An ontology of off-task behavior in a number activity: The body,space-time,and mathematics

Abstract

This article aims to interrogate the Cartesian rationality determining current early childhood mathematics by highlighting the irrational aspect of mathematics learning, which is usually underemphasized and even devalued by the dominant discourse. Using Deleuze and Guattari’s concept of refrain as the method, the article explores the unfolding emergence of a child’s off-task behavior in a number activity as it conceptualizes the child’s body in relation to its surroundings. The article also explores how this conceptualization of off-task behavior helps to rethink mathematical ability and conceptual construction in responding to and elaborating on research on embodied mathematics. Based on these explorations, the article hopes to contribute a strategy for the vision of adopting a more democratic manner in mathematics learning.

A child in the dark, gripped with fear, comforts himself by singing under his breath. (Deleuze and Guattari, 1987: 311)

Introduction

When reading these words, I think about Jim (a pseudonym), a seven-year-old boy with whom I once spent every Thursday afternoon for over six months in mathematics classes at the Clubhouse, an after-school program. In class, he was occasionally noncompliant with my instruction, such as changing the activity’s procedure or place, inappropriate use of objects as part of the assigned task, or instances in which it was unclear whether he was on-task (Godwin et al., 2016; Langer-Osuna, 2016). His off-task moments compelled me to rethink some educational convictions that I had been believing to be self-evident since I was an educator. Was Jim deliberately pushing back against the class norms set by my instruction? Did these behaviors, disconnected from my expectations as a tutor, suggest some misrecognized aspect of mathematics learning? How did these behaviors happen?

Cartesian conceptions of mathematics learning as meta-narratives have been informing early childhood education for years (De Freitas and Sinclair, 2014; Klein, 2007; Parks and Schmeichel, 2014). They treat the human body and material objects as inert matter and controlled by the active human mind (Rotman, 2014: xv), orienting the materiality of mathematics learning with reference to a socially sanctioned whole of measurable or describable units.

Overrelying on the psychological or sociocultural perspectives premised on these meta-narratives, most research on children’s mathematical activities risk the reduction of the off-task body to an individual problem or celebration (Alldred and Fox, 2017: 1162), which sheds insufficient light on its emergence and educational value (MacLure, 2016). More than that, the academic discussions reinforce the ways that teaching has attempted to discipline how mathematics is conceptualized and implemented in the school curriculum.

As a result, these discursive practices render unmeasurable or indescribable moments—that contribute to the emergence of off-task behavior—likely lost in the meta-narratives, which silences the ontological dimension of how it is that so many young children do not do mathematics in the way that we desire, even if they could.

In order to overcome these limitations, this article draws on the Deleuzian-Guatarrian refrain as a research method (Lenz Taguchi and St Pierre, 2017) to explore specific off-task emergences in a number activity regulated by the rituals of discipline, controlling Jim’s learning to fit into my tutorial vision. The refrain—or repetitious and rhythmic patterns (Deleuze and Guattari, 1987: 311–347)—is an ontology of movement within and among open systems (Jackson, 2016). It has a catalyzing function to make the meta-narratives become the bearer of a non-discursivity (Guattari, 2000), and thus produce something new. I elaborate off-task behavior as the ontological movement which not only entails facial expression, body posture, movement, and utterance mobilized in an improvisatory form, but also attends to a multiplicity of assembled corporeal actions (MacLure, 2016).

In particular, I conceptualize Jim’s body with reference to the refrain’s key elements—rhythm and milieu (Jackson, 2016)—as providing the condition for the emergence of off-task behavior. This conceptualization also introduces the virtual—“an unrepresentable reality conditioning what happens” (Lorrain, 2005: 163)—to Jim’s number activity. In this sense, the study further investigates Jim’s mathematical ability and conceptual construction by asking what rhythms are (which comprise the visceral actions of embodied mathematics).

In order to make the refrain intelligible, my field notes perform as meaning-making devices (Duhn and Grieshaber, 2016), inviting the registering of the account of the unexpected behavior on my sensing body (Boldt and Leander, 2017) as a researcher when I pay visits to the data. In this way, my thinking of Jim’s number activity allows for becoming the refrain for the field notes encourage a “transcorporeal process” of engagement (Taguchi, 2012: 267), rather than mental activities in the mind of the researcher understood as seperate from the data.

In order to develop the features of the refrain that impact on mathematics learning, the study uses the well-developed conceptual framework deriving from Deleuze and Guattari (1987). This framework has been developed and applied in the work of Deleuze (1998, 2003) and Guattari (1995, 2000); social and feminist scholars such as Jackson (2016), Lorraine (2005), and Grosz (2003); early childhood scholars such as MacLure (2016) and Olssen (2009); and embodied mathematics scholars such as Rotman (2000), De Freitas and Sinclair (2014), and Roth (2016); as well as scholars in other fields.

I have organized the article to address these problems with the following questions: How does the refrain perceive Jim’s body? How is Jim’s off-task body rejected from the mathematics class? How does Jim’s body make the goal-oriented task fragmented? How might the refrain provide new insights into some aspect of mathematics learning? With these questions, the article is “an experimentation in contact with the real” (Deleuze and Guattari, 1987: 12). By “real,” I mean the process of how the body deviates from what Cartesian rationality describes and measures in current early childhood mathematics (De Freitas and Sinclair, 2014; Klein, 2007; Walkerdine, 1988). The study aims to interrogate this dominant discourse through the real flow of a child’s mathematics learning, which seems perturbing and trivial but connected to the body’s vitality and virtual becoming.

Jim’s body in the Clubhouse: a milieus assemblage

Jim had been in the program for almost two years when I first met him at the Clubhouse. The Clubhouse provided children in a low-income community with an after-school enclave in the southeastern United States. The Clubhouse was a white wood-frame house with a small front yard and a big backyard. Inside the house, there was a front room, a kitchen, and three activity rooms. Faculty and students from a local university offered improvisational and dialogical instructions on children’s freely organized activities. Meanwhile, the open-door policy enabled children to choose to attend or not. Besides that, children could stream at will from room to room and freely used most of the items at the Clubhouse when they popped in after school each day.

After he finished homework with an adult’s help, Jim, like other children in the program, played around at the Clubhouse. Sometimes, he took a pile of cards out of a small basket and played card games with another two boys on the big rug or somewhere in the front yard; sometimes, he and other children ran throughout the backyard while teasing each other; sometimes, he stepped into an activity room and watched his peers placing wooden blocks; other times, he strolled in the kitchen’s direction to grab some cookies on the table and then went away. Occasionally, he walked toward a corner and played a small music instrument leaned against the wall for a while.

As a new tutor from the local university, I spent several afternoons with Jim and his peers. When he heard that the newcomer could offer a math class from 3 p.m. to 4 p.m. every Thursday, Jim immediately registered for it. He came to every class and was the only student. (Field notes)

The refrain is the “shell of the body” (Bonta and Protevi, 2004: 133), suggesting the “rhizomatic” tendencies of Jim’s body (MacLure, 2016: 176) as it is unnecessarily bounded with the human skin. In order to better understand the point, it is necessary to attend to the rhythm of the milieus that make a body.

Deleuze and Guattari (1987: 304–317) write that each milieu is “a block of space-time,” “constituted by the periodic repetition” (313) that determines the norm or function of the milieu. However, “the periodic repetition” cannot always be metrically exact, as the milieu is always open to change to provide some temporary coherence and connectivity. When change happens, the rhythm shows up as it “ties itself together in passing from one milieu to another” (313) (I will elaborate on this later). Unlike the milieu, operating in homogeneous blocks of space-time, the rhythm thus connects heterogeneous blocks of space-time.

Deleuze and Guattari (1987) conceive the body in terms of a milieus assemblage, including internal milieus (cellular formation, organic functions) and external milieus (food to eat, ground to walk on, physical items to use). In Jim’s case, the Clubhouse is a milieu of multiplicities “made up of qualities, substances, powers, and events” (Deleuze, 1998: 61), breeding the external milieu for the configuration of Jim’s body in mathematics class. The external milieu may emerge from the physical set-up, the open-door policy, the colors and aromas of food, or the shapes and sizes of blocks and cards. It may also arise from improvisational instructions, people’s free movement, the sounds of the indoor and outdoor spaces, or the disciplinary pattern of the mathematical activities.

The milieus assemblage’s boundaries are indeterminate until a bodily movement configures the body. In Deleuze and Guattari’s (1987: 314) view, Jim’s act is a territory staked by a refrain — a rhythm of the milieus that becomes expressive and territorialized. That is to say, the rhythm of the milieus assemblage generates a territorial refrain—that is an act and (re)configures Jim’s body— when the rhythm becomes expressive and territorialized. Thus, Jim’s body in the mathematics class is defined in terms of a milieus assemblage in which to produce a refrain or a behavior, including mathematical concepts, disciplinary pedagogy, teaching tools, Jim as a physical and symbolic existence, and other milieus in the Clubhouse, as mutually affecting one another and never fully formed or static. Then, what may become other milieus?

In line with Lorraine’s (2005: 160) explanation, these other milieus are “distinguishable” through Jim’s “feelings and sensations associated with the routines” at the Clubhouse, which renders the rhythm “homogenized into” his “lived experience” of the program. For over two years, Jim has freely enjoyed the space and materials offered by the program with what Jones and her colleagues (2016: 1144–1149) call practices—a spatiotemporal, fluid indoor/outdoor space and open access to material for playing. In this sense, Jim’s embodied experience with the schemed number activity, intended to discipline his body, suggests the emergence of the rhythmic process in the Jim–disciplines–mathematics–Clubhouse assemblage. Jim then carries out this regulated number activity in a manner in which the rhythm becomes territorialized into the sentient awareness of Jim, who has settled down into the program’s routines—for example, having the freedom to talk with an adult, walk into the front room, use the material objects, and play card games on the rug. It is likely that blocks of space-time related to those practices are called in the rhythmic process as other milieus.

Notably, the rhythmic process puts Jim’s in-class body on a virtual trajectory. To be specific, it renders the body comprise elements of unpredictable eruptions (Lorraine, 2005: 160–163) and assembles its past, present, and future (Grosz, 2003: 82). It is this disturbing process that leads to the occurrence of off-task behavior—namely, Jim’s past life (he exercised freedom with the practices), in the ongoingness of his present life (the class norm is disciplining his body), releases and actualizes virtual eruptions that were previously only implicit in the present. In this case, the virtual forces, which are “aspects of the past that constitute implicit forces of the present” (Lorraine, 2005: 164), unfold in the refrain’s occurrence through territorialization of the rhythm. Hence, the refrain actualizes the unpredictable eruption internal to Jim’s in-class behavior and introduces difference—namely, off-task behavior that functions to change the direction of on-task learning in that particular moment. In this sense, Jim’s body always has the “virtual and insignificant” tendencies (Drohan, 2013: 119) — that is, the body indoctrinated by disciplinary pedagogy, teaching tools, and mathematical concepts, and material objects can release itself from that indoctrination, because of the past and future inherent in the unfolding of the present. Also, these tendencies do not allow Jim’s body to break away completely from its already-existing form from which it desires to flee, because the milieus-assemblage offers provisional connections to set free the body's immanent variation always temporarily held in place, in line with Message’s (2010) argument. In other words, Jim’s body exists in a state of movement whereby it continually changes into something else; yet, it also keeps an internal organization that is temporary.

The rhythm of the milieus brings to the fore the fact that the refrain is an ontology of movement—a movement that territorializes the “body’s own existence” (Zembylas, 2007: 26) through the continuous deterritorialization of its deadening repetition. The following two sections discuss the becoming of this ontological force—the refrain.

Jim’s on-task behavior: familiar territory

The mathematics class at the Clubhouse provided Jim with activity-based exercises to develop his mathematical skills for completing homework and thus improving his academic achievements at school. The classroom, comprising a table and two chairs, was set in the program’s front room. The table was placed next to the window, through which the afternoon sun shed light. Not far from us, there was a big rug laid on the floor in the middle of the room.

At three o’clock on a bright afternoon, Jim showed up as usual. He began to examine a pen and a stack of dot cards that had been prepared on the table. The dot cards were placed in order, from the 1 dot card on the top to the 20 dot card at the bottom, which informed that each card had one more dot than the card before. This information was also shown by the design of dot patterns on the cards. I told Jim the procedure of a counting exercise: he was required to work on the cards one after another; when he had one card in his hand, Jim should write the number on one side of the card for the amount of dots that he observed on the other side. Jim followed the instructions as he picked up the 1 dot card, viewed the dot, and wrote the number down. He used the same way to work with the 2 dot card but stopped at the 3 dot card. (Field notes)

As indicated above, the rhythmic difference is never entirely free of “a given territory” (Parr, 2010: 69)—that is, on-task behavior from which off-task behavior deviates. This territory provides Cartesian thinking of mathematics learning, in which “gridded plots” are offered to make Jim’s learning “occupy precisely located parcels of land” (Deleuze and Guattari, 1987: 440, 441). This section shows how the “gridded plots” are at work to construct Jim’s learning, which fixates on the parcels, and how the construction turns out to prompt Jim’s body in becoming different.

The number activity is designed to promote Jim’s self-regulated learning and develop his numerical conception through mental representation and subitizing, as suggested in the Principles and Standards for School Mathematics in the USA (National Council of Teachers of Mathematics, 2000). I attentively implemented it mainly via the organized curriculum materials, the classroom’s physical layout and my oral instruction.

To be more specific, the arrangement of the cards and the design of the dot patterns, with my oral instruction and the location of the desk and chairs, intends to invite Jim’s eyes, ears, hands, and brain to constitute “a mathematically-structured mind” (Roth, 2016: 92). As these organs join in the constitution, Jim’s body is molded into a docile substance, held in the disciplinary pattern of corporeal operations: perceiving the physical characteristics of a dot pattern on a card, recognizing a quantity of the dot in that group, and writing the number paired with the pictorial representation. All of the operations are rigorously structured, creating an abstraction process from the world through the hand and into the mind (Piaget, 1952; Vygotsky, 1978). Also, the abstraction process, unfolding in a set of structured actions, should take place repeatedly, which leads to a firm belief that Jim’s performance should repeat the abstraction process exactly. Thus, this strategy is perceived as being able to unfold real learning for Jim.

However, for Rotman (2000: 78–132), the disciplinary pattern of corporeal operations is merely a figment of the imagination rather than a reality. He points out “two golden tricks” underlying the regime of Cartesian rationalism: Plato’s notion of numbers as eternal entities and Aristotle’s “endless coming-into-being of the numbers” (121). These “golden tricks” offer the ideological basis for the on-task territory, where the given numbers are not merely separated from the surroundings, but are separated in a given, unchangeable fashion. In this sense, the on-task territory becomes an “imagined scenario” of individual performance regarding linear causation that is displayed in the step-by-step procedure. Therefore, Jim’s behavior is identified as on-task when his learning process fits into the imagined formation of numbers, and this doctrinal procedure entirely determines how Jim should deal with a dot card and a numerical concept at a particular moment.

Jackson (2016: 185) writes that “fitting-in is desire.” The on-task territory makes Jim’s body “over-coded with molar constraints” (185) in seeking homogeneity, recognizability, and stability. It indoctrinates Jim’s learning by having it marked with a set of describable or measurable units prior to his encounter with mathematics (De Freitas and Sinclair, 2014). For this territory, representation is the law (St Pierre, 2016). The milieus assemblage is molarized as it generates clear boundaries not only among the milieus but also among the components of each milieu. All of them come to be fixing, representing, and signifying entities, incorporated into a dictatorial regime concerning what to learn and how to learn. As a result, these entities—the curriculum materials, the physical layout, the human bodies, the discourses, and other things in the mathematics class—work coherently as fragments that create a base for disciplined mathematics learning.

In particular, the territory depicts familiar but often taken-for-granted images of mathematics and learners. Mathematics—neutral, stable, and untouchable (Rotman, 2000)—functions as the rationale for disciplining the mathematics class (Rotman, 2014: xiii). Being self-disciplined, mathematical learners can learn rationally, responsibly, and flexibly (Klein, 2007; Sugarman, 2015). Together with other fragments, they enter the on-task territory of molar coexistence that exudes energetic expectations and surveillance.

A Cartesian split between the active human mind and inert matter is fundamental to the on-task territory’s desire. It creates truth effects through the disciplinary forces that expect, supervise, and judge the materiality of Jim’s learning with reference to a priori categories of on- and off-task behavior. In doing so, it renders Jim intentionally conscious of pushing unmeasurable or indescribable moments outside the on-task territory, in response to “the fixed relations of similarity and difference afforded by the logic of representation” (MacLure, 2016: 176). Thus, Jim comes to be a normative subject—“the object of pedagogical and psychological discourse” (Walkerdine, 1988: 202)—orienting himself entirely in terms of the familiar depictions of mathematics and learners. Hence, the dynamics and diversity of the body languish in its normalization.

However, a territory always remains uncertain and unpredictable (Message, 2010). It is a Cartesian illusion that the contextual variants of the number activity are in controlled and repeatable conditions (Grosz, 2003: 82). After Jim predictably works with the 1 dot card and the 2 dot card, the contextual variants become anti-predictive and transgress the expectations of rational ethics—that is, the control generates an affect of discomfort and prompts the desire for home or bodily comforts. This desire leads to “cracks” enacting their own ethics (Tesar and Arndt, 2016: 197) when Jim stops at the 3 dot card. The virtual becoming of Jim’s body is released.

Getting the goal-oriented task fragmented: unfamiliar territory

Squinting, Jim said to me in an affirming tone, “Can we go down there?” He pointed at the rug close to us, where the sun shone on the floor. “Yes.” There was much sunlight from the window on his face. On hearing “yes,” Jim left the table with the pen and sat down on the rug. I grabbed the unfinished dot cards and followed Jim onto it.

When I put down the stack of the cards in front of him, Jim patted them gently with his left palm until the stack became a pile where the cards loosely rested. Facing the cards, Jim lay on his stomach with his upper body propped on his elbows; this position seemed comfortable for him on the rug. However, he pulled out the 15 dot card instead of the 3 dot card.

Jim looked at the card for a moment, and then turned the card over to the blank side; he tried to write a number. Suddenly, he stopped writing, left the number unfinished, and turned back to the dot side. Rather than merely looking at them, Jim counted the dots one by one with his finger pointing at the dots, eyes moving around over them, and head nodding gently in time with the counting. When he completed counting, he turned over the card to the blank side again and wrote down the number 15. Impressed with Jim’s devotion to the updated exercise, I lay down beside him. (Field notes)

The cracks are refrains. For Jim and me as his tutor, with attempts to discipline the number activity, they are unfamiliar territories. They disrupt the unfolding of the expected learning procedure and change the location of the desired learning place. Yet these disruptions give Jim’s body permission to experience indeterminacy without being caught in anxiety (Guattari, 1995). Along with the experience come three interlinked aspects in making a refrain, an expressive, territorialized rhythm (Deleuze and Guattari, 1987: 311–347): creating an order, making a space, and opening to the outside. The aspects show how the rhythm combines heterogeneous blocks of space-time and produces Jim’s unexpected behaviors. They also indicate what Jim senses as his body becomes rhizomatic and unpredictable.

In the first aspect, the refrain creates the beginnings of order in chaos. Chaos is more like a feeling of disorientation (Jackson, 2016: 188). Jim’s disorientation arises through his squinting, when a sudden change occurs in his body, coded with the rituals of discipline. Such rhythmic turbulence suggests that Jim has some hesitation in continuing to work at his assigned location. It emerges when the rhythm assembles forces from the basic routines of the class, the position of the chairs, the sunlight coming in from outside, and Jim as a biological and symbolic system. The refrain is now fragile and “in danger of breaking apart at any moment” (Deleuze and Guattari, 1987: 311). This turbulence, which is makeshift, comes about within an uncertain process, generating a sense of insecurity. In this aspect, Jim’s squinting is toward home.

In the second aspect, the rhythmic refrain is territorialized as it marks the boundaries of a home. The rhythm becomes expressive when Jim breaks his silence, coded as an appropriate behavior in class: “Can we go down there?” This determined expression is a disruptive event that divests Jim’s body of any connection to my envisioned learning process. It appears when the rhythm moves into a new field, which touches not only the physical and symbolic characteristics of the instructor and the rug, but also a hubbub of excited conversations from those who move about randomly in the Clubhouse. Through the disruption, Jim orients himself toward a sense of comfort rather than the predetermined stoppage of my desired space-time—doing the 3 dot card while sitting at the table. In the pursuit of bodily comforts, the territorialized rhythm cancels out anomalous interactions among the milieus to keep outside the territory the chaotic forces emitted by them. Thus, this territorial refrain makes Jim feel alive and at home.

In the third aspect, the refrain opens a way out of the territory by continuously drawing up new forces from multiple milieus in the Clubhouse. My oral response, the floor, the unrestricted human movements, and other blocks of space-time join in the rhythmic passage to form a new temporary assembling, where Jim’s walking toward the rug replaces his breaking the silence at the table. In this aspect, the refrain becomes a line of flight, opening onto the outside. It is about moving beyond Jim’s personal world of the already-established territory to universes where his personal constraints disappear, and Jim’s body is carried along on something new. Here, the refrain is building up a new route toward home.

The three interlinked aspects describe the becoming of a refrain as “a passage between two states of deterritorialization” (Jackson, 2016: 183). They involve a relationship of continuous energy between the deterritorializing tendency of an emergent rhythm and the maintaining potential of a territorialized experience. More importantly, they shed light on rhizomatic movements immanent to Jim’s body. As Jim’s body lives with the Clubhouse, the play of forces from the program and his body renew the refrain every time and drive it toward eternal difference. Difference is “an event that is joyful in itself; it is not the difference of this being or for this end” (Spinks, 2010: 85). In other words, refrains allow the body to stay satisfied and vibrant by continuously breaking up the settled block of space-time with the unpredictable block of space-time. And unpredictable space-time entails movements, of which Jim is part, in all directions (Boldt and Leander, 2017).

As the refrain occupies the territory of walking and soon leaves it, Jim’s sitting marks a new rhythm associated with the rug. The rhythmic process continues to become expressive as it creates another territory of patting the cards with Jim’s palm, by connecting my body going after his and the stack of cards placed in front of him. Yet Jim’s patting quickly comes apart, which allows the constitution of a new territory where his stomach and elbows are closely associated with the rug.

The refrain, in its continuous openness to the outside, triggers an affective response—the grabbing of the 15 dot card—when one of the many available virtualities is actualized to introduce difference in this specific moment. Random, risky, and precarious, the grabbing comes from other milieus with different functions, perhaps including playing card games with the two other boys on the big rug or somewhere in the front yard. In its unfolding emergence, the fortuitous grabbing expresses a specific, rhythmic relation to the disciplined activity of Jim, who is expected to grab the 3 dot card instead.

In order to reinforce this newly territorialized refrain, the body recreates physiological conditions for its unfolding in a series of newly emergent reactions of viewing the dots, writing the number, stopping to count, and rewriting the number. These improvisational reactions assemble movements of Jim’s eyes, fingers, head, and trunk, further undoing the schemed rules and procedures of the activity.

As seen from the above, the refrain takes place in the sense that the rhythmic flow consists precisely in this putting-into-variation of the constants of on-task behavior and making the goal-oriented task fragmented. In other words, it is not the schemed activity but the rhythmic flow that carries on Jim’s body, a flow along which Jim’s body travels from one sense of comfort to the next as the improvisational need arises. When the variations happen, Cartesianism, which commands the order and sequence of the number activity through my instruction, is no longer fundamental; its power turns into “a transient, fluctuating phenomen[on]” (Alldred and Fox, 2017: 1166). In those moments, everyone and everything become something else. Notably, the moments create Jim, not subject to my instruction.

Refrains: the real flow of Jim’s embodied mathematics

The Deleuzian-Guattarian refrain examines Jim’s off-task behavior with an emphasis on the body’s virtual becoming, which Cartesian thinking of mathematics learning—the psychological or sociocultural perspective—falls short of describing. This move is crucial. It disproves the conception of the human body and material objects as inert and the attribution of off-task behavior to the individual’s mind or will. As discussed in this study, it is not Jim, but the materiality of his body in relation to its surroundings that generates off-task behavior. More than that, the refrain creates the potential for what else early childhood mathematics is and can become with a child’s rhizomatic body.

Cutler and MacKenzie (2011: 63) write that the real challenge to Cartesianism is “to treat learning as an ontological rather than an epistemological problem.” The refrain reaches out to the real flow of mathematics learning in the way in which it engages a bodily awareness in the rhizomatic and insignificant, rather than relying on the priori learning process believed to be significant. Thus, this study goes further into research using the Deleuzian approach to embodied mathematics (e.g. De Freitas and Sinclair, 2014; Roth, 2016; Rotman, 2000). It experiments with addressing how particular virtualities and insignificances that remain dormant in the docile body are actualized through the disobedient body.

First, the refrain highlights the indeterminacy of mathematical ability. As De Freitas and Sinclair (2014: 140–171) argue, rhythm fundamentally defines a body’s mathematical ability through “the provisional and temporary presence of indeterminate organs” in the relations—that is, mathematical ability is not the pregiven capacities of individual cognition, but highly responsive and distributed across an extensive set of unstable material relations (157). In line with their study, Jim’s mathematical ability can be examined through the impromptu emergence of indeterminate milieus—entailing organs—in becoming a refrain.

Yet refrains expand the meaning of De Freitas and Sinclair’s (2014) mathematical ability. As an ontological movement, they allow inappropriateness or insignificance—erased from the priori structure of learning—to inhabit distributive ability. As Roth and Maheux (2015) point out, the process of mathematics learning in De Freitas and Sinclair’s argument is somehow orchestrated by the pure idea of bodily schemas. In other words, these bodily schemas would replace the actual flow of learning with the ideal flow blind to off-task behavior. Yet the refrain highlights the erased part as it attaches learning movement to the body’s indeterminacy, which the schematic flow cannot seize.

Take Jim’s hands in their encountering of the number 15, for example. In terms of the predetermined movement, their actual moves are disruptions for three reasons. First, their encountering of 15 comes after their encounter with 2 instead of 14. Second, they are reconfigured with regard to their capacity to rearrange the cards, pull out the 15 dot card, count the 15 dots, and write the number 15, which is different from the intentional configuration that involves grabbing the card and writing 15. Third, their encounter with 15 allows organs—Jim’s legs, trunk, and elbows, considered as illegitimate participants in the schema—to take part in reconfiguring their capacity

Nevertheless, these disruptions are where Jim’s mathematical ability comes into view. What is important here is that refrains imply “more than a mere open-endedness of the path emerging from one mathematical step (in thinking) to another” (Roth and Maheux, 2015: 274)—namely, when the move of his hands encounters 1, 2, or 15, what the end of the path or each of the steps looks like remains unpredictable and undetermined, owing to the ongoingness of the hands’ movement being continuously open to multiple milieus that are always assembling the present, past, and future. Thus, the hands’ continuous movement engages in the becoming of Jim’s mathematical ability. In this sense, refrains constantly redefine what is currently necessitated in mathematical ability.

Furthermore, this mathematical ability enables space-time number-related knowledgeability. Roth (2016: 88–95) reconsiders mathematical conceptions in terms of Deleuze and Guattari’s (1987) lines of flight. For Roth, lines of flight involve disruptions, such as trials and errors, in students’ interactions with curriculum materials; therefore, their appearance suggests that the actual learning process moves in a direction that is different from the already assumed one. Consequently, it is not the already-existing mathematical concept but mathematics-related knowledgeability that occurs in the interactions, which do not permit the ideal conceptions in the schema.

Echoing with Roth’s (2016) study, refrains remold the preexisting linear pattern into a non-linear flow, a flow that unfolds within and between Jim’s various bodily movements—shouting, pointing, walking, sitting, lying, propping, patting, withdrawing, counting, and nodding. These actions seem inappropriate or redundant, but generate a real learning process, in which a 15-related knowledgeability may develop. However, what Roth (2016) might not notice is the “knotted assemblage” (De Freitas and Sinclair, 2014: 34), a larger area in which the interactions take place. That is to say, Jim–number interactions always draw on spatiotemporal forces from their surroundings to act on his body in becoming 15. However, those forces are unfamiliar to and ignored by the prescribed conception.

Rotman (2000) suggests that these spatiotemporal forces may contribute to the appearance of so-called mathematical errors. As he maintains, operations with mathematical signs are undertaken by an implicit yet never acknowledged person “in accordance with time, space, and materiality in the presence of noise and error” (123). For Rotman, errors suggest the occurrence of embodied mathematics. As he further explains: “In the real world, your calculator batteries might run down. Or your pen might run out of ink … The important thing is that a boundary is there” (132). This boundary challenges the decontextualized assumption of mathematics made by Cartesian thinking. Refrains make the boundary plain as they situate a mathematical sign in multiple milieus of the Clubhouse. When these forces work on the ensemble of Jim and 15, the ensemble is “hooked to” (Massumi, 2002: 121) this spatiotemporal contagion, whereby the body may sense that it has learned something new. Thus, the 15-related knowledgeability is becoming through the moves and fluxes at those visceral points. In those imprecise repetitions, 15 becomes meaningful to Jim.

Coda

In the outcome-driven territory that characterizes mathematical activities, Cartesian thinking of mathematics learning generates rigid territorial impulses to make learning processes achieve the envisaged aims of taken-for-granted notions of mathematics and capable young learners. In challenging the meta-narratives, this study draws on the Deleuzian-Guattarian refrain, which allows rhizomatic movements to be immanent to a child’s body, as well as the processes of the researcher’s meaning-making. With this method, the study thus exposes the mismatch between the Cartesian intentionality to stay with prescriptions of mathematics learning and “lines and movements” (Jackson, 2016: 191) within a young child’s embodied relations with mathematics. This inconsistence throws much light on a body’s improvisational experimentations, which break up the unfolding of already existing schemas and transform present ongoingness in the unfolding of an event.

The illumination furthers the larger project of what counts as mathematics in terms of bodily awareness by adding to this account a novel and joyful layer that might be downplayed for its presumptuous divergence from schematic behaviors. As shown here, the content knowledge that encounters the body in movement is no longer predetermined by the human mind, but formed through the constant generation of mathematical meaning that relies on corporeal efforts toward uncertainty, triviality, and vitality. More than that, the way that the article highlights embodied mathematics suggests “a more horizontal representation of the relation between human and nonhuman in order to be more to the style of action pursued by each” (Bennett, 2010: 98). In other words, appreciating those bizarre, trivial actions in the construction of mathematics entails horizontalizing relations between a learner, mathematics, curriculum materials, and other milieus that are both perceivable and unperceivable, in impromptu participation in learning.

However, it is by no means to romanticize mathematics learning by suggesting that children would learn mathematics without well-planned scaffolding. In fact, scaffolding is important, but with a more dynamic and complex approach to it as comprising momentary, specific, and various problems in need of construction and response. The refrain is about “case-by-case logic” (Olssen, 2009: 83), where the focus is not on what children know globally in the plan, but on their mathematical living in the moment. It is from here that all potentialities could work. Therefore, we need to understand how the potentiality for the emergence of a mathematical concept in play is interrelated with the potentiality for the rhythmic unfolding which continuously connects the multiple milieus as they work their way in and out of space-time.

Hence, this ontological analysis of off-task behavior is a political and ethical move to avoid homogenizing and thus disenfranchising young learners of mathematics, whose subjectivity always embraces irrationality, ambiguity, and variety. Given this change, we might step back a little more often and ask ourselves what might happen if we thought and lived with movement as researchers or teachers, rather than raising ourselves to some already established higher ground from which to regulate and stabilize. What if we were to make affectively charged space-time for the body to multiply and intensify connections to mathematical objects as a part of becoming-mathematics? This article hopes to contribute a strategy for the vision of adopting a more democratic manner in mathematics learning, as suggested by the National Council of Teachers of Mathematics (2000) in the USA.