to ensure this agreement is reached early on (problem solving), or leave it relatively unstructured, so that students must negotiate what the task itself is (problem finding);  •  an agreement (which may be negotiated throughout) about what member contributions may be useful, and about what conventions may already be in place;  •  a task that requires (and promotes) interaction between all group members and that is flexible enough that members' roles may change if needed;  •  positive (or at least neutral) relationships between group members so that all members are likely to contribute, regardless of status, and that an atmosphere of trust and respect is developed and maintained.  As Sawyer's focus is on flow as a result of group creativity in general, he only briefly mentions how his ideas might be applied in educational settings (2003c, pp. 183-187) and»does not specifically discuss group flow in the classroom. In Chapter 4, Borko and Livingstone (1989) also note how experienced teachers make more effective use of improvisation in the classroom than beginning teachers do.  6  36 Section 1,1 will describe his theory in more detail, extending his descriptions of improvisation in jazz and theatre to parallels that may arise in mathematical classrooms and applying this to the data obtained in Session 1.  Collective Learning Systems As Sawyer's model of group flow does not adequately consider the nature of a collective, I turn next to the ideas of Davis and Simmt (2003) and the complexity of learning systems. Stating "that mathematics classes are adaptive and self-organizing complex systems" (2003, p. 138), they argue that "complexity science has highlighted that, by attending to particular matters, a teacher can greatly increase the likelihood of complex transcendent possibilities within the classroom" (2003, p. 145). They propose that there are five conditions necessary for learning systems to develop and maintain themselves: a) internal diversity (providing a variety of ideas that allow for innovation), b) redundancy (providing a common ground for group members so there is coherence), c) decentralized control (relating to the nature of collective interaction - when no one is "in charge" no one can predict what outcome will emerge), d) organized randomness (so that diversity and redundancy find a workable balance), and e) neighbour interactions (the exchange and intermingling of ideas). This theory and its application to classroom groups, will be explored more thoroughly in Chapter 4, Section 2. Related to complex learning systems is the idea of collective understanding. It is described by Martin and Towers (2003) as a group situation where, although individuals are making contributions, a single image is emerging. This suggests that a group can learn through the growth of collective understanding. In their exploration of collective mathematical understanding, Kieren and Simmt (2002) argue for it being a  37 dynamic and recursive process, mathematically and culturally embodied, and posit the use of a fractal (and fractal filaments) as a effective simile for representing its complex characteristics.  The Need for This Study Although there are studies about group work in mathematics classrooms, there are very few that approach it from a systems perspective. There are studies that examine the presence of flow in the classroom, a few specifically looking at mathematics classrooms, but none that I can find that look at group flow and its characteristics. M y study seeks to address these gaps. Schools by definition are all about groups, both large and small, and, ideally at least, about learning. It is only in recent years that education researchers have used a systems lens in considering collectives as whole entities rather than a collection of individuals and, to date, there has been little study of the phenomenon of group flow in classroom situations, although the two appear to be a natural fit. Teachers are always seeking new ways to reach and motivate their learners, and being able to identify when a group of learners is "on," and thus performing at its peak, could offer helpful feedback for teachers in determining which practices are most effective.  38  CHAPTER 3: METHODOLOGY In considering the question of the characteristics of group flow, I felt that a descriptive, non-interventionist, field-based study would be most appropriate. M y purpose was not to create group flow, but to observe classroom proceedings as they normally occur, and observe group flow should it arise.  Procedure The research involved conducting group tasks while regular Grade Eight mathematics classes were in progress. The first round of taping took place in November of the school year in another teacher's regular classroom. There was only one taping (Session 1) and it lasted approximately 40 minutes. Students were seated in large groups although they were allowed to work with others or independently as they chose. The video camera was mounted on my shoulder, and I moved around the perimeter of the classroom designated for those students participating in the study, focusing on whatever behaviour I found of interest. This was meant to be an exploratory session. The second round of taping took place in late spring of the same school year but with a different Grade Eight mathematics class, my own. I chose to change the focus of the videotaping to my own class because I felt my presence as an observer would be less obtrusive (i.e. my students were used to me), and once my students became used to the cameras being in the room, they would quickly return to the normal behavioural routines. There were two video cameras recording the activities of two groups (which varied in size from four to six members) in the classroom per session. Two additional sessions, each lasting between 30 and 60 minutes, were recorded, one session per week for a period of two weeks, with the day of the week  39 and the time of day varying with each session. Session 3 was also audio-taped as a backup due to the high noise level of the groups working off-camera. During Session 1,1 was behind the camera and mobile throughout. During Sessions 2 and 3, each camera was stationary, mounted on a tripod, and focused so that only the participating groups were in the frame. The camera was close enough to record the facial expressions and gestures of some (but not all, due to camera position) students, but far enough away to keep them all in the frame, with the exception of one group in Session 3 who had one member who was sometimes off-screen due to my inability to to position that particular camera physically any further away from the group. To reduce their self-consciousness around the cameras, I invited students to help set up and take down equipment. The mathematical tasks were chosen to provide opportunities for collaboration within the groups. In Session 1, the students worked on fraction puzzle sheets (see Appendices), a typical review activity for that particular class. A s the "answers" were available on the sheets, students were able to focus on discussing how to arrive at an answer rather than on whether a particular answer was correct. In Sessions 2 and 3, students worked on open-ended Problems of the Week, a task with routines that were familiar to the class. During the course of the school year, students had come to understand that they were to work on Problems of the Week as independently of me as possible - i.e. that I would clarify information provided in the question if they asked, but would provide no hints about what to do, and no suggestions as to whether a proposed answer might be right or wrong. This established routine meant that students expected me to leave groups alone (barring any behavioural issues), whether a group was being videotaped or not.  40 In all sessions, students each received a puzzle sheet (in the case of Session 1) or a Problem of the Week sheet (in the case of Sessions 2 and 3) but were provided with no additional materials other than their own personal supplies and notes.  Participants The research took place at a middle school in a large suburban school district in British Columbia. This age group is.known for its high energy and enthusiasm for socializing, making its members ideally suited for working in groups while tackling math tasks. The school has a multicultural population with wide-ranging economic backgrounds and family situations. In order to further create a "fair" distribution of student types within a class, there is a class-building process which could be considered to be proportional stratified random sampling. A n index card is prepared by their Grade Seven teachers for each student which lists gender, perceived "academic ability," certain behaviours (disruptiveness, organization), special needs (ESL, learning disability, "gifted"). These cards are then used to distribute these students into "balanced" heterogeneous classes, meaning that there is an effort to distribute academic ability, behaviour issues, etc, evenly amongst the classes so that no one class would be deemed as preferable. The tentative class lists are then reviewed by the school counselor, school administrators, the student services teachers, and the Grade Seven teachers to check that this balance has been achieved. In both classes that were used in the research, students had a history of being in cooperative learning classrooms - in fact, one of the school's goals in that and previous years was for staff to implement cooperative learning strategies in their instruction - and the students seemed very comfortable with the way the activities were set up.  41 In the class taped for Session 1, the students who had returned their consent forms to their classroom teacher were seated in groups on the side of the classroom designated for videotaping. According to the teacher, Mrs. Shug, these students were of a variety of mathematical abilities. Within their designated side of the room, students participating in the study were able to sit and work with whom they pleased. The students on whom I largely focused the videotaping were chosen because they were active in the task, audible, articulate, and seemed relatively unaffected by the presence of the camera. Mrs. Shug later confirmed that their behaviour during the session was typical of their behaviour in regular math classes. The class taped for Sessions 2 and 3 was one of my own math classes, selected because the class was relatively homogeneous in math ability (i.e. almost everyone was working at or close to grade level) and behaviour was not an issue (i.e. class management during the taping would not be a problem). Students were chosen for the groups being taped based on their willingness to participate in the study, their articulateness, and their previously demonstrated interest both in mathematics and in collaborative work. In terms of their mathematical abilities, the students ranged from average (a final grade percentage in the low 70's) to high (high 90's). Although they were not allowed to choose their own groups because not all members of the class had agreed to participate in the study,.I made an effort to ensure that they were seated with peers with whom they felt comfortable, and with at least one other group member of the same gender, to create a group where all members would feel comfortable offering contributions to discussions. Although, my original intention was to maintain the same member composition for each group throughout the sessions, due to the time of the school year (spring) there were frequent absences due to out-of-school events and a few students ended up being switched in and out of groups both to keep the number  42 of group members at a minimum of four and to maintain positive interactions within the group. A l l students' names have been changed to pseudonyms to ensure confidentiality.  Role of the Teacher/Researcher In Session 1,1 was not the class's regular math teacher (although I was familiar to the students as a teacher at the school and had taught some of them in another subject area) and was strictly an observer during the session, moving around the room to videotape participating students who caught my interest. In Sessions 2 and 3, the cameras were set-up on tripods and were stationary. As classroom teacher, I was limited to the extent to which I was also researcher/observer. I made an effort to limit my interactions with the groups being taped, only approaching them if a group member had raised his/her hand to ask a question, and keeping my responses as brief and non-directive as possible. While the groups were being taped, I was away from the camera working with students in other groups, or occasionally giving direction to the entire class (particularly at the beginning and ending of the class periods). Although I looked at brief clips of the videotapes of Session 2 after the taping to determine if the lighting and audio components were adequate ,1 did not view the full 7  videotapes until all of the sessions had been completed. Because I had little contact with the groups being videotaped, and was not behind the cameras, I had no awareness as to whether or not group flow might be occurring during any particular session as they evolved. The cameras were set up to focus on participating groups and turned on at the beginning of each session; they recorded non-stop throughout the class period, and they were turned off at the end of the allotted time. As mentioned, the audio quality was not good - due to the small classroom size, groups of students were quite close together and the talking of groups off camera was quite audible at times - and audiorecorders were added for Session 3. 7  43  Transcription of Data I transcribed excerpts of the videotape for Session 1 in November of the school year, and transcribed the videotapes for Sessions 2 and 3 during the summer 8  following the taping. For Session 2, which featured the Train Problem, I transcribed the entire videotape of Cara's group, but only excerpts of the tape of Shannon's group since those students spent much of that session quietly writing out their results and there was little in the way of further discussion. For Session 3,1 transcribed the entire videotape and audiotape of Anik's group's work on the Locker Problem.  9  Because gesture and intonation of speaking were important in determining whether or not group flow was occurring, my transcriptions used a format with two columns: one column to show what was said and the other to describe what was done. Transcription notation included the following: forward slashes (/1) to indicate when students were speaking simultaneously, as a way of highlighting synchronization; words that were enclosed in parentheses ( ) showed where I was uncertain about what was said; "Unknown" was used when I could not identify who had spoken; and (Name?) when I was not certain if that particular person had been the speaker. During the process of transcribing, I was very aware that to transcribe was essentially to create a new text with a point of view since it would be impossible for any transcriber to capture everything that occurs in a video. And a video, in turn, only gives one angle of what is occurring at the time - not everyone in the group is visible, This transcription was for a paper I was writing for a graduate-level course. It was after this course was completed that I decided on the topic for my master's thesis and collected more data that spring through further videotaping. Another group that was taped working on the same Locker Problem accomplished little together three students in the group were consistently distracted by students outside the group, two of them even physically leaving the group for a short period of time, and their conversation was largely social; the other two boys in the group quietly went about their work independently and, aside from a couple of brief discussions to clarify directions, had very few collaborative conversations together. As there was little interaction related to the math question to consider, and obviously no evidence of group flow, I chose not to transcribe or analyse the video and audiotapes. 8  9  44 not every word spoken is audible, particularly i f more than one person is speaking at once. That being said, it is my hope that in repeatedly reviewing tapes and editing transcripts over a period of time, with breaks in between, and using other tapes when possible to provide another source of information (for instance, when two cameras were taping, sometimes the conversation of one group was also picked up by the other camera), I was able to create an accurate and fairly objective transcript.  Gathering Data The taping for Session 1 took place in November of the school year. When I first viewed this videotape I was looking for patterns of behaviour that might be related to improvisation, an analogy for mathematical discussion I had been reading about for one of my graduate courses. It was not until I had seen the videotape a few times that I began to focus on a particular pair of girls, their physical and verbal actions bringing to mind research by Gordon-Calvert (2001) and Forman and Ansell (2002), their apparent absorption and satisfaction in their task bringing to mind Csikszentmihalyi's work on flow (1997), and their apparent one-mindedness about their work reminiscent of the ideas of Davis and Simmt (2003) about collective learning systems. This paper resulted in my initial conception about what group flow might generally look like in a classroom situation. Later, after the paper was submitted and I had decided to do further research on the topic, I became aware of and read Sawyer's work on group flow (2003c). I chose not to interview my students from any of the sessions for a couple of reasons. M y main reason was my subject was the group as a whole, and I wanted to approach the data from the outside, as would a teacher or other observer who was witnessing behaviour in a classroom, in order to answer the question, "Is that group  45 flow? How can I tell?" A s argued earlier, one member's apparent experience of flow does not correspond to a group experience of flow, and might in fact lead to misanalysis, so self-reporting would not be helping in addressing the research question. What I needed instead was something that would give me access to the collective's thinking. According to Yrjo Engestrom. there is an advantage to observing a collective rather than an individual: One of the most persistent methodological difficulties of studying thinking has to do with access to online data from thought processes. When thinking is defined as a private, individual phenomenon only indirect data is accessible. Thinking embedded in collaborative practical activity must to a significant degree take the form of talk, gesture, use of artifacts, or some other publicly accessible mediational instrumentality; otherwise mutual formation of ideas would be rendered impossible. Collaborative thinking opens up access to direct data on thought processes (1994, p. 45) . 10  I focused my attention on what was directly observable: what students said, how they said it, their body language, facial expressions, and gestures, and the speed, and possible synchronization between members, of all of the above. This will be discussed further in the Analysis section of this chapter. The other reason I chose not to interview my subjects was that I was very wary of any affect I might have on my students' behaviour during the sessions. I felt that an oral interview or a written questionnaire given immediately after each session might inadvertently give students "hints" about how they should be behaving during  Although, as Zacks and Grave warn (italics theirs). 1 0  "spoken words do not  equal  thoughts in the mind' (2001, p.  265)  46 the next tapings, a concern because I had planned to videotape the same students on more than one occasion.  Analysis After spending the spring gathering more videodata, I began viewing the new tapes to see i f I would observe similar behaviour and could thus generalize from Session 1 's single case in order to make stronger arguments for the observable characteristics of group flow. Initially in analyzing this data, I attempted to use coding methods similar to those used by Sfard and Kieran (2001) and Goos et al (2002) in order to track the ideas and contributions of each student. The day I abandoned these coding systems, I wrote something in the margin of the transcript that I happened to be working on which summed up my frustration": "Whose idea is it anyway?" That's when it really struck me that group flow is not a group of individuals experiencing flow but is instead an entire entity in flow, and that following the transcript in a linear fashion - as in, "this produced that and this produced that" - has its limits for revealing what may actually be happening. I turned to a more descriptive method. Transcribing the video had forced me to "slow down" (Zack & Graves, 2002, p. 240) and really listen to what the students were saying, watch who was doing what and when, and, perhaps most importantly in terms of group flow, notice times when students were saying the same thing or , performing the same gesture simultaneously. Once the transcription process was completed, I continued to view and review the videotapes over a period of weeks, sometimes at different speeds, making dated handwritten notes. During each viewing, while keeping the original notes unaltered, I elaborated on and revised my ideas, and  Martin and Towers report similar frustrations in their attempts, ultimately abandoned, to code their videodata (2003). 11  47 also generated new notes. This resulted in a collection of notes which enabled me to go back and look up earlier ideas for reference or comparison. Gordon-Calvert writes that the "communication of mathematical ideas is in large measure nonverbal in terms of its reference to symbols, diagrams, and drawings on the page, as well as the gestures which include pointing to, as well as talking about" (2001, p. 96), and, as mentioned previously, I was looking for behaviour similar to that displayed by Mandy and Hannah in my original data to see i f it could be generalized as a description of the observable features of group flow. One of the most striking things about Mandy and Hannah was their often synchronized behaviour, which I categorized as being verbal and physical. O f this kind of entrainment , Condon wrote, 12  A speaker's body is precisely synchronized with his own speech across multiple levels. And between human beings there is an exquisite, rhythmic synchronization.... the listener's body moves in rhythmic organizations of change which precisely reflect the speaker's speech. This is observable even in infants. Human communication is fundamentally synchronous and rhythmic. Synchrony and rhythm are primary aspects of human individual and interactional behaviour (1986, p. 75). Condon's work suggested to me that there was a connection between the physical and verbal synchronization I was witnessing in that they each could be feeding off of and spurring on the other, helping to generate a sense of "togetherness" for the parties involved. Entrainment is the process of external synchronization. In physics, for example, two tuning forks of the same frequency can end up driving one another. In biology, groups of fireflies show entrainment when they gradually begin blinking in unison. The human body, when isolated from external light stimuli, runs on a 25 hour clock; humans in regular conditions become entrained by the cycle of light and dark to the 24 hour day (Strogatz, 2003). 1 2  48 In order to determine conditions that might encourage the achievement of group flow, I turned to Sawyer's model of group flow which suggests that there needs to be a balance between the nature of the group's goal and the shared structures. In determining the structures for each session, I considered the academic background of the students, their past experiences with the type of tasks they were performing, and the nature of the tasks themselves (this is further detailed in Chapter 4). I also observed the students' behaviour, particularly when considering what conventions and ready-mades were being used - for instance, what routines were they following and which ones recurred; what pieces of knowledge seemed to be assumed and which ideas had to be negotiated before they were accepted by the collective. The conditions Davis and Simmt propose for the development and maintenance of learning systems helped to provide a further framework for my examination of the nature of the students' interaction. In considering the condition of redundancy, I looked for instances of revoicing (Forman & Ansell, 2002), while for decentralized control I looked at instances of how the group related to the teacher and to each other as sources of authority. These in turn connected strongly with neighbour interactions - were all students contributing ideas? Which of these ideas were being considered? - which also related to Voigt's concept of "themes" (1995). And this in turn connected with the condition of diversity - were group members simply restating the same ideas or were they adding new considerations. Although Davis and Simmt's conditions are so intertwined in nature it was sometimes difficult to decide if a certain aspect of a session would be best described by a certain condition, this actually served to highlight the complex nature of collective work.  49 Reporting Results In the Results section, I have chosen to report the sessions in a nonchronological fashion in order to explore if the characteristics of flow change according to the degree the task is structured. For that reason, I begin with Mandy and Hannah and their worksheets (which also happens to be the first session taped), move on to the Locker Problem (which was actually the third session taped), and finish with the two groups that worked on the Train Problem . 13  This is also why the audiorecorders used in the Locker Problem seem to "disappear" for the Train Problem - it wasn't until after the taping of the Train Problem session that I realized the need for the audiorecorders and thus introduced them for the next session (the Locker Problem) 13  50  CHAPTER 4: RESULTS Section 1: Structured Mathematical Task  intra clip  Mandy and Hannah are two Grade Eight students working together on fraction review worksheets. Another girl at their table calls Mandy's name over and over again but receives no response from either Mandy or Hannah who continue to talk and work together. A second girl at the table reaches around Hannah, who seems oblivious to the physical contact, pokes Mandy in the shoulder with a pencil a few times and, still  51 getting no reaction, begins jabbing Mandy in the ribs, finally rousing her attention. It is just a math worksheet - why are Hannah and Mandy so absorbed in it?  Initially I had intended this taping merely to be an exploratory session while I learned how to manager video camera in a classroom setting. While I did record the activities of a few different pairs of students in this particular classroom, I found myself immediately drawn to Mandy and Hannah, who were sitting with a group of approximately eight other students, returning to record them again towards the end 14  of the session. At first it was simply because, despite the noise of the busy classroom, they were very vocal and I could hear them fairly clearly. But there was also something very compelling about their behaviour in comparison to that of their peers, and it was not until I viewed the tape later that I could put my finger on exactly what it was - for brief periods the two girls were strikingly synchronized in their movements. When I played an excerpt of the video in one of my graduate seminars the following week, the reaction of my classmates was much the same: "Well, now would you look at that...." Sawyer (2003c) argues that this kind of interpersonal synchronization is an important part of group flow. He quotes jazz musician Melba Liston as saying that during a synchronized performance, "everyone can feel what each other is thinking and everything. Y o u breathe together, you swell together, you just do everything together, and a different aura comes over the room" (2003c, p. 44). According to Sawyer, for group flow to happen while jazz groups are performing, a type of parallel processing needs to occur. The musicians need to play their own instruments without pausing, yet at the same time they need to listen to and watch their group members so  1 4  A couple of students switched back and forth between tables during the taping.  52 they can respond immediately. Another musician interviewed by Sawyer stated, "You have to be able to divide your senses... so you still have that one thought running through your head of saying something, playing something, at the same time you've got to be listening to what the drummer is doing" (2003c, p. 44). This is one of the reasons that Chicago blues bands often use the first set as a preliminary warm-up before the headlining singer or soloist comes out to join them on stage - the band is given an opportunity to develop an interactional synchrony so they've got "the feeling" for one another during the main part of the performance. Improvisational theatre groups work in a similar fashion, participating in group exercises to warm themselves up off-stage, potentially working themselves towards a state of group flow (of which there is no guarantee), while the audience members are still getting settled into their seats. Sawyer (2004a, 2004b) considers the discussion that occurs in a classroom situation also to be a kind of improvised performance. Although the focus in his writings is on how the teacher's participation influences the students' performance, it stands to reason that the students' discussion, without the presence of a teacher, is also an improvised performance. Researchers such as William Condon (Condon, 1986; 15  Sawyer, 2003c) have argued that interactional synchrony takes place between any two or more people who are interacting, so it makes sense that it could also take place within a group of math students. Might Mandy and Hannah's brief periods of physical and vocal synchronization then be considered evidence of them experiencing collective flow? In this section, I will use Sawyer's model to analyse Mandy and Hannah's behaviour, starting with a discussion of each of the structures he proposes in his  Indeed, Sawyer has argued that all everyday conversations do have an element of improvisation involved (2003b). 15  53 model of group flow and how they apply to this session, and followed by an analysis of a portion of the transcript where I will outline why I believe the two girls may be experiencing collective flow.  Pre-existing Structures According to Sawyer's model, "group flow is more likely to occur when the degree to which the group must attain an extrinsic collective goal is matched by the number of pre-existing structures shared and used by the performers" (2003c, p. 167). For an extrinsic goal, Sawyer gives the example of the task facing a team of businesspeople who need to resolve a budget shortfall by the end of a meeting. He notes that for improvisational theatre groups there is no extrinsic goal; instead the goal is intrinsic - "to perform well and to entertain the audience" (2003c, p. 168). In a mathematics class, a task where a specific outcome is expected to be performed, such as the completion of the questions on a specified page of the textbook, would be considered an extrinsic collective goal. On the other hand, should the students be given an open-ended mathematical problem where no specific solution path was prescribed, the collective goal would be more (although perhaps not entirely) intrinsic. The pre-existing structures Sawyer mentions include the following: a known outline of the performance, predefined roles, conventions, and ready-mades. I will begin by briefly discussing each of these, and then will suggest how they may apply to the taped session.  Outline of Performance According to Sawyer, one of the types of pre-existing structures that may need to be in place for group flow to occur is an outline of the performance that all group  54 members are aware of ahead of time, although there is still room for improvisation within this outline. The more extrinsic the goal, the more structured and less flexible the performance outline will be. Sawyer (2003c) notes how jazz performances in small-group jazz start with a song that has been arranged in advance, and then move into group improvisation that is based on this song. For instance, in playing jazz standards, which have a 32-bar chorus structure, the group will begin by playing the melody of the song "straight" giving the audience a chance to recognize the song. The group then moves into a number of solo improvisations using this 32-bar chorus, with each soloist being the primary improviser during his/her turn and the rest of the group supporting the soloist with their improvisations, either echoing the soloist's ideas or suggesting new ones which he/she might develop further. Another example of performance outline that Sawyer mentions takes place with improv. theatre. In improvisation, groups often perform short games in front of their audiences. The games are usually five minute scenes that have a set of rules that mark the resulting dialogue as being different from regular conversation - for instance, in one game the main actors speaking the dialogue must allow themselves to be physically posed by other actors during the scene, essentially behaving like puppets. The name of the game and the rules are announced by an emcee, who then asks the audience to contribute ideas to help begin the game, usually things like a type of job, or a setting. The actors are well aware of the expected outline of their performance: there is a time limit, their scene must be guided by the rules, and the actors must open with the audience's suggestions. In the case of this particular taped session, the math class's focus was on reviewing fraction concepts that the class had been working on for the past few weeks - specifically, conversions between "improper" fractions and mixed numbers, and the  55 operations of adding and subtracting fractions. The required task involved a package of worksheets, each one requiring students to complete fraction questions that required the repetition of the same algorithm in order to discover a series of letters that could eventually be unscrambled in order to solve a word puzzle (the answer to a corny joke). This task was a very structured one, allowing little room for improvisation. There were a specific number of questions per sheet and each question involved the invoking of the same algorithm to solve it. The answer for each individual question was immediately available at the bottom of the sheet (although the students had to look around a bit), and these answers were further verified by the completion of a word phrase which answered the riddle posed at the top of the page. Solving worksheets like these ones was a regular activity (i.e. it was not set up especially for the videotaping), and the class was familiar with both the math concepts involved and the format of the sheets themselves; Mandy and Hannah were so familiar in fact that they had already started the worksheets before Mrs. Shug had finished giving the class their instructions for the period.  Predefined roles Another pre-existing structure, according to Sawyer, is predefined roles: in order for group flow to occur, each performer must have a clear idea what part he/she will play. For instance, in a small group's jazz performance, each performer takes a turn as soloist and knows that, in that role, he/she will be able to choose the style in which the group will play, as well as take the lead in introducing ideas with which the group will work. In turn, the rest of the group knows that their role is to support the soloist by following his/her lead. But Sawyer (2003c) also points out that with improvisational groups the roles are necessarily flexible as the groups find and solve  56 problems; so the members of the jazz group might also offer the soloist ideas with which to work, although it is at the soloist's discretion whether or not to accept the ideas. As for improvisational theatre, where the roles are possibly less-defined than in jazz, Sawyer interviews actor Pete Gardner about the benefits of rehearsals for establishing trust and habits of interactions between actors. Rehearsals result in "a sensitivity that comes with knowing each other," Gardner says. "[Y]ou wouldn't be as attuned [with a stranger] and you wouldn't be hearing the differences in their voices as they're changing and as they're saying things" (Sawyer, 2003c, pp. 64-65). Within a classroom setting, students experience this same kind of "rehearsal" on a more informal level simply by spending time with and getting to know each other , their 16  relationships and level of interaction always being in process. The classroom session took place in Mrs. Shug's Grade Eight mathematics classroom in early November. The class had been together long enough for students to establish preferred working relationships with various classmates and, during this particular activity, although students had been assigned to work in specific large groupings of eight to twelve people (defined by who had agreed to appear on camera), within those groupings they were allowed to work with whom they chose. According to Mrs. Shug, Mandy and Hannah both achieved good grades in math and frequently chose to work together, although they did not seem to socialize together much outside of class. This suggested that they had developed a good working relationship. Their behaviour towards one another during this session, which will be described in greater detail later, also suggested a high comfort level with one another.  1 6  Many instructors build "ice-breakers" into their introductory lessons for this purpose.  57 Conventions In his work on creativity and systems, Csikszentmihalyi (1988) argued that each creative endeavour is based on a domain which contains a body of shared conventions and techniques. Mathematical researchers such as Cobb (1992) and Voigt (1995) might call this body taken-as-shared, mathematical ideas, conventions, and concepts that are a shared part of the socio-mathematical culture of the classroom. Although it can be difficult to tease apart the idea of conventions from that of techniques, conventions might be defined as being largely social practices or, as Howard Becker (2000) put it in his discussion of jazz improvisation, "etiquette," while techniques or "ready-mades" (Sawyer, 2003c) might be considered chunks of knowledge of one kind or another. To begin with, conventions will be discussed. Sawyer defines conventions as "a set of tacit practices governing interaction in a group" (2003c, p. 168). In his discussion of the etiquette of improvisation, not only of jazz but of actors and even of scientists solving a problem, Becker writes that people take "some elements of what they will do as given, not subject to change during the course of the improvisation, leaving others as what it [sic] will be okay to vary and work with as they perform together" (2000, p. 173). For instance, in a math classroom conventions may include explaining one's work if another classmate disagrees with a particular answer, or helping someone who is stuck on a particular question. These elements, whether in a math class, an improvisation group, or in any group of people who are working together, are rarely openly discussed; they are developed over time, as group members become familiar with one another, and this development is on-going, a process rather than an end product. In jazz, for instance, Sawyer (2003c) notes that musicians are expected to know, and play without rehearsal, a large number of jazz standards; they are expected  58 to know standard chord substitutions and when to use them; they are expected to recognize when a soloist is reaching the end of a solo and act appropriately. He also notes other conventions such as: the leader gets to choose the song and the key of the song; each musician gets to perform a solo; the soloist gets to choose the style for the group while he/she is playing; and no musician should put another in a spot where he/she might be embarrassed or have to perform something he/she is unable to do. Sawyer also cites examples of conventions in improvisational theatre. For the performers, their "standards", are the basic improv. games they learn as part of their training, or techniques like how to indicate that one is ready to take over another performer's part during a skit (i.e. tap him/her on the shoulder). Conventions, both in jazz and improvised theatre, provide a kind of structure that's hidden to the neophyte audience member, and they are rules that "emerge informally in a community of practice, over the years with continual experimentation with what works and what doesn't work" (Sawyer, 2003c, p. 52). In math classrooms, the conventions would vary according to the level of the students and the expectations of the staff and the school's community . For instance, 17  a Grade Nine student might be expected to show his/her work step-by-step for an algebra solution. Or a Grade Five student might be expected to access mathematical manipulatives to help show his/her ideas about solving a problem to his/her group members. Because Mandy and Hannah already had experience working with each other and with the task at hand, they were comfortable enough with the general conventions involved in completing the sheets to actually start working on the sheets independently before the teacher had finished giving the class its instructions. As In the city where I live, certain communities are known to have very strict expectations about the amount of homework their children should be bringing home, while for others homework is less important (and, in some cases, less possible). The teaching staffs have made adjustments accordingly. 17  59 well, the girls quickly fell into a pattern of interaction which suggested that they had already established their own conventions as work partners: each of them worked out the same problem at the same time, and they discussed the question between them as they worked until one would call out an answer, and the other partner would either confirm or dispute this answer. If both partners agreed on an answer they automatically proceeded to the next question; questions that were particularly easy were sometimes not discussed at all and they moved directly to the question following. If there was a disagreement, there was some discussion about the calculations involved until the situation was resolved. If an impasse was reached, the girls appealed to an outside authority (Mrs. Shug) for help, which they did twice during the taping. Then it was back to work. According to Sawyer, for goal-oriented groups "efficiency is a major concern" (2003c, p. 168). As mentioned, the fraction worksheets provided the girls with a very specific goal and structured task to complete in order to achieve it, and certainly their working conventions provided them with an efficient manner with which to accomplish their task.  Ready-mades According to Sawyer, ready-mades in improvisation are "short pre-composed motifs or cliches" (2003c, p. 112) and they are symbolic of shared cultural knowledge. In improvisational theatre, performances ready-mades may take the form of cultural symbols that may be gestural (for example, someone shrugging their shoulders to suggest they don't know), visual (someone holding a steering wheel to suggest he is the driver), or verbal (someone using a well-known catch-phrase associated with a certain celebrity to suggest she is portraying that person). In jazz  60 improvisation, ready-mades may take the form of stock musical phrases - Sawyer notes that musician Charlie Parker was known for having "a personal repertoire of 100 motifs, each of them between 4 and 10 notes in length" (2003c, p. 112) - that musicians might draw on in creating an improvisation. In visual art, one might think of ready-mades as set images that may be arranged in a collage as the artist chooses, depending on what he/she wishes to express. What is important about ready-mades is that they represent units of information or ideas that can be easily shared with the audience, and that they aid in communication. In the case of a mathematics classroom, ready-mades can take the form of shared mathematical knowledge and concepts. In the case of this session, as this was a review exercise, the algorithms of converting mixed fractions to improper fractions (Sample question: 2 7/9 = • / 9 ) , and vice versa, were well-established and thus could be considered ready-mades. There was no negotiation required between the girls about how to do the conversions; the algorithms were accepted and used throughout the session. Mathematical concepts involved in these algorithms included operations of addition, subtraction, multiplication, and division, the definitions of fraction types, and fraction equivalency, suggesting that, in mathematics, ready-mades may be nested in other ready-mades as mathematical concepts build upon other concepts. Again, 18  Hannah and Mandy did not have to negotiate any of these meanings during the session.  A n d certain students may prefer to rely on certain kinds of ready-mades - for example, preferring to use decimals in their calculations rather than fractions - depending on what they find easiest to work with or more meaningful.  61  The Performance of the Mathematical Task Indications of Group Flow Sawyer writes about how in one improvised activity, jazz musicians "trade fours" or "trade eights" - where "fours" or "eights" refer to the number of measures that the soloist plays before the next soloist begins. It's a fast-paced exchange and, rather than start with completely new ideas, each soloist continues playing in the way the previous soloist did, but tweaks it slightly in order to transform it. Writes Sawyer, these performances seem to work because the performers are closely attuned to each other; monitoring the other performer's actions at the same time that they continue their own performance, they are able to quickly hear or see what the other performers are doing, and then to respond by altering their own unfolding, ongoing activity.... [ This] demonstrate^] a form of what is often called interactional synchrony (2003c, p. 37). Throughout the videotape, Mandy and Hannah were focused on the task before them, demonstrating periods of interactional synchrony. Seated beside one another, they were continually writing, stopping only when they needed to discuss an idea in more detail. It seemed at first that they were working independently except for two things: their physical behaviour and their almost constant dialogue about their work. A closer examination of the tape, made it clear that these two girls were always communicating their ideas to one another in some form, to the point where they appeared to be of one mind about the particular sheet that they were working on, a mind that was almost completely dissociated from the other students in their table group. It was the nature of this intense communication that signaled that these girls  62 may have been collectively experiencing flow. I will now discuss the synchrony that was observed, which I have categorized as physical and verbal.  Physical Synchrony In her work about mathematical conversations, Gordon-Calvert (2001) discusses how students demonstrate their intentions to work together on an activity in part through their gestures and physical postures, and her ideas help to flesh out what physical synchrony might look like in the setting of a mathematical classroom. As Mandy and Hannah each record calculations on their sheets which they later refer to, they point to one another's sheets as well as their own, they count-off visually using their fingers, and they continuously discuss with each other the work in progress. Nonverbal communication requires a lot of visual interaction. Sawyer notes how musicians and actors both use visual attention to establish group flow. Musicians observe each other carefully which helps them to anticipate what comes next - he gives the example of how a pianist might raise her arms before playing a certain type of chord (2003c, p. 45). And actors, while improvising a scene, maintain steady eye contact, even when performing actions that in real life wouldn't require that kind of visual attention. Similarly, Mandy and Hannah, although seated beside one another, frequently glance at each other and the worksheets, directing a steady stream of attention to one another as they work. This eye contact seemed to help keep them connected and working on the same questions at the same time. Their tablemates, in contrast, do not exhibit this kind of behaviour. The girls also demonstrate what Gordon-Calvert refers to as the "togetherness" of their working relationship (2001, p. 95) through their physical proximity. Physically Mandy and Hannah were sitting very closely, their shoulders a couple of  63 centimetres apart and often actually touching, the intimacy of the space suggesting that they had a high comfort level working together. When the girls had more focused discussions, their physical proximity increased - they turned towards each other, their faces only a few centimeters apart, their shoulders touching, their outside arms either encircling their worksheets or perhaps gesturing to their partner's paper. In fact, Hannah seemed to use her binder as barrier between herself and the students on her left, further protecting the space she shared with Mandy. Mandy, who didn't set up any barriers, was more likely to interact with the other students sitting at the table during the session. As they did their mathematics, Mandy and Hannah's physical work appeared to be synchronized. Each girl's writing frequently stopped and started at the same time as her partner's; occasionally, when they both realized they had made a mistake, their erasing occurred at the same time as well. The echoing of gestures continued further - their working posture matched, they tended to follow each other in rocking back and forth in their chairs, and they occasionally performed grooming gestures (i.e. pushing hair back out of their faces behind their ears) simultaneously. Viewing the video tape at four times faster than the regular rate provided a more global perspective (or a sense of greater distance from the situation), making this physical synchrony even more striking. In his microanalysis of films of the behaviour of listeners in the mid-1970s, William Condon noticed a startling synchrony between a listener's movements and the articulatory structure of what the speaker had said (Clayton, 2003, p. 309). His ideas suggest how the continuous dialogue between the girls (to be discussed in the next section) may have contributed to the synchrony of their movements.  64 Verbal Synchrony Not only can people's physical behaviour be observed to be in sync, but so can aspects of their speech. In the case of Mandy and Hannah during this session, the synchrony of their verbal behaviour was also striking. In this section, I will be analyzing a portion of the transcript from the session using ideas from Forman and Ansell (2002), Davis and Simmt (2003), as well as Gordon-Calvert (2001) to point out ways in which the girls' speech was synchronous as well as how it suggests group flow is occurring between them. Because the episode took place in a classroom containing approximately 32 students and three adults, many of them talking, the girls' dialogue was sometimes difficult to hear on the video and thus the transcript of their dialogue makes their talk seem more fragmented than it actually was.  Double click on icon for video clip of transcript below  & transcript clip  Transcript  [1] [2] [3] [4] [5]  Mandy: . Yep. 'kay now, 6 goes into 31 five time... five times Hannah: (39?) (inaudible) Mandy: 21 with a remainder of 8  Commentary The girls are converting improper fractions to mixed fractions on a puzzle sheet: for each question on the sheet the puzzle letter, e.g. O, is then followed by the actual fraction to be converted, e.g. 31/6. Each question that the girls work on appears in bold below.  Question:  O  31/6  Hannah: ... and 12 (inaudible) Question: Mandy: You have to divide that down... you divide it by 4, you  N  20/12  65 divide that by 4 and get 2/3, and one and 2/3... Right there... that's what you get... 16 goes into 26 once. Hannah: Once? Mandy: Now you've got to... that's 10 times so that's 10/16 so you divide by [8] Hannah: 2 [9] Mandy: You're... 2 [10] Hannah: Yep. [6] [7]  [11] Mandy:  Question: N 26/16 [5] Mandy makes afingercounting hand movement as she does "16 goes into 26 " [7] Finger counting hand movement.  [10] points at sheet Question: D 6/4 Points at Mandy's sheet again.  8... (inaudible) One and one half... That would be...  Question:  O  28/7  7 goes into... 4 times.  Question:  H  45/15  I know this one. No. Hannah... [12] Hannah: 4 [13] Mandy: No. That goes into 45 three times [14] Hannah: That goes into 12 goes into 9 twice, 12 goes into 9... and once [16] Hannah: Once with 3 left over. [17] Mandy: 1 and 1/3. 1 and 1/3  Question: N  12/9  [15] Mandy:  [17] writing  In this episode, unlike the conversation of others sitting at their table, Mandy and Hannah's discussion rarely strayed from the math at hand. In fact, they were focused on the same question at the same time, suggesting a similar level of concentration. And they employed a similar vocabulary to express their mathematics (for example, in the exchange above, "goes into," and in other passages "times") suggesting a shared discourse. That in their discussion one girl was as likely to come up with answers as the other suggested that they were each finding the math work equally challenging.  66 Their syncronization was further shown in how Mandy and Hannah echoed each other's words, completed each other's sentences, and occasionally uttered the answer to the calculation on which they've been working. Transcript clip (doubleclick on icon)  & C:\Documents and Settings\laynie\My Dc  Commentary The girls begin bottom section of same worksheet which involves writing each mixed number as an improper fraction. Each question that they work on appears in bold below.  Transcript  Question: H [23] [24] [25] [26]  Mandy: Hannah: Mandy: Hannah:  2 2/3  That's 8, 8 over 3 Are you sure? Yeah. It's right here. H . . . Question:  T  2 3/8  Question:  U  2 11/24  [27] Hannah: 8 x 2 = 16... 19.. .{inaudible)... [28] Hannah: 24 times 2 is 48 plus 11 is... 59 [29] Mandy: Yep [30] Hannah: 59... over 24... (that 100?) [31] Mandy: Oh, that would be 9 over 2 is [32] Both: [33] Mandy: At the N (end?)  [34] Hannah: Y e p N . [35] Mandy: N  Question: A 4 'A [32] Simultaneous speech [33] Mandy indicates where to write the puzzle letter on the sheet. [34] Echo of "N. " [35] Another echo of "N. "  [36] Both: 5 times 9... 50 over 9 [37] Hannah: Right here after the U . . . close after the U . . .  Question: D 5 5/9 [36] Simultaneous speech  [38] Hannah: 7 plus 5 [39] Mandy: 12 over 7 is here  Question:  [40] Both:  Question: T  17... 17over 10 is...  O  1 5/7  1 7/10  67 [40] Simultaneous speech [41] Hannah: 19 over 12. It's right after 0...(inaudible)... Right? [42] Mandy: Yep. [43] [44] [45] [46]  Hannah: Both: Hannah: Mandy:  9 times 4 is 36 It's right below Yeah, I got it. P.  [47] Mandy: 2 over 7 right after that [48] Hannah: Right after the beginning  Question: E  1 7/12  Question: P  9 A 3  Question: O 3 1/7 [48] Echo of "right after. " They don't discuss the next question H 3 %  [49] Hannah: 5 times... 23 over 5 [50] Mandy: It's at the end  Question: A  4 3/5  [50] Mandy is referring to where on the sheet to fill in the answer. [51] Mandy: 37 over 5 [52] Hannah: It's right near the beginning  Question: E  7 2/5  [52] Hannah is referring to where on the sheet to fill in the answer [53] Mandy: What's 4 times 16? Um it's going to be 4 more than... 64!  [54] Hannah: 64?  [55] Mandy: I'm pretty sure. [56] Hannah: Yep. It's 64.  Question: Y 4 3/16 [53] Mandy stops writing and seems to address the entire group. Starts writing after she finishes speaking. [54] Echo of "64 " as Hannah appears to be writing a calculation at the side of the worksheet, perhaps to confirm the validity of the answer. [56] Hannah continues writing, then when she confirms it's 64, she echoes "64 "one more time and then erases what she has.  Each girl's talk not only echoed the other [(33, 34, 35); (47, 48); (53. 54, 56)] but there were times when they spoke simultaneously (32, 36, 40, 44). Davis and Simmt  68 note, "Sameness among agents - in background, purpose, and so on - is essential in triggering a transition from a collection of me's to a collective of us" (2003, p. 150). Within their collective, the girls felt free to think aloud, exchange ideas, and occasionally argue (lines 88-90, cited later) their way through their math. They were working within a collaborative zone of proximal development, defined earlier in this study as a collective intellectual space in which students can share ideas, in this case the correct calculations of fraction conversions. Revoicing (Forman & Ansell, 2002) was another way that Mandy and Hannah seemed to develop and maintain their collaborative zone of proximal development and it was used by the girls in a few different ways. It seemed to be used to repeat an idea so that the listener could simply hold it in her mind longer, which was, at times, what Hannah and Mandy appeared to be doing when one of them repeated what the other had said at the same time that she herself wrote it down (33-35). Revoicing also seemed to be a way to acknowledge that the second speaker had heard the first speaker's idea (and, in this case, questioned it), and then to show that the second speaker supported this idea (Forman & Ansell, 2002, pp. 53-56). Finally, revoicing was employed by the second speaker as a launch pad from which to then elaborate upon or criticize the idea of the first (Forman & Ansell, 2002), as Hannah seemed to do later in the session when she questioned Mandy's calculations.  Double click on icon for clip of transcript below  & Second transcript clip  Commentary  Transcript [88] Mandy:  38 thirtieths, (inaudible) thirtieths. Divide by... 1 and 2  [88J As this clip opens, Mandy  69 fifteenths...  [89] Hannah:  1 and 8 thirtieths.  [90] Mandy:  1 and 2 fiftheenths. See I told you... Whoa. Why do I have 1 and 2 fifteenths again? I know. It's not 1 and 2 fifteenths, (inaudible)  [91] Hannah:  [92] Both:  4.  [93] Mandy:  4 fifteenths. I divided it wrong Hannah. 1 and 4 fifthteenths.  [94] Hannah:  is saying her calculations aloud and then repeating "38 thirtieths " to herself as she writes. Hannah is working with her fingers as she does her own calculations. [89] Hannah stops writing as she says this and then continues writing again. [90] Mandy stops writing as she notices her mistake. [91] Hannah stops writing There is a bit of inaudible conversation between them as they both stop and think about the calculation. [92] Simultaneous speech. They both start using their pencils again and stop at pretty much the same time.  Through revoicing, Mandy and Hannah seemed to be able to provide each other with the frequent feedback required to facilitate the flow of communication between them. Gordon-Calvert's (2001) notion of re-pair  19  may also be considered as a way  of maintaining the collaborative zone of proximal development. Re-pairing appeared frequently in Mandy and Hannah's dialogue, seeming to be a way of reconnecting with each other after they had been working separately, in general with Hannah tending to re-pair by asking questions ("Are you sure?" and "Right?") while Mandy re-paired by addressing Hannah by name ("You have to add these two together Hannah"). The exchange cited above occurred after a period of frustration when they had been working largely separately for a couple of minutes. Mandy seemed to use  Gordon-Calvert defines re-pair "a purposeful effort... made to reenter into conversation with one another - to pair again.... Reestablishing their interactivity through questions addressed to the other, such as, "What'd you do?" and "Do you know what I mean?" act as occasions to re-pair and to reenter into conversation with one another" (2001, p. 97). 1 9  70 re-pairing in both her question and her use of Hannah's name as a way of reestablishing their working relationship.  Overview of Session Involving a Structured Mathematical Task This section of my results focused on a pair of girls in a Grade Eight math class completing puzzle sheets as a review exercise. They were familiar with the format of the sheets, (suggesting a known outline of performance), they were familiar and comfortable with one another (suggesting predefined roles), they had an established working rhythm that was evident as soon as the teacher introduced the task (suggesting shared conventions), and they knew the math algorithms and skills required to do the calculations (suggesting ready-mades): all examples of pre-existing structures according to Sawyer's model of group flow. The girls could have been working independently, as the rest of their peers in the class were, but they weren't ; they'd chosen to work on the sheets jointly and 20  their behaviour as they did so suggested that they were experiencing group flow. Mandy and Hannah's physical behaviour indicated full concentration, which is an element of flow. Their body postures, close physical proximity, and lack of attention to distractions all suggested that they'd isolated themselves from the rest of their tablemates and were intensely focused on each other and their work. Their physical behaviours (echoed gestures, frequent visual attention) and speech patterns (shared vocabulary, completion of each other's sentences, repeated words and phrases, and simultaneous speech) suggested not only an intense and continuous communication  Sawyer writes, "Some combinations of performers are more likely to attain group flow, others less likely, and this remains true across many different performances" noting that these performers have what's known as chemistry (2003c, p. 48). Exactly why Mandy and Hannah were the only students in the class who chose to work together throughout most of the period, and in such a close manner, is a good question but one that lies beyond the scope of this study. Please see Conclusions for ideas as to future research about group flow. 2 0  71 but a like-mindedness about their task. They seemed to have found, as a jazz musician might put it, their rhythm. Their joint delight in telling the rest of their group the answer to the first puzzle and the immediacy with which they begin work on the next worksheet suggested that they had found the challenge level of the work satisfactory (i.e. not too difficult yet not too easy) and were interested in continuing, yet another indication of flow . 21  That they were able to achieve group flow suggests that the conditions of their activity - the balance between the highly structured task and the extrinsic goal of completing the puzzle sheets - were favourable for achieving a one-mindedness in pursuing this activity.  Later in the session, the girls appeared to derive much satisfaction from the discovery that while they were now on the third worksheet their tablemates were only part way through the second one. 2 1  72  Section 2: A More Open-Ended Mathematical Task  Intro clip for Section I (click on icon)  22  C:\Documents and Settings\laynie\My Dc  Within a small group of four students, Anik and Sandi have been struggling with an open-ended mathematics question. Suddenly Anik appears excited and introduces a new idea; Sandi becomes equally excited as they check and double check to see if the idea works. Their group mates become roused by the activity, but Sandi and Anik focus on only each other as they work, passing a sheet back and forth between them, speaking more quickly and in an increasingly fragmented fashion, turning slightly away as if to shut the other two girls out. Finally, they both lean 23  back in their chairs, smiling. A s Sandi begins to point out, they don't have the answer yet but - and here Anik interrupts - they now know how to get it. Anik waves her arms in the air and pounds her desk with her hands in triumph.  " The audio portion of this videoclip is not good - the recording of this session was supplemented with a separate audio recorder - but it shows Anik (wearing red) and Sandi's (in pink) physical movements as they arrive at group flow (372-395 of transcript, cited on page 86-88). Crystal is wearing black and Khona is in gray. This latter part does not occur in the clip attached. 2 3  73  Although it has taken a lot longer to get there than it did for Mandy and Hannah, Anik and Sandi appear to have reached a state of group flow in their work together on a mathematics problem. According to Sawyer's model, a collective only has the possibility of reaching group flow when there is a balanced relationship between the number of shared structures and the nature of the collective goal. In the last section, Mandy and Hannah were completing fraction worksheets, a highly structured task with a specific extrinsic goal, and as they did so, their physical and verbal behaviours suggested an experience of group flow. In this section, a session will be described where a slightly larger group of students was given a more openended problem, one where the goal was less extrinsic and there were fewer pre-set structures in place. According to Sawyer's model, group flow should still be possible since the shift in the nature of the goal towards being more intrinsic would balance with the reduction in the number of shared structures, meaning that the students are still in a situation where they might achieve group flow. After using Sawyer's model to begin the analysis of this session, I will then turn to Davis and Simmt's outline of the conditions required for the development and maintenance of collective understanding in order to make connections between the experience of group flow and the growth of collective understanding.  Sawyer's Model: Pre-Existing Structures Outline of performance The session took place in the researcher's own classroom in the spring of the school year. Each student in the class (including the members of the group observed) received a sheet of paper with the Locker Problem. The problem read as follows:  74  A newly built school has 1000 lockers, numbered 1 to 1000, located down one side of a really really long hall. The lockers are unlocked and the doors are shut. A student walks down the hall and opens the doors to all 1000 lockers. A second student follows and closes the doors of the even numbered lockers. A third student changes the position of the doors numbered 3, 6, 9, 12, .... (that is, opens closed doors and closes open doors). A fourth student changes the position of doors 4, 8, 12, 16,.... This process continues until the thousandth student changes the position of the door of the locker numbered 1000. Which doors are open? The class had worked on these Problem of the Week style problems regularly (although not weekly) throughout the school year so the format was a familiar one. Students were expected to write a description of their thoughts in attempting to solve the problem, and the custom was that the students were given time in class to work together towards a solution before (or while) doing this write-up, and were expected to finish the write-up at home. In the regular class setting, the students were given the choice about with whom and where in the classroom they worked, and had the option of changing partners at any time. However, due to the restrictions of who had agreed to be videotaped for my study, the expectation during this particular session was that students were to work with those with whom they were seated and remain at that location. There had been other tapings previous to this session, so students had become familiar with this change. Each Problem of the Week question was unique: they were unrelated to one another, nor were they necessarily related to the content in the students' course of study in their current mathematics unit. Thus, unlike with Mandy and Hannah's task, there was no expectation that students should use particular facts, methods or strategies in determining their solutions to the Problem.  75  Pre-defined Roles The videotape of this session focused on a group of four Grade Eight girls. Two of them (Sandi and Anik) were close friends, a third (Khona) also hung around with the first two, and the fourth (Crystal) had some social connections with the other three both inside and outside of school and appeared to be well-liked by them. The four students were comfortable working together and their personal dynamics were positive before, during, and after the session. The Problem of the Week did not involve the assignment of specific roles that one, for instance, might find in more traditional cooperative learning groupings ("leader," "scribe," and so on), so the students were free to take on and cast off roles informally as the session evolved. The girls had not been grouped together in mathematics class before the session so they had not had time for their roles within the group to evolve prior to the taping. Researchers such as Lloyd and Cohen (1999) and Yamaguchi (2003) have noted that students are well aware of each other's academic status within a classroom, and this seemed to be the case during this session. Both Sandi and Anik had a history of high grades in math and math concepts appeared to come easily to them; Khona's marks were generally good and she too seemed to grasp math concepts without much difficulty; and Crystal's math grades were also good although she worked very hard to get them and sought extra help from the teacher on a regular basis. Thus it was not surprising that Sandi and Anik led most of the discussion, and Crystal spoke the least. However, the emphasis of the Problem of the Week was on process and communication, and each of the students took turns leading the discussion at different times during the session.  76  Conventions As described in the last section, conventions are practices governing social interaction. Some of these likely pass unnoticed, since they derive from the students' past experiences together both in school (Had they been in the same class in previous years?) and in the community (Had they been on the same soccer team? Did their parents socialize with each other?). A more obvious source of conventions would result from being in the same class with the same math teacher for most of the school year resulting in familiarity with each other, the teacher and her expectations, the class as a whole as it developed its personality (Bowers and Nickerson, 2001), and familiarity with the environment and available resources. As well, the students had been videotaped previously and knew what to expect and how to behave in front of the camera, although for this particular session the presence of the audio tape recorder was new. The students also appeared to be very familiar with the practices of cooperative group work, which had been a teaching focus of their school for the past few years. They took turns speaking, contributing ideas, and asking questions, and each was willing to do her fair share of the work. There were occasional pauses for thought, or to complete some independent work, but each student seemed aware that the purpose of the session was to work together and to bring ideas from this independent time back to the group. Even Crystal, who at times seemed unable to keep up with the group's discussion, listened attentively throughout most of the session.  77 In this session, some of the conventions were provided by the written instructions for the Problem of the Week, which were the same as those for each Problem assigned throughout the year: For this problem, provide the following: Problem Statement Write a concise statement of the problem. Write clearly enough so that someone picking up your paper could understand exactly what you were asked to do. Plan Tell what you did to prepare to solve the problem. How did the problem seem to you when you first read it? Consider what you are asked to find, what you know, what you need to know, and what strategies you can use. Is this a problem like any others you've done? Before you begin to work on the problem, make a guess at the answer to the problem. Work Explain in detail what you did to solve the problem. Use charts and graphs where appropriate. Tell what worked, what didn't work, and what you did when you got stuck. Did you get help from anyone? What kind of help? Answer State your answer(s) to the problem. Does the answer make sense? Could there be other correct answers? Compare your final answer with your original guess. What did you learn from this problem that could help you to solve other problems?  24  The emphasis in all Problems of the Week was on the problem solving process: being aware of the origin of one's ideas, and being able to communicate these ideas clearly and logically. This emphasis was reflected in the manner in which the students approached the problem. For instance, fairly early in the session the students offered predictions about how many lockers would be open, a convention that had been promoted by the instructions for the problem. Through their experiences with Problems of the Week, the students had developed the convention of recognizing the  from Tsuruda(1994)  78 need to find a strategy with which to deal with situations that could potentially involve a lot of calculations. For example, in this problem the girls established early on that they did not have to write out all 1000 lockers in order to find a solution. This was something they reminded themselves of several times throughout the session, particularly when they were trying not to be discouraged by the apparently large scope of the problem. Related to this was the strategy of looking for a pattern, a convention that the girls mentioned a few times during the session.  Ready-mades Ready-mades are cultural symbols and chunks of knowledge/information that are already agreed upon and can easily be shared between people. In the case of this session, ready-mades included mathematical concepts that students had previously been taught - including factors (which they consistently mislabeled during this session as "multiples"), prime numbers, and fractions. Unlike Mandy and Hannah's session where the worksheet questions evoked specific ready-mades (the relationship between an improper fraction and a mixed number for instance), these ready-mades were not specifically prescribed for doing this question but were brought up by the girls themselves as they sought different ways to approach solving the problem. These ready-mades also provided some of the shared vocabulary that the girls used ("multiples," "prime numbers").  Nature of Goal The girls had fewer pre-existing structures to work with than Mandy and Hannah had with their fraction worksheets, but correspondingly their goal was less extrinsic. From the written description of the question, they knew they needed to find  79 the specific number of locker doors that were open, and they knew that all the information they needed to work with was contained within the description. Beyond that, they were free to work as they wished - there was no set number or type of calculations to perform, no specific letters to unscramble. The girls were problem solving, and in the process, they were problem finding as well, as they discovered and dealt with smaller problems on their way to solving the main problem posed by the question sheet. The students' goal was still extrinsic, but the means by which they 25  reached it were far more flexible. Thus, according to Sawyer's model, the group was potentially in a position to experience flow - the reduction in pre-existing structures, compared to Mandy and Hannah's task, was balanced by a reduction of how structured, or "known," the goal was.  Characteristics of Group Flow In this session Sandi and Anik appeared to experience group flow but it did not appear quite the same as Mandy's and Hannah's experience. Part of the reason for this difference, aside from the students' different personalities and working styles, is that due to the nature of the problem they were solving, the locker group's collective zone of proximal development was far less static than Mandy's and Hannah's was. As the girls found and solved different smaller problems, the nature of the larger problem shifted, their collective zone of proximal development shifted accordingly, and the level of collective understanding of the group needed to be renegotiated. To consider this further, it is helpful to situate the girls' session using the conditions for collective understanding as outlined by Davis and Simmt (2003) - redundancy, decentralized  Although Sawyer argues that problem finding and problem solving are dichotomous - problem finding involving searching for a problem to solve in the way, for example, that Picasso improvised on his canvas, and problem solving involving "starting with a relatively detailed plan for a composition and then simply painting it" (2003c, p. 106) - in this session the two were intertwined. 2 5  80 control, internal diversity, organized randomness, and neighbour interactions - as they help to expand some of the ideas introduced in Sawyer's model of group flow. A s each of these conditions is outlined, excerpts of the transcript of the Locker Problem session will be used to provide an illustration. Following this, I will begin a more chronological discussion of the session transcript to show the kind of negotiations that took place within the group as it moved towards the potential of group flow.  Davis and Simmt's Conditions for Collective Understanding Redundancy According to Davis and Simmt, redundancy refers to "duplications and excesses of the sorts of features that are necessary to particular events" (2003, p. 150) and they note the importance of this kind of sameness within a collective: it allows group members to interact with one another and "maintain coherence" (2003, p. 150); and, should a member somehow fail in a task, it enables another member to make up for it. Davis and Simmt note the importance of the kind of sameness that is developed through redundancy - because it helps group members develop and maintain a common ground, it is "essential in triggering a transition from a collection of me's to a collective of us" (2003, p. 150) Redundancy was a crucial factor in the first half of the Locker Problem session, when the students were determining what exactly was going on in the problem, to enable meaningful interactions between them. Transcript  Commentary Teacher has justfinishedreading the problem aloud to the class and has said that she '11 give students some time to discuss it before she asks for general questions. Sandi, Anik, Crystal and Khona are sitting at a cluster offour desks. Crystal is writing, others in group are sitting.  81  [1] [2]  [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]  Sandi: I don't get the question Anik: But she just said that, um ... open all the even numbered lockers, then it says the third student closes all the open doors and opens all the closed doors, then he opens the closed ones. Sandi: Nooo Khona: What? Sandi: Okay, First... Khona: Oh, so he closes Sandi: First student...Okay, first student opens them Khona: Sandi, does this mean he like, does the first student open them? Anik: First student opens a0 of them Sandi: Yeah, so he opened all of them Unknown: Second student Unknown: I think I got it. Unknown: Closed all of the uneven ones Sandi: All of the even ones Khona: Uneven ones Unknown: No he closed the even ones Sandi: Then the third one, Khona: Oh  [8] Echo [9] Echo [10] Echo [13] Crystal starts watching conversation  Anik began by trying to explain to Sandi. Sandi cut her off and started to talk it out for herself, essentially making the same point (a redundancy), despite interruptions from Khona who was also having difficulty understanding the situation. Sandi summed up the problem: Commentary  Transcript [19]  Sandi: Um, numbers 3, 6, 9, 12, you know up like that, if they were open he'd close them, and if it were closed he'd open it, whatever those numbers are.  Anik asked a question to ensure that she understood what Sandi was saying. Commentary  Transcript [20] [21]  Anik: Like the 3? Sandi: Yeah... multiples of 3  82 [22]  [23] [24] [25] [26] [27]  Sandi: Yeah. And then fourth student multiples of 4 change the position to either... Anik: And then there'd be the multiples of 5 [23] Idea of multiples is Sandi: Yeah repeated. Anik: Six Sandi: And the multiples of every other [26] Idea of multiples. number, up to 1000 Anik: Oh my god  At this point, thanks to verbal repetitions/reiterations of the ideas, Anik and Sandi both appeared to share a common view of what was happening with the position of the locker doors, although occasionally during the session one would say "closed" or "open" when she actually meant the opposite. This would give them a common base on which to build. Crystal had more difficulty grasping the situation, as shown later in the session.  Transcript [83] Crystal?: [84] Sandi: [85] Crystal: [86] Sandi: [87] Crystal: [88] Sandi: [89] Crystal: [90] Sandi:  Commentary We have to do 10 and then divide it by 100. There's one open locker and then there's 2 and Open all of the even ones. So it's closed... it closes all the even ones. Oh right. The third one. Opens. No no no no, don't, (laughs) Closed, open.  [86] Idea repeated.  [89] Crystal is leaning towards Sandi and trying to help her with the diagram. [90] Sandi indicates that she doesn't need help and then goes back to her diagram and starts verbally listing which lockers are open or closed.  Sandi and Anik explained the situation to her (while, at the same time, continuing to talk to each other about another aspect of the problem [149-151]).  83 Transcript  Commentary  [143] Crystal: [144] Sandi:  Why is it closed? Because number 7, 17, number 1 came up and opened all of them. [145] Anik: The 17 kid came up to 17 and closed it. [146] Crystal: Why did he close it? [147] Sandi: Because because he had to change its position. [148] Crystal: Ohhh, I got it, yeah. [149] Sandi: So right now [150] Anik: So they would all be closed? [151] Sandi: Open closed open (...etc) th  [152] Crystal:  So would the fifth change? Do the switchy thing with all the fifth lockers? [153] Sandi: Yeah. Every fifth locker. [154] Crystal: And then they'd do the same thing with the sixth locker? [155] Anik: Yeah. [156] Crystal: Whoa.  [147] Idea repeated.  [151] Sandi is audibly listing the lockers in her diagram again. [152] Crystal repeats idea in own words.  [154] Crystal repeats idea in own words.  After this explanation, and another iteration of the idea, Crystal put it in her own words first (yet more redundancy) but she finally seemed to understand it, as demonstrated by her reaction of "Whoa" (156) and her comment slightly later about how if the question were only asking about the position of each locker door it would be much easier to solve ("I wish the (inaudible) open or closed" [161]). Khona also required the backup that redundancy provided. A t first she appeared to fully understand what was going on with the positioning of the locker doors as soon as Anik and Sandi did because she moved on to the idea of a pattern of multiples. Later, however, when she was caught up in the idea of a fraction pattern and had moved away from the information given by the question, she too needed to have the explanation of the locker door positioning repeated to her. Transcript [357] Khona:  Commentary Number one opens all of them,  84  [358] Sandi: [359] Khona: [360] Sandi:  second one opens half of them, third one the [sic] fourth of them, the fourth one opens the [sic] eighth of them. No no no, that's not what it means. Yes it is. [360] Repetition of door That's not what it means. The position idea. second person changes the position of all the even lockers. It doesn't mean they just open it. If number four's already open they'll close it.  In these excerpts, it is evident how redundancy, through repetition/reiteration, helps to keep all members of the collective together so that they can build a common level of understanding. Redundancy makes it possible for one individual to compensate for another's failings. For instance, if one is unable to express an idea clearly, another member can back her up by saying "Oh, I know what you mean," and then proceed to explain it to the rest of the group in another way, just as Sandi did [19] when she summarized the problem's situation in her own words. Redundancy also can help someone hang onto a new idea. Towards the end of the session when Sandi was working with Anik on a new idea about the number of factors, there was a reference to the slippery nature of new ideas. Transcript [387] Sandi: [388] Sandi:  [389] Anik: [390] Anik: [391] Sandi:  Commentary Okay, if you go to number 30 and find out how many /numbers can go/ into that, then that's the definition, 'kay wait, so, so, so wait, if it's even, what did you say? /numbers go/ If it's even, wait, write this down. I'm losing my thought Anik.  [388 & 389] Overlapping speech indicated with // lines.  85 The direction to write something down that was about to be said may have demonstrated an awareness of how repetition in another form (in this case, written as opposed to spoken) can function as kind of a safety net. A little later, even after they had talked it through, Sandi suddenly did not seem sure about the reasoning and was careful to double check. Transcript  Commentary Anik is writing, the other girls are sitting and discussing.  [440] Sandi:  A couple of those sheets that shows, like, up to 1000 what those numbers will, all prime numbers will be closed. [441] Khona: So all prime numbers are closed. [442] Anik: All. [443] Sandi: No. [444] Crystal: What? [445] Unknown: All lockers. [446] Sandi: Are we sure about that? [447] Crystal?: All prime numbers. . [448] Khona: That'dbe 200 and 3. [449] Sandi: [450] Anik: [451] Sandi: [452] Anik: [453] Khona: [454] Sandi: [455] Anik: [456] Sandi:  But that's only because 1 and itself goes into it. Yeah. So that's just the same as the other thing. Yeah. So if all prime numbers are closed that leaves 760 lockers. Wait.. .even numbers is that followfing] what we said? Yeah. Okay good, just making sure there. Whew. Okay.  [448] Khona starts arranging sheets on desk.  [456] Sandi leans back in chair.  Again, restating arguments appeared to help Sandi assure herself that her new understanding matched that of her group. Finally, conventions and ready-mades, which were used throughout the session, are two aspects of Sawyer's model that would fall under the condition of redundancy, as both are things that were shared between members. For instance, there  86 was no negotiating between these students about whether or hot to look for a pattern, or whether they should only consider a sample of the lockers rather than all 1000; there appeared to be a shared assumption that these conventions were the way they should proceed. This was the same with such ready-mades as prime numbers, "multiples," and fractions - the definitions were largely unspoken (although there was a bit of discussion about the correct name for prime numbers). This kind of unspoken agreement helped to build a collaborative zone of proximal development within which they could interact.  Decentralized Control Decentralized control, as the name suggests, is when there is no one in charge of the group or the activity. In this session, there was no coordinating agent for the group. The central authority of the teacher, the traditional organizing figure of the classroom, was minimized as much as possible (although it was still present, which was evident when I called the class together for two general discussions, one to introduce the problem and a little later on to clarify what was happening with the position of the locker doors as each student in the problem traveled down the hallway). I was not immediately available to the group, although in time I did respond to the signal of a raised hand, and when I was with the group I attempted to minimize my input. Transcript How many prime numbers are therein 1000? [233] Teacher: I don't know off the top of my head. I should know but I don't [234] Anik: You should know them. [235] Anik: Do you know whether or not this equation has anything to do with prime numbers? [236] Teacher: Do I know whether or not it  Commentary  [232] Anik:  [236] After speaking, Teacher  87  [237] Sandi:  does? (smiling)Yeah I do know whether or not it does. Does it? That's the prime numbers up to 20. So it's closed, closed, closed (etc)  leaves to help another group. [237] "Does it" is called after the teacher. Receiving no response, Sandi continues working with group  This limited teacher input was not just for this study but was the way classroom Problem of the Week sessions had been run all year , so it also functioned as a 26  convention. In this session, no specific roles were assigned to the students for the solving of this problem, and this was evident in how the students took turns suggesting ideas, and criticizing or encouraging the ideas of others. There were no preplanned steps; the "system" decided what worked and, as a result, some ideas were adopted, others rejected, and still others later resurfaced to be tried again. The collective understanding of the group emerged as the session proceeded and could not have been predicted, never mind controlled by any centralized agency.  2 7  Internal Diversity Diversity is always present in a group of people, no matter how homogeneous one attempts to make it, simply because the group will always be made up of different individuals who have different experiences, ideas, and ways of thinking. This diversity has its benefits. As the Locker Problem was a new type of question for the students, with no set procedures or past experiences from which to work, internal diversity played an important role in this session, evident in the variety of ideas which were explored "Far from being seen as a matter of decentralized control, mathematics is more often regarded as the embodiment of regimented, top-down, knower-independent knowledge" (Davis and Simmt 2003, p. 153). Problems of the Week have been structured to try to work against this tradition (Tsuruda 1994). Davis and Simmt further argue that "the phenomenon at the center of each collective is not a teacher or a student, but the collective phenomenon of a shared insight, similar to what Bowers and Nickerson (2001) call a collective conceptual orientation" (2003, p. 153) 2 6  2 7  88 factors/multiples, fractions, and prime numbers - and the variety of manners in which these explorations took place. A s was mentioned previously, the four students differed in their mathematical abilities, but they also differed in their approaches to math. One of the ways this was demonstrated was through their work styles. Anik tended to work with the problem's text, reading and rereading the question. When Sandi asked the group i f anyone had a way to do the question, Anik promptly picked up the question sheet and replied, "I can do the easy thing - highlighter, anyone have a highlighter? I'm going to highlight all the useful information" (65). Both Crystal and Sandi preferred to draw out the situation, with Sandi ultimately creating the drawings that the group worked with to establish patterns, while Crystal's drawings and other written work were used by her alone. Khona's preferred method seemed to be listening to and questioning other's ideas and then talking these ideas through, which sometimes forced her group members to consider their own thoughts more carefully. None of these styles was privileged over the others; all contributed to the process of working through and building a collective understanding of the mathematics of the problem.  Organized Randomness According to Davis and Simmt, organized randomness is a structural condition which helps to provide a balance between redundancy and diversity, a balance required for collective understanding to emerge. The rules for this session were both explicit and implicit, both social and mathematical. The students were restricted to working within their group and could only receive a very limited amount of help from the teacher. They had to work within the boundaries of the problem, using the information provided. They needed to explain their ideas clearly to their group  89 members and to negotiate possibilities in a fair way. They were aiming to find a specific answer, and their solution had to make mathematical sense. However, the students had a number of possibilities open to them as well: students could play whatever role(s) they wished to within the group, they could use whatever resources were available, and they were free to approach the problem from whatever direction and with whatever strategies they pleased. Some of the rules - and possibilities for that matter - were predefined, while others were negotiated within the group throughout the process, particularly those involving roles and strategies. Whether explicit or implicit, predefined or negotiated, social or mathematical, the rules provided a foundation for the task on which the students could then use their creative energies to build.  Neighbour Interactions Neighbour interactions are not people bumping into each other, but ideas bumping into each other in the form of statements, questions, predictions, intuitions, and other forms presented by the students as they negotiate their understanding, each form highlighting different interpretations and representations of the information provided in the question. In order for that to happen, students need to have access to each other's ideas as well as feel comfortable and confident enough to share their own and offer interpretations of others. Furthermore, there must be a "sufficient density" (Davis & Simmt, 2003, p. 157) of such interactions, Davis and Simmt argue: "concepts and understandings must be made to stumble across one another" (2003, p. 156) in order for "rich interpretative moments" (2003, p. 157) to occur. In this session, the high comfort level the girls appeared to share with each other helped to provide a safe collaborative zone for the interaction of their ideas, and certainly they  90 shared a variety of them. The development of this zone and the results that followed from that will be considered next in a more chronological discussion of the session transcript.  Performance of the Mathematical Task  Establishing a Collaborative Zone of Proximal Development The session began in a situation very similar to that of an improvisational session; once they felt they had established what the question was about, the girls looked for a way to begin. Transcript  Commentary  [35]  Sandi:  [36] [37]  Crystal: Sandi:  [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]  Anyone know how to start this?  [35] Anik reaches behind Khona along counter apparently to get a sheet ofpaper  I say we just... I always just start writing all the numbers out but there's a thousand here and I'm not going to write a thousand numbers Anik: I could write down a thousand numbers but I don't want to Khona: It would take you Sandi: No, it's more than that because you have to like Anik: Yeah Anik: Okay, so... Khona: I need a piece of paper [43] Crystal starts writing (doodling?) Sandi: I don't want to do this Crystal: Anyone found this easy? [45] Khona flips through her Unknown: Yeah.... binder Anik: Otherwise the way I thought of it I'd have the answer already, it's all the evens.  91 In this exchange they briefly considered, and then rejected, the idea of writing down all 1000 numbers, agreeing that it would take too long. This convention, an implicit recognition that they needed to find a strategy to make dealing with these kind of numbers easier, also provided them with an initial common cognitive working space.  Themes Began to Be Introduced Anik had suggested an idea in which she did not seem to have much faith [47] and it was not immediately considered by the others. There was a pause. Then Khona seemed to pick up on what Anik had said. [48]  Khona:  Wouldn't it go to multiples of 100 then?  It was not clear at first if by "multiples" that Khona actually meant "multiples" in the correct mathematical sense, but the way in which Sandi and Anik started and continued to use the word throughout the session suggested that to the girls it really meant "factors." Although from a mathematical perspective, their use of the word 28  was incorrect, it did point to the existence of a shared vocabulary for the collective. There was some discussion about the idea of multiples. Sandi, who had started writing, appeared to pick up Anik's point about the even lockers being closed. [60]  Sandi:  And then the second one closes the even numbers, right? 'Kay^ so there's none remaining  However, the group stopped again. [61]  Unknown: I'm, like, totally confused.  Anik offered to highlight useful information and then Sandi asked her to report on what she'd found [68]. Crystal then triggered a round of predictions by the group which put more ideas on the table. There were now three themes (Voigt, 1995)  This is the meaning it will be given for the remainder of this section.  92 circulating: the trick answer (Crystal: "I betcha... I think in the end none of them will be open" [70]), fractional proportions (Anik: "I think like every second a quarter of them will be open" [74]) and "multiples" (Khona: "5, 10, 15, 20, 25, 30 and then it'd be..." [81]). The trick answer theme did not reoccur, but the other two themes were now in play.  The Loss of a Collective Member It was at this point, approximately one third of the way through the session, that Crystal's participation in the mathematics discussion diminished, perhaps because she still appeared to be having difficulty grasping what was happening to the locker doors in the question. Her contributions to the discussion became limited to asking which position certain locker doors were in, suggesting that she might have a resource sheet listing the prime numbers up to 100 on it, and pointing out - once she understood what was going on with the locker doors - in which position certain locker doors were. At one point she said, "I wish the question (inaudible) open or closed" [161], which suggested that she wished the question was about whether or not certain doors were open or closed, something with which she was more comfortable. Finally, when Khona complained that she was not sure what was going on, Crystal admitted, "I've been lost for a little while" [384]. Physically, although she continued to write, draw and willingly listen to the rest of the group's discussion, there were also times that she was looking around the room to see what else was going on. No one commented on this, unlike when later in the session Khona was briefly gazing around, seemingly having lost her depth as well, and Sandi immediately called her back into the conversation ("Are you doing anything Khona? Are you having fun? (laughter)" [354]). Although Crystal was still officially a member of the group, she no longer  93 seemed to be actively participating in the collective understanding that was emerging. It seemed that the collaborative zone in which the others were working had moved beyond a place where she could meaningfully contribute, and she withdrew until much later in the session.  The Slow Negotiation of Collective Understanding Although the first instinct of an observer in such a situation might be to try to credit certain students with originating certain ideas, this would soon prove to be an impossible task (Martin and Towers, 2003). How does one determine exactly where an idea starts? Who do you credit when: an idea is introduced, dropped unacknowledged by the others, and then resurfaces some time later but uttered by another group member; one idea morphs into another; two ideas are combined to form a different one; one idea is divided into parts which are pursued separately. Even as the idea is being introduced it becomes the property of the collective. This was certainly true of this session. The "multiples" and the fractional proportion themes were the main two in development, occasionally occurring in parallel and even intertwining. For instance, in the following exchange, while Sandi . was drawing out what happens when the second student reaches each of the lockers and exploring the idea that he closes every even numbered locker, Khona was persistently presenting the idea of fractions. Transcript  Commentary  [91] [92]  Anik: Sandi:  Is the first one open? Closed, open  [93]  Khona:  The first one opens them all, and then the second one opens all the even, so that'd be half. So that's how there'd be half  [92] Sandi appears to be talking to herself as she works and continues to do so as others talk.  94 [94] [95] [96] [97] [98]  [105]  Khona: Anik? Unknown: What about number 1 ? Unknown: No that stays Unknown: Anik? Anik: Then it would be a quarter of them Sandi: Yeah, then it opens Unknown: Anik? Anik? Sandi: Open, closed, closed, closed... Sandi: That's what it would be for the first 10. Open closed closed, (etc) Anik: Maybe that pattern would go on forever Sandi: It's open, closed, closed (etc) and then it stops, it goes to 1 Anik: Maybe that's the pattern  [106] [107] [108] [109] [110] [111] [112]  Sandi: I'll try it then from ... Khona: What's '/ of a •/•? Anik: (1/8?) Khona: A '/ would be 4/8's Unknown: (some counting to 64ths) Sandi: I have to redo it Anik: 128,250.  [113]  Anik:  [114]  Sandi:  [115] [116] [117]  Khona: Sandi: Khona:  [118]  Sandi:  [99] [100] [101] [102] [103] [104]  [105] Anik reaches over to Sandi's sheet and points at it. Crystal leans back but continues to watch Sandi  2  2  So he closed all the even ones? Yeah because they were open from number 1. Number 3 changes the positions, (inaudible) So it's 1000 divided by And then 1000 divided by 2 and then 1000 divided by 4 Shhh (laughs)  [109] Khona starts writing. Anik watches her. Crystal looks on briefly but then looks back to Sandi Sandi is writing. Anik and Khona are watching her. Crystal is writing. [114] Crystal is looking over at Sandi's sheet and continuing to write  Pause in discussion for approx 20 seconds.  The fractional idea was present (although mostly in the background since it seemed to be only Khona who kept resurfacing it) until Khona brought up the fractional proportion idea one last time and Sandi finally dismissed it (358-360). The "multiples" idea proved to be the one that was accepted by the collective to work with  95 in their collaborative zone of proximal development  and thus became part of the  developing theme. For instance, it triggered another contribution by Anik: [120] Anik: Wait, wait. I know what we can do (inaudible) we know (inaudible) don't have any (inaudible) like 17, you can't times any number to get 17 (inaudible) is that right? Anik referred to a kind of number which Khona was able to correctly name ("Prime numbers, are they called prime numbers?" [128]), a label which was then taken up by the collective. This in turn divided the theme into two more strands - one based on applying the properties of a prime number to the situation and another where the girls tried to determine how many prime numbers exist between 1 and 1000. Their conversation seemed to stall after they estimated that there were 250 prime numbers, meaning that 250 lockers must be closed for sure, an idea about which Sandi still had her doubts. The exchange that followed showed the importance of being able to clearly communicate ideas (Kieran, 2001) in order for the collective understanding to continue to grow. Sandi recognized her inability to clearly express her thoughts: Transcript [275] [276] [277] [278] [279]  [280] [281]  Commentary Khona: Sandi:  So far 250 are closed for sure That's like how many more? (laughing) Khona: So we have got 750 doors to check Crystal: Open or (inaudible) closed? [279] Sandi reaches to Anik's Sandi: Closed. But I don't even sheet. know if there is 250 prime numbers. Cause one of those numbers, like 2, it's an even number. Should we take that one out? 2 is prime because if it Anik: So it's 20 Sandi: And then, I don't know, like some of these numbers, and like 5,1 think it's not going  This also follows Saxe's notion of sociogenesis, which occurs when a collective adapts appropriate forms that individual members introduce/generate in order to accomplish goals (Saxe, 1991). 29  96  [282] [283]  Anik: Sandi:  [284] [285]  Anik: Sandi:  [286] [287]  Anik: Sandi:  [288] [289] [290]  Anik: Sandi: Anik:  [291]  Sandi:  [292]  Anik:  [293]  Sandi:  [294] [295]  Khona: Sandi:  [296] [297]  Anik: Sandi:  [298] [299]  Anik: Sandi:  [300]  Anik:  [301]  Sandi:  to be perfectly for all 1000 numbers Well it works, 5 goes into No, but like, 1 don't know some of these numbers 1 don't think they work No, it's not a prime number Can't you tell us how many are in 1000? Or does somebody want to figure that out? It's 'kay wait. Can I just see this for a second (inaudible) Nothing would (inaudible) (inaudible) No no, that's right, that's right. Okay, okay, just a second, 'kay let's ... That's right, that's right, it's 240. Because no number...prime, and like this, the prime number, if the second person comes they change, they change it (to the side?). But nothing would change... it works I know, I know what you mean but 1 think So there's 750 left Yeah I'm not even going to make (any sense?) I can't explain I can't explain what I'm thinking right now but So we know (Do you want?) somebody do it up to like 250? (laughs) I think we have to look for a pattern Yeah but I don't know where to start  [283] Sandi leans back in her chair. [285] (Sandi has leaned back in her chair even more and appears to be addressing someone out of the frame, perhaps the teacher? In her second question she is addressing the group. [287] Sandi reaches to Anik.  [291] Sandi takes notebook away from Anik.  [293] Sandi looks down and rubs her forehead.  [299] Sandi leans back in chair  It was only when Anik returned them to a topic within their collaborative zone of proximal development that Anik and Sandi started writing and, shortly thereafter, began discussing ideas again, with Sandi continuing to draw and talk about the  97 patterns emerging in the locker door positions as each "student" traveled through the hallway. Throughout this part of the session, Anik and Sandi's similar body postures, and their sharing of materials (a notebook, a sheet of prime numbers, the sheets on which they're each writing) suggested that their individual thinking was more closely aligned with each other than with the other girls, but Khona was still participating in the discussion, helping to shape ideas through her comments and suggestions.  Moving Towards Group Flow Counting the locker doors in Sandi's drawing that were open or closed, as though Anik and Sandi were traveling through the same loop of ideas again, was an example of a kind of meaningful inefficiency - sometimes one must struggle through the same material again (inefficiency) to establish common ground (redundancy) in order to work towards a new level of understanding. Anik and Sandi had established a comfortable collaborative zone of proximal development and were able to begin working there. Khona had withdrawn from the conversation, having had her last contribution (fractional proportions again) talked over and ignored by Sandi, and was now only observing. Anik and Sandi were currently the only two members of the group who were still absorbed in the work. As they discussed who was going to determine what had happened to certain locker doors, a boy from another group started calling out that the answer was 600. Commentary  Transcript [337]  Sandi:  [338]  Anik:  [339]  Sandi:  What do you want me to go up to? 30 or? (inaudible) (sheperhaps [338-346] Anik and Sandi are suggests that Sandi does all writing. Khona is watching. Crystal continues to work on own. In the of them) background a boy from another Are you joking? (laughter) group has been calling out that the Oh my.  98 (pause) [340] Anik: [341]  Sandi:  [342]  Anik:  [343]  Sandi:  [344] [345] [346]  Anik: Sandi: Anik:  [347]  Sandi:  [348]  Anik:  [349] [350] [351] [352] [353]  Sandi: Anik: Sandi: Anik: Sandi:  answer is 600. The answer would be a long list of numbers. Obviously she's (meaning the Teacher) not going to make us, like, write down a whole bunch of numbers, like it says What is the actual point of this? It says how many, or, which doors are open Yes So there's not that many. Yeah, I know but he's saying (inaudible) Yeah, 1 know but is he saying 600 doors are open or number 600 is open I don't know but I think more than one door would be open Yeah But I think less than 600 Yeah Would be open Because she's saying which doors she's not going to make us write down 600 doors. 'Kay I'll go up to 40 I guess.  [346] Referring to the boy calling out.  Sandi's comment [341] indicated that they were aware of the conventions - prior experience with Problems of the Week had shown that there would not be a long list of numbers for an answer. Their minimal reaction to what the boy from the other group had been calling out was interesting. At first it seemed that they had not even heard him, but then they casually considered and dismissed what he had been saying, and then carried on with their own calculations. Their high level of absorption in their task suggested that they were entering a period of group flow. This was briefly interrupted by a brief discussion with Khona about fractional proportions again, and then by an assertion by Crystal about which locker doors the second person closed, but it resumed shortly afterwards.  99  Arriving at Group Flow Sandi and Anik seemed able to achieve group flow while they worked through the concept of the number of factors. At this point, Crystal was working on her own and Khona had fallen mostly silent, leaving Sandi and Anik to have an almost uninterrupted exchange of ideas. At the beginning of this exchange, Anik introduced a new idea, moving their collective mathematical understanding to a new level. Transcript  Commentary  [370]  Anik:  [371] [372]  Sandi: Anik:  [373] [374]  Sandi: Anik:  Wait, I just figured something out, look. What? [372] Anik uses her pencil to If you figure out how many emphasis her points and her other times any number... 1000 hand to hold up various numbers of goes into each number, like, Say you pick the number 20. fingers. Crystal leaves. Khona is You know 2 can go in it, into looking around. it, 4 can go into it, 5 can go into it, 10 can go into it, and if I'm right that's it. So you know four numbers can go into it. Right. [374] Sandi suddenly leans back It starts open, or it starts close, so it will end closed. Or, I mean it will end open.  Sandi's recognition of the possibilities of this idea was immediate. Transcript  Commentary  [375]  Sandi:  [376] [377] [378]  Anik: Sandi: Anik:  [379]  Sandi:  Oh my god, you're so smart [375] Crystal returns Anik. Closed, open, closed, open Yeah. It will end open. And if it's [378] Anik waves hands in air. odd then it will end the same as it started. So it will be even if the number of numbers that goes into it is, I don't know if I'm making any sense 'Kay you know what? [379] Sandi gets sheet from Anik  100  [380] [381] [382] [383] [384] [385] [386]  Khona:  You're smart. It's not making any sense to  me. Sandi: No no no Unknown: Sandi is... Sandi: She's right, she's right. Crystal: I've been lost for a little while. Sandi: Oh my god you're right, you're right. Anik: Yes!  [383] Anik thumps her hands on desk, obviously pleased  [386] Anik waves her arms high in air  Once they agreed that the idea was working, the slipperiness of this new idea was then acknowledged. It seemed that it needed to be more thoroughly processed before it could belong to their collective of two: Transcript  [387]  Sandi:  [388]  Sandi:  Okay, if you go to number 30 and find out how  Commentary After her initial positive reaction, Sandi paused to double-check if the idea made sense. [387] Crystal grabs a calculator from Sandi's desk and then gives highlighter to Khona [388&389J Some simultaneous speech.  many /numbers can go/ into that, then that's the definition, 'kay wait, so, so, so wait, if it's even what did you say? [389] Anik: /numbers go/ [390] Anik: If it's even, wait, write this [390] Anik starts writing down. [391] Sandi: I'm losing my thought Anik. [391] Sandi hides her face in hands [392] Anik: Even briefly [393] Sandi: You're so smart [394] Anik: If, if it, no, if it started [395] Anik: If an even number can go into it They paused. Then, as they both wrestled with the new idea, they appeared to move further towards a state of group flow. Transcript [396]  Sandi:  Okay wait no, odd number is going to be open. It has to include the first person. First person, I didn't write it down, I just thought of something, so, so wait  Commentary [396] Sandi reaches across to Anik's desk. She writes a bit. At the end of her statement she takes Anik's sheet.  101 Anik and Sandi now shared their personal spaces more frequently by each reaching across and touching the sheet on which the other was working, or passing the sheet back and forth between them, something neither of them did with either Crystal or Khona during the entire session. Transcript [397] Anik: [398] Sandi: [399] Anik: [400] Sandi: [401] Anik: [402] Unknown: [403] Sandi:  Even number Right here Can go into it Number 1 is Stay open or closed? Stay open. So if an even number can go into it, in the number 20 right? [404] Anik: It will be open. [405] Sandi: It will be open, yeah [406] Anik: Okay so [407] Sandi: No, it will be an odd number goes into it too, because, because here at number 1 [408]Sandi: I didn't write it but number 1 went to number 20 and opened it and I didn't write that down, 'cause [409] Khona: Number 1 opens all of them [410] Unknown: I know, I know  Commentary [398] Sandi seems to be talking to herself [402] Sandi returns Anik's sheet..  [404] Anik erases.  [408] Sandi reaches across and points at Anik's sheet using her other hand to hold it down.  Unlike Crystal or Khona, neither Sandi nor Anik were distracted by people outside of their group throughout the session; now they appeared to be shutting out Crystal and Khona as well. Transcript [411] Anik:  Commentary Number one, so I'm saying so they all started closed, they all started closed [412] Sandi: Yeah then number one went [412&413] Some simultaneous on and /opened/ speech. [413] Anik: /opened/ [414] Sandi: And I didn't write that down. So that's one open, closed, open, closed, open. So if it's an odd number it goes into it [415] Unknown 'Cause I'm doing a (inaudible) too right? [416] Sandi: So it's 5. So if an odd  102  [417] Anik: [418] Sandi: [419] Anik:  [420] Sandi:  number goes /into it/ /into it/it will be open. Yeah But even amount of numbers will be...(inaudible) into it, the position of the locker.  It will be closed. So now you should find out, (laughing) what goes into every number up to 1000.  [418] Sandi leans back. [419] It sounds like Sandi is also speaking at the same time as Anik but, if so, her words can't be discerned nor can Anik's. When Anik finishes speaking, Sarah starts writing.  As they continued to work as a pair, each girl's speech began to echo the other's phrases and they began to speak in a shorter, more fragmented style with a quick interplay of phrases as though they were finishing each other's thoughts. They also voiced some phrases at the same time [416-417]. This kind of simultaneous voicing also occurred just prior to this exchange [388-389], again between Sandi and Anik. Finally, there was a sense of satisfaction, perhaps even exhilaration on Anik's part, a sense of inclusiveness, and a sense that they had briefly forgotten about their surroundings (when they appeared to realize that the video-camera was still taping), all further indications that they may have been experiencing group flow. Transcript [458] Anik: [459] Sandi:  We got it. That's so good. I We don't care about that right now. We don't have the answer but [460] Anik: But we know how to get it. We know how to get the answer Sandi! We got it! (laughing) [461] Sandi: 'Kay [462] Anik: [463] Sandi: Oh right, we're being taped (General laughter)  Commentary  [460] Anik waves her arms in the air and pounds her desk.  103 It was not clear i f Khona, who had witnessed Sandi and Anik's discussion, was included in this new level of collective understanding. She mentioned [409] that "Number 1 opens them all," but it is not clear i f she said this because it was stated in the original question that the first student to go down the hallway opened all of the lockers, or because she realized that 1 was a factor of all of the numbers 1-1000. Her comments after this did not refer to the number of factors either, but instead circled back to the discussion about prime numbers and how many lockers had prime numbers. This seemed to indicate that she was still working with that particular idea. Because the Locker Problem task was relatively unstructured compared to the worksheet task of Mandy and Hannah, it took a comparatively longer time for the pair to reach group flow. The students, although working with some specific pre-existing structures, first needed to foreground common conventions, and ready-mades in order to negotiate a collective zone of proximal development. Once this zone was in effect, they could begin to introduce new ideas for the collective to discuss and shape. Unfortunately, because of the difference in mathematical abilities, it was not long before the group moved into zone of proximal development where Crystal could no longer participate, leaving her out. Khona was able to participate longer but eventually Anik and Sandi's discussion left her behind as well. This may not have been due to mathematical ability in this case: it may have been that Anik and Sandi were "on the same wavelength" in terms of what they were thinking about and interested in and, when Khona withdrew from the discussion, they were finally able to close in on each other. It was interesting to speed up the videotape at this point in the session to clarify Sandi and Anik's physical behaviour patterns: the two girls literally did close in, increasing their physical proximity by leaning towards each other, sharing objects,  104 and reaching across to touch the sheet on which the other had been writing. They truly seemed to be in their own world. This physical behaviour was similar to that of JVlandy and Hannah, although, perhaps because Anik and Sandi were seated at a greater distance and across from each other, rather than side-by-side like Mandy and Hannah were, it was not as mirror-like. The results of this session suggest that the conditions were appropriate for the possibility of group flow. The decreased number of structures (in comparison to session one) was balanced by the more intrinsic nature of the goal inherent in the open-ended mathematics problem. The conditions of redundancy, increased diversity, decentralized authority, organized randomness, and neighbour interaction were all intertwined creating a situation where a variety of ideas could be explored and expanded. Maintaining this exchange, Anik and Sandi were not only able to create a collective understanding, but they were able to work together to move this understanding to a new level that they may not have been able to reach without their combined input. Their excitement and delight in this discovery reached a point where they were not only echoing each other but finishing each other's sentences. It was as though one person was speaking - they truly appeared to be in the same zone, and in flow.  105  Section 3: Open-ended Question with Fewer Pre-Existing Structures  A class is working in small groups on an open-ended mathematical problem. The group led by Shannon "gets" the answer within 10 minutes, with a minimum of discussion and interaction and no indication of a group flow experience. Another group made up of four equally contributing girls has a very long and wide-ranging discussion about the problem. A t the end of the 40 minute period, the girls have not yet "gotten" the answer, but a pair of them has shown characteristics of group flow as they have worked their way through an assumption about the problem. These two groups are working on the exact same problem: why do they not have similar flow experiences?  In this study so far, when a mathematical task was very structured, a pair of motivated students was able to establish a collective understanding and from there move on to apparently reach group flow. In a less-structured mathematical task, a group of four was able to establish a collective zone of proximal development, but only two members proved to be able to work within it to a point where they were able to reach a collective level of understanding where they could wrestle with new ideas and, in this process, exhibit characteristics of experiencing group flow. This section will describe the experiences of two groups of students who were working on the same open-ended problem at the same time, who were in fact located right beside each other, and yet who achieved very different results. I will begin by briefly situating common aspects of the session in relation to Sawyer's model. I will then discuss the background of  106 each group followed by a description of its performance of the task, beginning with the group that quickly "got" the right answer. Finally, I will use the works of Sawyer, and Davis and Simmt to explore why two groups, while working on the same task with the same pre-existing structures, were observed to have different flow experiences.  General Discussion of Session  As with the session described in the last section, this session took place within my own classroom in the spring of the school year. The question, again, was a Problem of the Week, with the same conventions in terms of group behaviour and the end-product of individual written solutions. As with the Locker Problem described in the previous section, students had been assigned to work in specific groups but every effort was made to make sure they were working with others with whom they felt comfortable. Students were used to the format of Problem of the Week, had experience working on open-ended questions, and they were used to the presence of the video cameras in the 30  classroom . The problem itself was unique and this particular one (the Train Problem) had fewer implicit strategies (conventions) than the Locker Problem had. It read as follows: One day Bob was walking across a bridge. He knew that a train was due to come across the bridge soon, but he thought he could make it across in time. It was sunny  Although this was the first "official" taping for this class, I had completed a practice taping the previous week in order to test using two cameras simultaneously and to accustom my class to the presence of the equipment. Audio recorders were not used in this session as I had not yet found a problem with sound quality.  107 out, and there was a great view from the bridge, so Bob walked very slowly, enjoying the view and savouring the lovely spring day. He was 2/3 of the way across the bridge when the sound of a train whistle jolted him back to reality. A huge locomotive pulling tons of boxcars was coming directly at him at 45 kilometres per hour! Using all of his powers of mathematical thought and analysis, Bob immediately figured that he could run directly ahead and get to the far edge of the bridge at the exact same instant as the train. But he also knew that he could run back in the direction from which he had come and get to that end of the bridge at the exact same instant the train overtook him. How fast does Bob run? (from Tsuruda, 1994)  Unlike the Locker Problem, there was no "find the pattern" convention embedded in the Train Problem. In one sense, this made it a less daunting question to solve because there were fewer numbers to work with and fewer calculations to perform. The goal of finding a definite answer was still present: perhaps it may not have been clear that there was one definite answer , but it was implicit in the question that Bob 31  ran the same speed regardless of the direction in which he traveled and both groups assumed this without any discussion. Because the "find the pattern" convention was absent, this appeared to free students to try a variety of approaches and, because the question seemed to be less restrictive in terms of large numbers and expected calculations (ready-mades), there tended to be a wide-ranging discussion of ideas when groups worked on this question. O f the three sessions videotaped, this was the 32  one task that most approached the improvisational side (fewer shared structures, less extrinsic goal) of Sawyer's model. In past classroom experiences, I have had students make arguments for there being two speeds depending on Bob's direction of travel. Again, this comment is based on my past experiences as a teacher with using this question in Grade Eight math classes. 3 1  3 2  108  Group One: Shannon, Richard, Peggy, and Jack  Background These four students appeared to get along well, although none of them were close friends and normally they were not grouped together in class. Richard had been designated as a "gifted student" by the school district and, although his strengths lay more in the sciences and social sciences than in mathematics, he likely had the most experience in his group with problem solving tasks. Jack, Peggy, and Shannon were students who normally achieved low B grades in math class, but were quite conscientious about doing their work and participating in group tasks. Shannon was very friendly, out-going, and self-assured. Peggy was also friendly but much quieter. Jack very much wanted to be liked by his peers and was sometimes teased by others about the way he acted (even during this videotaped session by students outside of his group), although Shannon often came to his defense in such situations. The group was videotaped, but there was no audio recorder to supplement the videorecorder so some of the students' discussion was lost to background classroom noise.  109  Performance of Task Videoclip (click on icon)  9f  C:\Documents and Settings\laynie\My Dc  Early in the session, Shannon came up with an approach to solve the problem and most of the ensuing discussion centred on her attempts to explain it to the other members of the group. A s the session opened, she took charge almost immediately,. reading the question aloud as the rest of the group listened. Richard and Jack offered some initial predictions about how fast Bob was running: Transcript [7] [8]  Commentary Richard: Shannon:  [9] Jack: [10] Richard: [11] Shannon:  45 km/h (inaudible) he runs V2 of that, the bridge is there and he runs V2 way. He (inaudible) Vi He's 2/3's (inaudible) 45 divided by 2 (inaudible) I'll draw it out. There. There's our bridge.  After a brief interruption for a whole class discussion about the problem, the group returned to work, briefly pondering the question's situation before Shannon introduced her idea. Transcript [24] Richard:  Commentary  [25] Shannon:  How long would it take the train to make it to the end. That's what we have to figure out (inaudible) (inaudible)  [26] Jack: [27] Shannon: [28] Shannon:  The ratio is 2 to 3 right? (inaudible) Bob is (inaudible) Bob  [25] Shannon is off-camera at this point and seems to be holding a conversation with someone across the room.  [28] Jack is beside Shannon and leans towards her.  110  [29] [30] [31] [32]  Richard: Shannon: Jack: Shannon:  [33] Jack: [34] Shannon:  wants to get (inaudible) the train is coming at 45 km (inaudible) and it says he's going to get there at the same time (inaudible) (inaudible) (inaudible) is 15 (inaudible) divided by 2 is 15 (inaudible) Ahhhh (inaudible) So 45 divided by 3 equals 15 (inaudible) three sections of (inaudible) Bob's on the second section so then Bob (inaudible)  [29] Peggy laughs [32] Shannon is writing [34] Richard leans way forward again  Richard, who was located kitty-corner to Shannon in their seating arrangement, was very attentive to the explanation, kneeling in his chair and leaning way over his desk so he could follow what Shannon was writing and saying. Peggy, barely visible on-camera, was also leaning forward in her seat. Transcript [35] Richard: [36] Shannon:  [37] Peggy: [38] [39] [40] [41]  Richard: Jack: Shannon: Jack:  [42] Richard:  Commentary How many sections? Three. One, two, three. And then, this (guy?) the train (inaudible) 45 (inaudible) 15 divided (inaudible) 45. So Bob would have to run 15 km to get to the end. This guy (inaudible) 45 (inaudible) That's confusing. Sorry (laughing) I sort of get it. I don't. How do you figure it out? [41] Jack takes his sheet as he (inaudible) 2/3''s (inaudible) begins to speak, gestures to 2/3's Shannon's sheet as hefinishesthis statement. He's interrupted by another student passing behind his and Shannon's chairs and he doesn't continue speaking. He watches her return to her seat and then starts watching his hand as he twirls his pen. He does not appear to be attentive to the rest of the discussion. So, because Bob has, was at [42] Richard turns Shannon's sheet  ILL  the third section and train (inaudible) Bob, he has (inaudible) so we have to divide it by 3? [43] Shannon: [44] Peggy: [45] Richard: [46] Shannon: [47] [48] [49] [50]  Richard: Peggy: Shannon: Jack:  Yeah see (inaudible) (So he needs to run 1 /3 of the) (1/3) (inaudible) the train's two sections of (inaudible) (inaudible) Oh, I get it. Oh yeah, she gets it! Ohhhh.  around towards him. Before he begins speaking, Richard carefully reads what Shannon has on her sheet and scratches his nose (he has allergies and scratches his nose frequently). [43] Shannon is likely still speaking as Peggy and then Richard speak but they are closer to the mic.  [50] Peggy begins writing. Richard continues to watch Shannon  The group's attention (aside from Jack's) was focused on what Shannon was saying and what she was writing on her sheet. The discussion centred on clarifying Shannon's ideas, with Richard and Peggy each posing a question to see i f they were following Shannon's argument. No new ideas were introduced or explored, and there was no dispute about the veracity of Shannon's approach. It was not clear when Jack said "Ohhhh" [50] i f he actually had understood Shannon's arguments or i f he was just making a socially appropriate noise. Transcript [59] Shannon: [60] Peggy: [61] Shannon: [62] Richard: [63] Shannon:  Commentary I'll ask her if we're right but she never tells us answers She just goes (inaudible) Which is actually a good thing because then (inaudible) All you have to do is just say I have the answer I just have to check (inaudible) matches Bob needs to run (etc)  Shannon is referring to the teacher.  Shannon reads aloud what she has written When Shannon is done there is a bit of inaudible discussion then Richard takes the sheet to copy. Shannon calls the teacher over and explains what they have as an answer. Teacher wonders if it makes sense and they all cheer.  112 Seemingly content that they had the answer, the four students spent the next several minutes quietly writing up their individual solutions, although Shannon left her desk for a couple of minutes to go explain her ideas to another group across the room (disregarding the convention of staying with her group). Once she returned, she worked on her write-up for a few minutes. The arguments of the group next to them (Anik, Sandi, Crystal and Cara, whose session will be discussed next) were audible. It was not clear i f Shannon's next comment was sparked by something she overheard of these arguments, or by something she had thought about as she was writing. Transcript [65] Shannon:  [66] [67] [68] [69]  If Bob's running, and the train's going 45, once the train catches up to him (wouldn't it?) pass him? Peggy: Yeah Jack: No, he's going to run Richard: (inaudible) run him over (inaudible) Shannon, Richard, and possibly Peggy: (inaudible)  [70] Shannon:  [71] Peggy: [72] Jack: [73] Jack:  [74] Peggy: [75] Shannon:  I'm going to still stick to my answer. The train's coming at him (which) means that the train's going this way and Bob needs to get to here or to get to here (inaudible) thirty (inaudible) (inaudible) the exact same instance as the train. So The track is here He's running up here. He just wants to get the heck out of the way. I think what we have is right. (? - said to Jack) Okay, okay (to self)  Commentary [65] All look up from writing  [69] Overlapped comments. Can't distinguish words. Richard uses his fingers to walk along desk. [70] Shannon points at her own sheet to indicate where Bob needs to go  [72] Jack points at Shannon's sheet [73] Richard starts writing.  Approximately 10 minutes into the session, the group appeared content with their current answer and spent approximately the next 20 minutes writing up their  113 individual solutions and chatting about other things. Occasionally each of them briefly would leave the group, but all returned and continued working.  This group can easily be characterized as working well together, since there was a friendly atmosphere and no conflicts. They were able to come to an agreement that Shannon's answer was correct - the contributions that each student made to the discussion suggested that they had reached a collective understanding about the problem's situation and solution - and seemed satisfied with their results. Despite all this, there was little indication that group flow had occurred. One characteristic of group flow that was absent was an intense level of collective concentration. Although there was obvious concentration on an individual level as each student wrote up a solution, there was little evidence of the kind of intense one-mindedness that has marked the instances of group work described elsewhere in this paper. As mentioned, during their work on the problem both Shannon and Jack were obviously distracted by others in the room, and once the group had solved the problem, each member physically left the group for brief periods of time. None of the sporadic discussion that took place within the group as they were completing their written solution was about the problem at all. Finally, there was no synchrony in gestures or in speech between any of the group members. They were respectful of one another during the discussion, and their body language suggested an openness to ideas, but there was nothing in how they behaved that suggested that any of them felt especially connected to any other member(s) of the group. Any increased physical closeness appeared only to indicate a desire to better see what Shannon had written on her sheet (for instance, Richard's kneeling position in his chair).  114  I will now discuss the background and describe the task performance by the second group, one that had a very different experience in solving the Train Problem. As this group's transcript is quite a bit longer than that of the first group, I will make observations related to flow as the description proceeds, rather than waiting until the very end of the description.  Group 2: Anik, Sandi, Crystal, and Cara  Background A l l the members of this group were diligent students. Anik, Sandi and Crystal were also in the group that worked on the Locker Problem and their mathematical backgrounds have already been discussed (Anik and Sandi were strong, while Crystal tended to be weaker). Cara was a strong student with high math grades, but she sometimes lacked confidence in her mathematical abilities. A l l four girls were on  115 friendly terms and shared a common interest in sports, while Anik and Sandi were good friends and hung around together outside of school. Both Cara and Anik were leaders within the classroom. Given the girls' personalities and their academic interests, it was not surprising that they worked well collaboratively. Although Cara and Anik spoke most often, both Sandi and Crystal were frequent contributors. In general, the girls followed group performance conventions: they took turns speaking, actively listened to and encouraged what others had to say, and were willing to explain their own ideas. The group was videotaped but there was no audio recorder to supplement the videorecorder so some of the students' discussion was lost to background classroom noise.  Performance of Task Throughout the session, the girls took turns offering suggestions. Their body positions (open posture, leaning back) suggested relaxation and openness, even very early in the session. Their smiles and laughter showed that they were enjoying the exchange, and provided encouragement (feedback) for others to continue their contributions. As they spoke, there was a shared repetition (echoing) of phrases that showed both an appreciation for other's ideas and a willingness to build upon these ideas, sometimes in a playful manner. For instance, when Anik suggested that the question's character Bob simply lie down on the tracks to avoid being hit by the train, all three other girls quickly took up the idea: Transcript [41] [42]  Commentary Anik:  He could lie down on the track Crystal: Yeah Anik, lie down on the tracks (sarcastic tone)  116 [43]  All:  [44]  Anik:  [45]  Sandi  [46]  Cara:  [47]  Anik:  (lots of laughing and overlapping inaudible comments about this idea) [44]Anik gestures with both her I know but it goes over him thumbs that the train passes overhead. [45] Sandi leans back in chair, big (inaudible) lie on the track smile. [46] Cara extends her arms in You'd have to (inaudible) front of her and squeezes them together as ifshe were trying to squeeze herself between two railway tracks. [47] Anik also squeezes her arms (inaudible) together.  The pace of the conversation quickened as the students warmed to the idea. The phrase "lie (down) on the tracks" was repeated as each girl worked with the concept and there was much appreciative laughter as the idea was increasingly treated as though it were actually feasible. This initial burst of group flow was reminiscent of the process of improvisation that Sawyer (2003c) describes in terms of the concentration (all girls are facing inwards towards and focusing on each other, effectively shutting out the rest of the room), the expressed enjoyment (laughter, smiles, and relaxed body positions), the group-mindedness (all were on the same topic, all were listening to the others, all were building on other's ideas, and all were able to interact quickly and spontaneously), and the flexibility of roles. This opening sequence, although short-lived, suggested that the girls already had social conventions established that would enable them to be equal contributors once the subject turned towards mathematics. As the group moved into a more serious discussion of the question, they needed to establish a common zone of proximal development from which they could work to negotiate a collective understanding of the question. In the following exchange, Anik's diagram helped to launch a debate about where exactly the train was  117 located compared to the bridge, an aspect of the question these group members needed to agree upon if they were to be able to continue discussing the question. Transcript [64]  [65] [66] [67] [68] [69]  Commentary Anik:  He ran. I know but (inaudible) the train would catch up to him if he ran backwards (inaudible) off the bridge the train would be there. So it'dbe2km... Anik: Okay here, this... so that's 2 km. The train is right (inaudible) bridge Crystal: (Not on the bridge yet). Sandi: /By the time he gets there (inaudible)/) Crystal: /Like right here. Not on the bridge yet/ Sandi:  [71]  He would get there, the train would get there (inaudible) Sandi: the train isn't starting there. It's starting um like Crystal: Like right there.  [72]  Sandi:  [70]  Okay, if you can like, um  (pause in conversation)  [64] Anik is pointing at her question sheet, stops, uses her pen as the bridge and uses afingeron the other hand to indicate the train. [65] Anik turns her page around and draws a bridge on it. [67] Sandi reaches over to point at Anik's sheet. [68] Crystal reaches over to point at Anik's sheet and then leans back.  [71] Crystal points at Anik's sheet. It is difficult to see but apparently she is pointing at a drawing of a bridge. [72] Sandi moves back to point at her own sheet. She looks at Anik who is reading from her own sheet and Sandi stops talking and goes back to writing on her own sheet.  Crystal and Sandi immediately responded to Anik when she appeared to say that the train was at the very end of the bridge, and their comments weaved around each other as they sought to prove that the train actually had to start further back. Although Crystal and Sandi's comments were quickly paced, intertwining as they occasionally talked over one another, which earlier has been argued as being symptomatic of flow, this discussion did not amount to flow: they were working together to create a common understanding for the collective, with Sandi and Crystal trying to persuade Anik about the train's position (i.e. to establish it as something that should be taken-  118 as-shared) and that she should join the rest of the group's thinking; they were not bringing the collective understanding to a new place. A l l of their physical finger-pointing suggested the establishment of a physical site representative of their site of intellectual interaction, in this case, Anik's diagram. Transcript [73] [74]  Crystal: So if the train starts there (inaudible) Anik: What I was saying, the train would have to start here. I was saying the train would start.  Commentary [73] Crystal reaches over to Anik's sheet, extending both arms. [74] Anik uses her pen to point at her own sheet.  Another exchange also demonstrated the importance of physical sites for intellectual interaction as two members in particular within the group, Anik and Cara, worked to establish their own collective zone of interaction. There was a lot of physical gesturing as they struggled to negotiate a collective understanding of the situation in the problem.  Videoclip (click on icon)  &  C:\Documents and Settings\laynie\My Dc  Transcript  Commentary The group beside them has just announced loudly to the teacher that they have the answer. Sandi glances over but returns to her own work. No one else in the group appears to notice the outburst  [80] [81]  Cara: Cara:  [82]  Anik:  [80] (Cara points to own sheet) (inaudible) (inaudible) so (inaudible) so (inaudible) it'd be 5 km in 2 minutes [82] Anik, leaning forward, points (inaudible) the train's here at Cara's sheet (inaudible) one hour (inaudible) an hour this is a minute and 15 divided by 2 , (inaudible) to 2 minutes because you have to  119 [83]  Cara:  [84]  Anik:  [85]  [86] [87] [88] [89]  So then if we know  I was saying the train is going. He would have to run 2 km and the train Anik: 'cause I was thinking would have to go 3 so I was thinking he would have to go 3 km Cara: What if Crystal: Except for, Anik (inaudible) Anik: I'm saying that (Anikpoints at her own sheet) Cara: (inaudible)  [98]  Anik: I know but (inaudible) Crystal: (inaudible) Anik (pause) Cara: So if we know that he runs 3 km that means (inaudible) 2 km Cara: (inaudible) km to run back Anik: (inaudible) Cara: I thought he runs 3 km. Anik: He walks along the bridge and he walks 2/3rds of the way, so that would be 2 km. Cara: So it should be 2  [99]  Anik:  [100] [101]  Cara: Anik:  [90] [91] [92] [93] [94] [95] [96] [97]  No, that's the train, not the boy. I thought it was the boy That's the train (inaudible) (it's always the train?)  [83] Cara's pen is pointing to the same spot as Anik's. [84] Anik's pen still pointing at Cara's sheet. [85] Anik points back at her own sheet. [86] Cara gestures in the air. [87] Crystal leans over and points at Anik's sheet. [89] Cara gestures in the air. Sandi points at Cara's sheet.  [93] Cara leans forward and points at Anik's sheet. [94] Cara leans back and points at her own sheet. [97] Anik points at her own sheet. [98] Cara sits up in her chair and starts writing. [99] Anik points towards Cara's sheet. [101] Anik uses her pen to trace a drawing on Cara's sheet.  During this conversation they took turns pointing, first at Cara's sheet, then at Anik's, then Cara's, until finally they were both pointing at Anik's sheet simultaneously. Reading just the description of the gestures from the session transcript of this exchange gives a taste of what it was like to watch the videotape on fast-forward with no sound as they negotiated. The amount of pointing of fingers and other gesturing was quite striking; it was like they were playing a physical game, such as badminton, that involved reciprocal action.  120  The concentration of the group as a whole during this exchange was quite intense. In fact, when the group beside them (Shannon's group) loudly called out to the teacher that they had found the answer, there was little reaction from this group. Only Sandi glanced up rather distractedly and then immediately went back to work, while the others in the group did not respond at all. There were also some interesting speech patterns present in this section as Cara and Anik established a working rhythm. Anik would first present her ideas, punctuating her speech with "I was saying" and "I was thinking." This was followed by Cara either restating her understanding of what Anik has said (usually prefaced by "So" as in "So, if we know that he runs km..."), asking for clarification ("What if?"), or stating her differing interpretation (prefaced with "I thought" as in "I thought it was the boy"). Although Anik and Cara's patterns of speech and gesture showed that they seemed to be working very well together during this exchange, they did not achieve group flow - they still seemed to be engaged in establishing a common understanding rather than building a new one. But they were starting to come close. The group had a long discussion about the physical realities of the problem, arguing about what assumptions they could make about Bob's speed relative to the train, and the relative positions of Bob and the train. A brief consultation with the teacher seemed to clarify that it was okay to use a hypothetical situation to test their ideas, but that the assumptions Crystal and Cara had been supporting were not possible (which is what Sandi and Anik had been arguing) and this left them feeling lost.  Transcript [195] Cara: [196] Sandi:  Commentary (inaudible) I don't even know where to begin. Me neither. We need something to start at. Anik just start us out with  121  [197] Anik: [198] Sandi:  something. I did! But it didn't work out! Do something! (laughing)  Immediately after this exchange, the group breaks into paired discussions. Sandi and Crystal tried to establish where Bob and the train would meet if Bob ran one way or i f Bob ran the other. Meanwhile, Anik and Cara were ready to experiment with a new set of numbers, with Cara suggesting that they assume that Bob could run 15 km/h. Transcript [208] Cara: [209] Anik: [210] Cara: [211] Anik: [212] Cara: [213] Anik: [214] Cara: [215] Anik: [216] Cara: [217] Anik: [218] Anik:  Commentary This way would take him Vi an hour (It would take him half an hour...?) How long would it take the train to go (inaudible) (inaudible) (inaudible) Are you saying that the train (inaudible) I'm saying that the train, yeah, (inaudible) 1 km (inaudible) ('cause the?) train (inaudible) How long would it, say it's 2 km (inaudible) So an hour. So 15 minutes so in 15 minutes you go 11 and A No, half an hour. It takes him half an hour to go 2 km(inaudible) hour to get to (inaudible) Half an hour (inaudible) times (inaudible) l  [219] Cara:  [220] Anik:  Once they established that they were talking about the same thing, using the same phrases established in their earlier working rhythm ("Are you saying", "I'm saying" and "So"), they began to recap their assumptions.  122 Transcript [241]  Cara:  [242]  Anik:  [243]  Anik:  [244] [245]  Anik: Anik:  Commentary  [252]  Anik:  [255]  Cara:  We have to assume that the train is 2/3rds of the way... and what they know We found out the boy could run, he would run, in 15 minutes he would run 2 km so so it would take him 15 minutes to get there and it would take him 7 Vi minutes to get there  So it has to take the train 7 Vi minutes to get there (excited tone) and then Cara you said (inaudible) km (inaudible) wait (inaudible) going 45 km/h or seven km... something like that (excited tone)  [245]Anik leanedfurther forward and touched Cara's sheet with her pen. Then new ideas began to surface, withfirstAnik getting excited  They began to work together more closely, reflected by them physically leaning towards each other. [259]  Cara:  [260] [261] [262] [263]  Anik Cara: Anik Cara:  [264] [265] [266]  Crystal: I'm confused. Sandi: What are you guys doing? Anik You'd say the train (inaudible) 8 km (inaudible) the train Cara: 8 km one way (inaudible) [267] Cara gestures with her hand. take them 2, 16,8 km each [268] Said in response to Anik's Cara: Yeah. confused look [269] Cara points at her own Cara: Divide by one 7 Vi sheet. Cara: That's one kilometre. [270] Cara points in the air with her other hand.  [267] [268] [269] [270]  So divide it you divide by IVi 60 minutes how long, no, divide by divide 60 by IVi TA So it's 8 (inaudible) 8 It's 8 km. Yeah, 8 km  [259] Points her finger in the air. Starts to use calculator. [260]Anik uses her calculator too. As they performed these calculations, they began to echo each other words, their sentences becoming shorter and more fragmented. Anik and Cara tried to ignore the interruption. Their understanding was building and they did not want to lose it.  Their movements continued to  123 mirror each other's. [271]  Anik  Oh, I thought you had to divide by 60. [272] Cara: (to Crystal) We're trying to figure out how far the train would go if the boy would go this way. [273] Anik: We want to figure how far it would go in one minute. One minute [274] Cara: (inaudible) okay [275] Anik: So we'd need 8 times 7 !4 [276] Cara: Why? [277] Anik: We don't want to know how far it goes in one minute, we want to know how far it goes in [278] Cara: 7.5 minutes [279] Anik: IVi minutes [280] Cara: 60? [281] Anik: 60. [282] Cara: 60 km. [283] Sandi: (to Crystal) Yeah, (to Anik) How did you get that? [284] Cara: Is this km or minutes? [285] Anik: That's minutes. [286] Sandi: (to Crystal) (inaudible) (smiling) [287] Anik: • (to Sandi) Just a sec. (to Cara) Write that down [288] Cara: (inaudible) once we get (inaudible) [289] Anik: We'll explain, we just have to figure it out. 'Cause we're like . [290] Cara: We're on the [291] Anik: We got it (inaudible) okay so [292] Sandi: (to Crystal) (inaudible) [293] Cara: (inaudible) 60 km [294] Anik: So just write 60 down. [295] Cara: Isn't it 60 minutes? [296] Anik: Km. 60 km/h. Oh, it can only go 45 in an hour. [297] Cara: Exactly. 60 min. What if it takes them 60(inaudible) if (inaudible) [298] Anik: (inaudible) (pause) [299] Sandi: (inaudible) [300] Cara: Take them 15 minutes to and (inaudible) [301] Anik: Let's explain it to them and  [272] Cara simulates drawing on her own sheet.  [274] Anik and Cara nod at each other. [276] Both Anik and Cara start using their calculators. [278] Cara points her finger in the air. [280] Cara starts writing.  [284] Cara points at her own sheet with pen.  [287] Anik leans over and points at Cara's sheet. [289] Fluttery hand movement. [290] Fluttery hand movement.  [298] Anik leans her cheek on her hand. [299] Anik holds herfingerup for Sandi to wait.  124 When Cara's ideas were coming quickly, she began to gesture more with her hands, in particular pointing one of her fingers and shaking it, and at one point (255) she became especially animated, seeming to cause Anik to withdraw from her attempt to pick up Cara's sheet. Both of the girls' voices grew louder, and their eye contact with each other became more frequent and longer lasting as they alternately explained and then listened to each others' ideas. Cara's syntax became more fragmented as the pace of her spoken ideas increased (267 & 259). The speed of their conversation grew, and echoing began as they started to do calculations and check each other's answers. The numbers 7 Vi, 8, and 60 became the touchstones of their conversation that they kept restating in order to check their common level of understanding: These echoes (261 & 262 & 263, 278 & 279, 280 & 281 & 282) provided immediate feedback that may have assured each that the other was thinking along similar lines. They were interrupted by Crystal and Sandi (264 & 265, 283, and finally 286) which, after Anik asked Sandi to wait (287), prompted Cara and Anik to explain to the other pair why they needed to wait, a report that not only featured them finishing each other's thoughts and sentences, but suggested that they were feeling good about what they're doing, that they recognized that they were making progress, and that they did not want to be interrupted - all signs of group flow [287] Anik:  (to Sandi) Just a sec. (to Cara) Write that down  [288]  (inaudible) once we get (inaudible)  Cara:  [289] Anik:  W e ' l l explain, we just have to figure it out. 'Cause we're like  (fluttery hand movements) (fluttery hand movements)  [290] Cara:  We're on the  [291] Anik:  We got it (inaudible) okay so  125 There were two more interruptions from Sandi (292 & 299) which were inaudible, one directed to Crystal and the other directed to an unidentified group member. It may have been these interruptions, or that Anik and Cara had met a rough patch in their thinking (296 & 297), that prompted Anik to suggest to Cara that they now tell Sandi and Crystal what they had been doing. In any case, the observed flow behaviour dissipated as Anik and Cara alternated turns explaining their ideas, which Anik appeared to find frustrating, perhaps because she was no longer speaking to someone who was in sync with her. Transcript [322] Anik:  No (inaudible) the train is right here (inaudible) it takes the train 15 minutes to get there [323] Sandi: (inaudible) 3 km [324] Anik: We know (inaudible) [325] Sandi: (inaudible) [326] Anik: No, that's the train, that's the train. Bob is only 2 km from the end. So it takes, they're together at the same time (inaudible) Bob (inaudible) two [327] Cara: (inaudible) now we're trying to figure out how far the train goes  [328] Cara:  Commentary  [326] When Anik is referring to Bob at end of her statement, she holds up two fingers.  Pause in conversation. All four students physically pull back. Anik removes her handfrom Cara's sheet. Cara seems to note the loss offlow.  (to Anik) I think I lost my track. The session continued for another minute or so until the school  announcements come over the public address system. Although Anik and Cara were unable to resume their momentum and the group flow had completely dissipated, the group now seemed to focus their attention on the ideas that the pair had been explaining, with Crystal particularly eager to participate in the discussion, while Sandi was writing down her own thoughts. This suggested that the collective flow experience that occurred between Anik and Cara had enabled them to develop and  126 then contribute new ideas to the rest of their group, thus contributing to the growth of the group's collective understanding of the question. In summary, Anik and Cara were observed to display an intensity of concentration and an increased synchrony of gestures and speech that have been characterized in this study as being indicative of an experience of group flow.  Discussion On the face of things, both groups had the potential to reach flow. Both groups were comprised of students who were friendly and open to one another, and who demonstrated an interest in solving the problem. Working with Sawyer's model, the reduction in pre-existing structures inherent in the problem was balanced by a reduction in the extrinsic nature of the problem, keeping both groups in the area of potential group flow. Why then was Group Two observed to achieve group flow while Group One was not? In reviewing Davis and Simmt's conditions for learning systems, it seems that decentralized control is an issue for Group One. First, there was some evidence that they did not fully regard themselves as a collective. Right after Peggy and Jack appear finally to "get" the solution, Richard and Shannon both use the inclusive "we" in describing the next action to take: [52] Richard: Now we all have to do is [53] Shannon: Now we'll write it right here [54] Shannon: and we'll actually do our write-up However, the relative importance of certain group members was reflected in what was said a little later in the session, in Shannon's exuberance when the group had agreed with her answer and was settling down to write up the solution:  127 [64] Shannon: Go Team! Go Team Shannon! Actually, Go Team Shannon and Peggy because she gets it too! She refers to the group as a "team," yet names the team after herself. And it was not clear why she left out Richard, who obviously "gets it" too. It was interesting, as well, that Shannon later referred to the solution as "my answer" [70] while Peggy referred to it in a more inclusive way [74]. Shannon's speech suggests that she sees herself as being separate from the others. Shannon's proposed solution to the problem is the one that is privileged throughout the discussion. When one member of a group dominates the other members in explaining a solution to a math problem and the others passively accept the explanation, this restricts the exchange of ideas (Yamaguchi, 2003). Although Shannon is a friendly presence in the group, and is certainly not aggressive in her approach to the others, it is clear that she has taken charge of the situation and the others are more inclined to try to understand her ideas, saying "I don't get it" rather than actually challenging them. Although Shannon is the most obvious authority within the group, there is an indication that the group regards the teacher as an outside authority. Once they have all reached an understanding of Shannon's idea, their next step is to call the teacher over [59-63]. Although Shannon and Peggy state that the teacher will not let them know if they have the right answer, Richard seems to believe that the teacher will i f one states one question in the right way. The students' cheering reaction to the teacher's response indicates that they feel they have received approval and all they 33  have to do now is write it up [52-54].  Although my intent as a teacher in my interaction with this group was to ask them to check if their answer made sense, my standard reply when asked if an answer to a Problem of the Week was correct, in reviewing the tape it seems what I said was interpreted as a confirmation that their answer did make sense. I now wonder what the effect of a more carefully worded response might have been - if they  3 3  128. Group One's appeal to the teacher raises another issue. Group Two also has discussion with the teacher, one that is inaudible , and the one that follows' Transcript [173] Cara:  Commentary Hey, let's just ask her  [173] Cara puts up her hand and looks around the room for the teacher. Pause in conversation. Anik extends her arm towards Cara who throws her the highlighter. Sandi and Crystal are both writing. Anik starts highlighting. Cara sorts her papers and starts writing [174-175] Cara points to Anik's sheet and taps her pen on it throughout..  [174] Cara: (to Teacher) So we assume the boy was here and the train was here like the exact same distance from that point then (inaudible) he'd be running 45 km (inaudible) is that possible? [175] Anik: (inaudible) [176] Cara: If the boy ran 45 km and the train went 45 km (inaudible)? [177] Teacher: (inaudible) [178] Cara leans back and [178] Cara: Exactly! shrugs her arms in air. Crystal also holds up her arms [179] Crystal tries to give [179] Crystal: Oh yeah Cara! Cara a high-five but Cara isn't looking and Crystal puts her hand back down [180] Teacher: I'm not saying (inaudible) (leaves) [181] Anik: I'm just saying that your answer is not physically possible. [ 182] Cara: She just said don't worry about the physically impossible. [183] Anik I know she said that for right now [184] Cara (Calling to Teacher):Ms A., what do you mean for right now? Like it's not part of the problem? (Teacher returns)  [185] Teacher: (inaudible) ^'ell it might be. It's a way to set up your thinking about what the problem is. A hypothetical situation. 1  [182] Cara leans forward and puts her chin in her hands [184] Cara turns away from group for a moment [183-187] Sandi and Crystal work independently through this [185] Teacher points at different spots on Anik's diagram  might have explored the problem further, or i f they would have shrugged it off and started their written solutions anyways. The effect of the teacher on a group's potential for group flow is beyond the scope of this study but it is well worth considering. If my memory is correct, they asked me to confirm that it does not indicate in the problem how far the train is from the bridge. The lively discussion prior to this excerpt revolved around whether one would need to be superhuman in order to run faster than a broken-down train. 3 4  3 5  129 [186] Cara:  How do we know whether to use it or not? [187] Teacher: You talk it out - is it going to work for this, or for this. [188] Cara: (inaudible) (your own case?) [189] Teacher: Yes [190] Cara: /Ooohhhhh/ [191] Anik: /See, if he's running 45 km/h, that's what I was saying, if he's running 45 km that way he'll get there faster than the train./ (Teacher leaves)  [190] Cara continues to say this through [191] Cara bangs desk emphatically with both hands.  Bowers and Nickerson have noted how two different classes can seem to have their own personalities: "Although the materials, instruction, and emphases of the sections may be the same, each class tends to develop slightly different ways of justifying mathematical arguments and valuing what 'doing mathematics' means (2001, p. 3). Judging from Group One's appeal to the teacher, "doing mathematics" meant getting a "right" answer. For Group Two, whose members did not even react when Group One audibly called the teacher over by saying they had the answer (this occurred around line 80 of Group Two's transcript), "doing mathematics" appeared to mean exploring the problem and making logical arguments. What Cara was seeking from the teacher was clarification about information about the problem and her thinking process. Anik did not address the teacher at all, but immediately started restating her own arguments about the situation. Sawyer's model proposes that a balance needs to exist between the number of shared structures and the nature of the collective goal. Although both groups were working on the same problem with the same number of shared structures, it is apparent that Group One's goal was actually more extrinsic. They approached the problem in an efficient manner, focusing on problem solving - find the right answer and complete the written solution quickly. In contrast, Group Two broke the problem up into smaller problems which they then hashed out, a process of problem finding.  130 Perhaps, Group One's more extrinsic goal was enough to upset the balance of the situation and thwart the possibility of reaching group flow.  It is impossible to know where this second group would have gone in terms of finding a solution to the train problem had they not run out of time. Their thorough discussions of the problem's situation and its possibilities suggest that, whatever their final result might have been, all members had been collaborating within the same zone of proximal development and had shared a collective understanding of the problem throughout most of the session. To return to Bowers and Nickerson's quote near the beginning of this section, for this group "doing mathematics" was a process of discussing, sharing and understanding; for the first group, it was a matter of having a "right" answer, regardless of what doubts they individually might have harboured about that answer. Perhaps this attitude is what enabled the first group to move towards a sense of one-mindedness, which Cara and Anik were then able to briefly take into a state of group flow, as evidenced by their intense concentration on each other, and their synchronized speech and gestures. It is curious, although not unexpected, that two groups from the same classroom would have such different experiences. Zack and Graves note that the emergence of a collaborative zone of proximal development through interaction does not always occur, even under optimal conditions, and cannot be orchestrated (2002); group flow, an even more allusive experience, is much the same.  131 CHAPTER 5: CONCLUSIONS In his theory of optimal experience, Csikszentmihalyi has described flow as "the state in which people are so involved in an activity that nothing else seems to matter; the experience itself is so enjoyable that people will do it even at great cost, for the sheer sake of doing it" (1990, p. 4). He has suggested that flow may occur when there is an inherent interest in the activity, as well as a match between the level of challenge that the activity offers, and the level of skills that the participant possesses. The concept of flow has been studied in a variety of educational settings, almost entirely focusing on the experiences of individuals. In his studies on group creativity, R. Keith Sawyer (2003c) has argued that while group flow and individual flow may both be considered emergent, arising from the activities of individual agents, they are not the same thing: individual flow represents the individual performer's state of consciousness while group flow is a property of the entire group and emerges from the interaction of its members. Learning that is self-directed is both meaningful and lasting, and to have motivated students who are enjoying their learning is likely the ultimate goal of all teachers. Group flow is a phenomenon that suggests that those groups that are experiencing it are at a peak level of performance, working in a collective zone of proximal development that offers a satisfying balance between skill level and challenge in other words, the group is learning. The implications of group flow for educators are powerful.  132  In this study, I applied group flow theory to regular mathematics classrooms i n three different contexts to answer the following questions: 1) what are the observable characteristics of group flow, and 2) what conditions may help to promote the experience of group flow i n small groups i n a regular mathematics classroom setting? A summary o f my results follows. In the first section o f my results, a pair of students who had an established working relationship worked on a highly structured task involving fraction worksheets. Using Sawyer's model to analyse the conditions o f the situation, I suggested that the balance between the structure of the activity and the extrinsic nature o f the goal provided conditions that encourage group flow. I then characterized the students' behaviour as being observable evidence o f group flow: this behaviour included increasingly synchronized verbal and physical behaviour, intense concentration, and a sense o f satisfaction. In the second section, a group o f four students worked on a less-structured task, an open-ended mathematics problem. After briefly positioning the situation using Sawyer's model, I then applied the conditions for emergent learning systems proposed by Davis and Simmt to further suggest what would encourage a collaborative zone o f proximal development to develop in a less-structured situation: redundancy, decentralized control, internal diversity, organized randomness, and neighbour interactions. I then characterized the behaviour o f a pair o f students within the group as being evidence of the occurrence o f group flow. The behaviour o f these students was basically the same as those described in the first section.  133 In the third section, I compared the behaviour o f two groups o f students who were working on the same math problem. One group, although finding the "right" answer quite quickly, did not display evidence o f achieving group flow. In contrast, a pair o f girls within the second group, which itself was unable to reach a solution before the end of the session, appeared to reach group flow while working through a particular aspect o f the problem. Using both Sawyer's model and the conditions o f Davis and Simmt, the findings suggested that the first group may have been unable to achieve group flow' because o f the more extrinsic nature of its goal (getting the right answer), and because certain figures (one group member and the teacher) seemed to be perceived by the group as having more authority than other members. The three sessions varied i n terms o f the nature o f the goal and the structure o f the activity, ranging from the first session where the goal was the most extrinsic o f the three and the activity contained the most pre-existing structures, to the last session where the open-ended task provided a goal o f a intrinsic nature and the least number o f structures. That all three sessions provided evidence o f group flow seemed to show that group flow can be achieved in a variety o f situations i f conditions are balanced. The one group that did not reach group flow did not appear to have the same kind of balance as the other group during that particular session did. This study also aimed to identify characteristics of group flow, and one observation that was striking was the nature o f the feedback that occurred between group members as they gradually synchronized their thinking. Individuals moved closer together, their eye contact became longer and more frequent, their gestures began to mirror each others, their speech became faster and more fragmented as their  134 communication became more efficient. A n d , finally, the exhilaration with the breakthroughs that sometimes accompany group flow became evident: the triumph o f Mandy and Hannah in being the first at their table to decode the riddle on the math sheet; A n i k and Sandi's delight in making the connection o f locker position with an odd or even number o f factors; A n i k and Cara's increasing excitement in almost having "it" and their expressed sense o f loss later when they had "lost [their] track". A s group members moved into collective flow, their behaviour was less o f that o f a collection o f individuals and more o f a single entity. The nature o f the signs observed in all three sessions suggested that group flow builds and ebbs. Groups with fewer pre-existing structures seemed to require more time in order to generate a flow situation, suggesting that a period o f negotiating was required to establish a collective understanding o f the task as a foundation with which to start and a zone of collaborative development within which to work. Group Two seemed to reach flow more easily (in non-mathematical conversations as well) with the Train Problem (which had the least number o f structures) than the group solving the Locker Problem. More study would be required to determine how dependent the achievement o f group flow is on the condition involving the number o f structures. Observations also suggested that group flow may only last varying amounts o f time, with Mandy and Hannah being able to maintain their flow for the longest period o f time. Perhaps this was due to the high level o f structure o f their activity, meaning that there were fewer other factors to manage or negotiate within the group. To determinie the minimum length o f time required before a state could be considered group flow would require further study, but I would venture that the period must be long enough that the  135  group members involved are able to communicate back and forth so that all members receive enough feedback to allow the flow state to be developed and maintained.  Revisiting Sawyer's M o d e l I've used Sawyer's model to outline conditions that promote the occurrence o f group flow. He suggests that there needs to be a balance between the nature o f the group's goal (i.e. how extrinsic it is) and the number o f shared pre-existing structures inherent in the situation. These structures include a known outline o f the performance, predefined roles, conventions, and ready-mades. Yet, on its own, Sawyer's model could not fully explain what was happening i n terms o f group flow in all the three sessions. For instance, it needed to be combined with Davis and Simmt's conditions for learning systems in order to explain i n Session 3, for instance, why flow occurred for Group Two but not for Group One which was unable to meet the condition of decentralized authority. Sawyer's model, although a good starting place for discussion, on its own is too linear and too static to successfully suggest the dynamic nature o f group flow, with its ebbs and flows, its interactive nature, and the nature o f engagement in a task. I believe it must be combined with another model, such as Davis and Simmt's, in order to more successfully provide a sense o f what group flow is. Another difficulty with Sawyer's model arises from the post modern notion o f framing. He argues that because group, flow is an emergent property o f the group itself, whether or not individual members achieve flow or feel that, as a group, they have achieved peak performance does not matter; this initially suggested to me that group flow is a phenomenon observable by the outside observer and that self-reporting would be  136 unnecessary. A s my study proceeded, the nagging question o f perspective arose. In an anecdote, Sawyer cites how a performing group might feel they had not performed well, but the audience might have thought the group was at its peak. But suppose not all members o f the audience hold this view: would the view o f these dissenters negate the possibility o f group flow having occurred? Suppose there was an audience o f several thousand, and only one person disagreed that the performers had achieved group flow. Would the majority rule, or would one negative opinion be enough to overturn them? Whose perspective is privileged in determining whether or not group flow has occurred? These questions suggest how complex a phenomenon group flow is.  Implications for Practice Perhaps the most obvious implication o f this study is that educators cannot expect 1  instant results when a group is assembled to work on a task. Aside from factors such as personalities, social status, and the relationships between group members, which are all beyond the scope o f this study, groups need time to negotiate what the task is, how to approach it, and what types o f contributions each member might make. Even given sufficient time, these negotiations are fragile - a friendly group such as Shannon's was unable to achieve flow, despite quickly finding the "right" answer to their problem because one member's contribution dominated the others. Educators need to be aware that the conventions with which groups work are taken from those that have been established throughout the year in the classroom, both ones that are explicitly set out as "rules" or expectations, as well as ones that are more implicitly determined by what teachers and students alike accept as reasonable behaviour.  137 A t the beginning o f the school year then, one might expect groups to function less efficiently because fewer conventions have had time to be established in a particular class and thus shared, meaning that students may be working with a m i x o f conventions that they have carried with them from past school experiences. Awareness that certain unspoken practices are in play may help the teacher to guide his/her students towards agreement about which conventions w i l l be most helpful for their particular collective. Although in this study I did not assign specific roles to the students, due to the school's focus on cooperative learning students had likely been exposed to the standard cooperative learning theory roles (scribe, encourager, leader, etc). Still, the participants in this study were able to work well together without resorting to the rigidity o f these kind of roles. Sandi, in particular, took on a variety o f tasks. For instance, i n the Locker Problem Sandi played central parts as a leader, a facilitator, and a scribe, among others, while in the Train Problem she tended to act in a more peripheral manner as a critic and then as a problem-solver working more independently from the group. This suggests that students are quite capable o f negotiating and playing the roles that best suit them given the task at hand without teacher intervention. It is important to assign students to groups where they feel a high level o f comfort and trust so that all members feel safe to contribute and develop a collective zone o f proximal development. In mathematics, the level o f a student's ability seems to play a particularly significant part; i.e. putting a l o w level student with a high level one in the same group would appear to be fruitless i f the task is not one that w i l l offer them both an  This confirms what author Harvey Daniels found in the 1990's when many teachers around the United States used the roles sheets provided by his Literature Circles program too rigidly: "in some classrooms, the role sheets... became a hindrance, an obstacle, a drain - sometimes a virtual albatross around the neck of book club meetings. What had been designed as a temporary support device to help peer-led discussion groups get started could actually undermine the activity it was meant to support" (2001, p. 13) 36  138 adequate opportunity to contribute, to be challenged, and to find a common level o f understanding. Crystal, for instance, was left out o f group flow experiences in the Locker Problem because she seemed to be unable to keep up with the negotiation o f mathematical ideas occurring between the others. In order for group flow to occur, students need to be able to provide each other with quick and meaningful feedback, difficult to do when one member has taken on the role o f the expert and is leading the others (again, Shannon's group provides an example o f such a non-flow experience). When a teacher uses heterogeneous groups, he/she w i l l need to ensure that the task is such that all levels o f students w i l l have something to bring to the table and thus the task either w i l l need to have the potential to be addressed in a variety o f ways, or w i l l need to have aspects that can be shared between group members in a manner that is perceived by all to be fair. Finally, even routine classroom tasks can promote flow i f approached in the right way. A structured task can be built upon - as Hannah and Mandy did when they developed their own challenges o f efficiency and speed in completing their puzzle worksheets. A n open-ended task can be structured as A n i k and Cara did when they assigned specific values to the situation to create a situation that had proscribed limits (Davis & Simmt, 2003). A s long as the task is o f interest and provides a balance between skill level and challenge, and the group situation allows for frequent feedback so that the group can negotiate ideas quickly and efficiently, the potential for flow, however fragile, is present.  139 Future research In his work on individual flow, Czikszentmihalyi (1988) noted that certain people have a personality type (autotelic) that made them more likely to achieve flow. Could the same also be said o f a group's "personality"? If so, it would be interesting to consider what factors would be influential in making a group more likely to experience flow. In this study, only pairs o f girls were able to achieve flow. However, this was a very small sample size and the gender composition o f the groups studied was determined largely by who was available and interested in participating in the research. In my own classroom experiences outside o f the study, I've informally observed both groups o f boys and mixed gender groups experiencing collective flow during mathematical problemsolving tasks. It would be interesting to investigate how gender is related to the ability to achieve collective flow. Does the age o f group members influence a group's ability to reach flow? Does a group's level of skill or knowledge affect their potential for collective flow? Is a group o f high achievers (in math or any other subject area) more likely to experience collective flow than a group o f low achievers? H o w does difference in ability between group members affect the collective flow experience? Does the perception of ability o f group members (both o f oneself, o f others within the group, and o f the group as an entity) affect the collective flow experience? This study was only able to document collective flow occurring in pairs, and a brief non-math related experience of flow within a group o f four during the Train Problem. One might argue that the larger the size o f group, the more difficult it would be to achieve collective flow simply because there are more different parts (i.e. people) to  140 synchronize. Still, from an educator's point of view it would be interesting to document under what conditions a larger group, perhaps even an entire class, could experience flow. Would the characteristics of collective flow experienced by a large group necessarily be the same as they are with a smaller group? For example, i f a group o f students is really "with" a teacher during direct instruction or a lecture style lesson, might that be considered collective flow too?  Reflections The pursuit o f this study has reinforced for me as a teacher how important negotiations are as students pursue various tasks. Although I had been aware o f what could be negotiated within a group (the meaning o f the task, strategies to approach the task, the establishment o f intermediate goals within the task) and had certainly witnessed it in my own classroom, I had not considered what is not negotiated (i.e. what seemed to be initially assumed or taken-as-shared by the group: conventions and ready-mades) and how much o f that is affected by the routines and behaviours my students and I develop in the classroom from the first day o f school. I had also not considered that negotiations have a dual nature: they are the means by which the group members orient themselves to one another; yet they provide an end in themselves in that they are the process through which a collaborative zone o f proximal development is established and maintained. A s well, I am now more aware o f how the perceived "authority" o f a teacher can help to shut down negotiations i f students view him/her as being the only one who can judge the validity o f an answer.  141 V i e w i n g the videotapes made it clear to me that negotiations occur on a number of levels. In verbal negotiations, group members explicitly stated their ideas and sought to clearly communicate them to their peers. They also asked questions, and offered suggestions, objections, and compromises. Tone o f voice, rate o f speech, and use o f vocabulary signaled meaning and intent. Visually, negotiations were represented in many forms. Expressive gestures included raised hands and clapping, while directive gestures involved pointing towards written texts and drawings. Body language involved physical distance from others, positioning (i.e. facing away from or towards certain people), and posture. Finally, facial expression and length o f eye contact were also observable cues. For instance, when students such as Jack and Crystal dropped out o f their groups' discussions, it indicated that they were no long participating not only due to their lack o f contributions to the conversation, but to their body positions (sometimes focused away from the group), their gaze (around the room rather than at their group), and their facial expressions (blank). A s a teacher, I want all my students to be participating in group activities as long as possible. Negotiations need to be able to occur freely between students are as they work together in a group setting. Students need to be in a setting where they have built trusting relationships before they w i l l feel free to explore ideas and take risks i n discovering new ones. I've become more aware, as a result o f this study, o f the importance of "ice-breaking" activities - activities that help students become acquainted with one another - and in activities that build positive social relationships between students throughout the year. A s well, in order for students to participate i n group negotiations they need a common background to start with, and as a teacher I am now  142 more careful to make sure that students have had a chance to develop some redundancies to work with, in terms o f mathematical ready-mades, before I providing them an openended task. Finally, in acknowledging the complex and slippery nature o f group flow, I am starting to accept that "control" o f the classroom situation on my part is impossible - not only can I not predict the outcome o f my students' activities, I cannot make my students interact i n a particular way as they work. 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