The Music of (Air)plane-Math

 
(Music by Mona Tynkkinen)

Air. A Whiter Shade of Pale. And now the above eight bars. Repeated musical movements in novel variations that hide a barely detectable Bach. A bit like the deterritorialization of math where an assignment sheet folded into a paper (air)plane is turned into an incentive to do math when a becoming-teacher declares, For each paper you finish, you may fold an airplane (C, p. 2). Math, folding (air)planes, dancing with a twerking reindeer, playing chess – the variations on what doing math entails seem endless. This is a look into what is happening during eighty minutes of ‘math’ and how one might explore and report changes that refuse to stay still and therefore evade naming. But first, musings about different language systems.   

Spoken and written languages enable reporting of local events across continents. Writing preserves what the impermanence of the spoken word fails to do[1]. Lexicon, interpunction, and grammar are the rules of language and the arbitrary sign-sound pairs we use to denote the signifieds in the world. Transmission of meaning is the main purpose here, and the aim is accordingly to represent what is ‘out there’ – as futile as that endeavor is, according to Deleuzian thinking[2].

Music, meanwhile, is less concerned with transmission of meaning. Music works through the affective registers that find their way straight into bodies. The frequency of sound physically touches non-/human bodies indiscriminately. There is not necessarily anything to understand in music, even though there is grammar to musical notation as well. Bare symbols combine into signaries that communicate complex information about pitch, duration, and rhythm. Expression moves through consonant and dissonant harmonies that change over time.

Different language systems, different possibilities, different grammaticality[3]. Take for instance the attempt to report the polyphony and flux of school life through written text, where events must be hierarchized and sequenced on the page whilst musical notation encourages relationality and asynchronous simultaneities. Or try reporting the transient relational architecting bodies slipping in and out of as part of living, where writing insists on naming and putting configurations in stasis to make them distinguishable, whilst musical notation has unnamed movement and change as its a priori. Or yet again, try reporting duration and velocity and language comes to hide temporal diversity in letters that hide in words, that hide in phrases, that hide in sentences, that hide in paragraphs. Musical notation, meanwhile, keeps them visible – and audible – through equally intricate internal structures (notes, phrases, bars) but through simpler signaries.

I propose we now move beyond comparison. We need not choose between systems. Deleuze and Guattari write that

the objection will be raised that music is not a language, that the components of sounds are not pertinent features of language, that there is no correspondence between the two. We are not suggesting any correspondence. We keep asking that the issue be left open, that any presuppose distinction be rejected. ([1980]1987, p. 96)

Let us refuse choosing and resist convention[4] and instead push boundaries and experiment. After all, the systems we plug our thinking and communication into provide different powers, so why choose when we can augment our affective capacities through experimentation with mixtures.

This mosaic plays with the transversality of fusing musical notation with writing[5]. The combination serves to augment the power to explore the problem of ‘how the contingency of the present affects teacher becoming – and how becoming-teacher unfolds in capricious and inexorable presents’. It does so by following becoming teacher during eighty minutes of math and asking, ‘how do encounters unfold and with what effects?’

This mosaic provides at least three different pathways through the text: 1) Read running text and all footnotes and you will get the onto-epistemological project together with the transposition between language and musical notation put in music terminology , or 2) stay with running text and make selected visits to footnotes and you will get the gist of the onto-epistemological project, or yet again 3) go straight to(Air)plane Math and experience eighty minutes of math told through music and musical notation. First, the context.

Covering for a Colleague

A becoming-teacher is sent to a classroom to cover for a sick colleague (teacher 2, T2) during eighty minutes of math. With eighteen minutes to prepare, becoming-teacher professes to another becoming-teacher to not feeling comfortable doing fractions. With no more than a stack of assignments in hand, becoming-teacher enters the classroom four minutes late:  

10.04, enters the classroom. Four children in the classroom. Children ask questions, becoming-teacher replies that becoming-teacher is about to answer in just a minute. A classroom assistant enters and announces that they will take three children to a group room. Becoming-teacher reminds the assistant to take copies of the printed assignments. […] One of the now four children in the classroom says to a classmate, Come on, I need to talk to you for a bit. The two children (C1, C2) leave the classroom before returning a minute later.  

The class consists of four children now that the assistant has gone elsewhere with three. Becoming-teacher points at the assignment and says to one of the children, You are going to do this, teacher 2 (T2) said so…The child (C2) answers, We’ve already done this. Becoming-teacher replicates, But you have to fortify it, so do it. The child answers, But I already know how to do this, ask T2. Becoming-teacher replies, Sure... before moving on to the next child.  (C, pp. 1-2)

Becoming-teacher enters, children exit. Children return. The class of four children and a becoming-teacher are now facing eighty minutes of what a schedule states is ‘math’. Immediately a child challenges the only materials becoming-teacher has to offer, whilst becoming-teacher insists that ‘you have to fortify it’ because another teacher has ‘said so’. But attempts to convince fail. The exchange intensifies and the child uses the same absent teacher and their long-lasting teacher-learner relation as a trump, which has becoming-teacher surrender with a ‘Sure…’ before walking away from the exchange.

With only four children in the full-sized classroom, bodies become peculiarly visible to one another. No gesture and no exchange can go undetected, and each begun and abandoned confrontation becomes a drama unfolding on an empty stage before its small audience. One need accordingly not be boisterous or loud to affect the assemblage. In fact, through all that happens, bodies dutifully try to perform the anticipated yet unscripted ‘math’-play.  Back now to the classroom.

Having just forfeited an exchange to a child, becoming-teacher now moves to the neighboring seat where another child immediately asks for help:

Check if it’s correct, the child asks (C1). Becoming-teacher replies, Have you simplified? The child says, I won’t do that, wherebybecoming-teacher replies, teacher 2 has said that you have to fortify this. (C, p. 2)

The child’s request that becoming-teacher check their work for  correctness generates a question instead of an answer. Asking if the child has ‘simplified’ indicates that becoming-teacher knows what the mathematical operation needs, but the child ‘won’t do that’. Or perhaps the child does not know how to simplify fractions? Either way, becoming-teacher enacts the child’s rejection as a question about willingness through the articulation ‘you have to’ – or perhaps becoming teacher’s interpretation is a way to avoid having to show the child? Regardless of why becoming-teacher shies away from showing how to simplify, the effects of not doing so become decisive; the child is left with no clues about whether the solving was correctly executed, and therefore no leads on how to continue with fractions. Moreover, becoming-teacher has just failed to perform a key teacher move, namely that of helping a child.  

During eighty minutes of ‘math’, becoming-teacher and three out of the four children can be followed struggling with the assignments at hand. In an institutional space where bodies have been taught that there is a teacher that knows and pupils that learn, a predefined subject with already solved problems for pupils to (re-)solve, the quintet now faces an extended period where familiar roles and rules have been cancelled. Not only must bodies create math material with nothing more than available means for a period of eighty minutes, but they must also territorialize new functions – invent roles to play. Because ‘teacher’ is not something one is, teacher is something one does. And with no territorialized teacher function in the classroom, there is no pupil function available either.

Coping

So “how [do] living beings create a space for themselves to maintain their existence”; how do they “stake out a place” for themselves “if not guaranteed a place in the world by some principled sovereignty or taxonomical destiny“ (Kleinherenbrink, 2015, p. 210)? Bodies territorialize territories through the enactment of refrains; that is, they perform minor acts that incorporate a selection of available matter to create temporary circles of order, Deleuze and Guattari propose ([1980]1987).

Before looking at the concept ‘refrain’, we will take a glance at what the quintet is doing to create such order in chaos: One child shoots a paper airplane made out of an assignment sheet; another child writes never-to-be-finished calculations on the whiteboard; a duo engages in incessant talking. A becoming-teacher, meanwhile, encourages children to dance with a twerking reindeer, fold yet another airplane, or simply do as told with reference to another teacher having said so. Or play chess on their laptops. The fourth child, meanwhile, knows how to do fractions but asks for permission to go sit and work in the adjacent group room with reference to the incessant talking from classmates.

In less than twenty minutes and with more than one hour of math to get through, there are now three children inside a classroom trying to organize chaos and one child seated on the other side of a semi-glassed wall – a visual actualization of resistance. Bodies enact refrains to create ‘purpose’ and ‘meaning’ by drawing transient ‘math’-territories to cope with eighty minutes of ‘math’; this “is the composition of one’s own [here: math-]world” (Buchanan, 2021, p. 98).

Refrains                                             

“The role of the refrain has often been emphasized: it is territorial, a territorial assemblage” (Deleuze & Guattari, [1980]1987, p. 312). Refrains bring together code from familiar milieus through mundane acts that create new mixtures through a form of transcoding (Deleuze & Guattari, [1980]1987). Becoming-teacher having to territorialize the teacher function through the creation of teacher-refrains, is another way of saying that math class is not math just because the schedule says so; doing math class entails the territorialization of math-milieu, teacher-milieu, and child-milieu through the production of teacher-refrains. The assemblage must accordingly be actualized as a math class, which is the joint work set out for becoming-teacher and children.

However, refrains do not merely territorialize territories, which they do; the refrains we repeat also entail component selection, exclusion, and that we organize a “calm and stable” center, “organize a limited space” and make transient homes for ourselves that keep chaos on the outside (Deleuze & Guattarian, [1980]1987, p. 311). In this circle of safety, the repetition of mundane acts serves to comfort us, but they also carry the power to deterritorialize transient homes, carry them away and create something unrecognizable out of the familiar as matter is turned into (math-)material (Deleuze & Guattarian, [1980]1987) –‘descending airplane’-matter turned into a ‘math incentive’-material, a ‘semi-glassed wall’-matter turned into a ‘resistance’-material. The refrain therefore combines into a material-doing.

Kleinherenbrink also emphasizes that this repetition must be thought of in accordance with Deleuze (and Guattari’s) overall stress on difference; this is to say that the repetition of refrains “is defined by variation” [6]  ([emphasis added] 2015, p. 225n1).This variation is in fact what I propose prompts the usage of musical notation in the first place, since notation builds on and enables showing how each actualized refrain builds on variation (in pitch, duration, rhythm etc.), not linguistic categorization.

The ensuing section offers ten repeated refrains enacted during eighty minutes of (air)plane-math. It was their repetition that made these otherwise seemingly mundane acts salient. To examine how refrains keep repeating and creating affect fluctuations in each encounter, I begin by bringing together language and musical notation. For those unfamiliar with musical notation, not to worry! The sense of ‘not quite knowing’ simply means that you have gotten a head start into exploring the most potent side of presentness and thus what I propose becoming-teacher entails, namely the terror of having to continue acting whilst on the threshold of not knowing.

Coping Refrains

Below is an overview of the enacted Coping Refrains. On the left side, refrains as musical signatures reported through musical notation. Refrain content is depicted on the right-hand side accompanied by an explanation of the intervals and rhythm of each musical signature.

Illustration 1: Coping Refrains

If combinations of letters can form ‘words’ that can be paired with sounds to form arbitrary relations with a world they conventionally are proposed ‘representing’[7], why not do something similar with musical notation (see Illustration 1)? However, instead of using words to explain and differentiate between different refrains, I use ‘musical signatures to present refrains which can be placed in sequence ad infinitum with other refrains to create a melody, andstacked vertically to create a harmony, so that each combination creates variation within a refrain.

Rather than proposing that musical notation and the sounds they are paired with (movement, pitch, and duration) can re-presentthe real, I suggest that the music musical notation creates has the capacity to express fluctuations in affect in a way that is “oriented toward an experimentation in contact with the real” (Deleuze & Guattari, [1980]1987, p. 12). Said differently, music has the capacity to affect again.   

Body as Instrument

Each human body actualizing the ‘math’-assemblage is accordingly presented as an instrument in an orchestra in keeping with the below outline.  

Bodies as instruments in the score (Air)plane-Math

Illustration 2: Overview of bodies as instruments presented as parts in a score[8]

Movement as Melody

Let us now return to the introductory scene where becoming-teacher is enacting challenge-refrains to get a child (C2) to do assignment sheets. This time the same exchange is paired with musical notation presented as a score, that is,a visualization of all instruments involved in playing a piece – or said in Deleuzian terminology, all bodies actualizing the ‘math’-assemblage.


Illustration 3: Challenge-refrains, becoming-teacher (BT) and child 2 (C2) [bar 8-12]

The challenge-refrain created by becoming-teacher (in red) is borrowed by child 2 (in green) before travelling between the two in an additional round (see Illustration 3). In each turn the refrain modifies through change of instrument and climb in pitch[9]. Musical notation enables illustration of how the effectsof affect build over time and change through each repetition. This is how musical notation shows the beforementioned variation within refrains.

Even though a challenge-refrain forms a melodic line carried between becoming-teacher and child 2 alternately, musical notation also keeps in view the other three children partaking in unfolding events[10]. The refrain is a combination of movement and selected components that irrevocably comes to interweave all other instruments as a chorus, a kind of background. Some instruments take the lead for a while whilst others turn into chorus before a new refrain reorganizes assemblage relations. The ontological force of presentness and movement can be shown as refrains augmenting and waning, before being exchanged by new lines, at all times securing the melody of (air)plane-math. Thus, refrains weave (math-)materiality.   

Transposition from Text to Music

A look now at the process of translating – or transposing[11] – from text and written language to musical notation.

Transposition from written text to musical notation is exemplified through the below scene (see Illustration 4) where a child asks becoming-teacher for permission to leave the classroom due to classmates incessantly talking. The scene covers a talk-refrain (C3 and C4) and the moment when the child (C2) tells becoming-teacher that two classmates (C3 and C4) talking is disturbing, and in so doing actualizes talking as discordance.

The text excerpt is from a mapping, the subsequent snippet from the score. Next to the score there is an account of the musical composition that explains how relational architecting builds relationality through a vertical axis by bringing together individual instruments (bodies) to form a harmony;in other words, how different notes sound together at a certain moment. 

Excerpt from mapping:

·    C3 and C4 sit and discuss schooling and grades [in a voice so loud that it carries through the entire classroom].

·    BT and C1 look at C1’s calculations. C2>BT May I sit in the group room, they [the two pupils discussing grades] are talking so loudly. (C, p.2).

Transposition to musical notation:

Illustration 4: Example of transposition from text to musical notation [bar 17]

Each unfolding encounter during eighty minutes of math has thus been carefully examined by asking ‘what are bodies doing with what/whom’, and ‘how are these doings affecting other bodies and unfolding events’[12]. Eighty minutes of math transposed into musical notation also transposes time; from eighty minutes taken as what Deleuze and Guattari ([1980]1987) speak of as ‘sequenced and linear time of chronos’, or clock-time, to music and the ‘fluid and non-pulsed time of Aeon’[13]. It is what happens in the unfolding scene that guides the number of bars[14] in musical notation, not how many minutes something takes[15].

Relational Architecting  as Harmony

Harmony, such as the vertical axis shown in Illustration 4 where multiple instruments create a joint sound, is thus one way of creating tension and release in music. ‘Harmony’ is about relationality and how tones played by multiple instruments create distance in pitch amid the sounding instruments, so called intervals. These intervals affect each sounding instrument differently while simultaneously creating a consonant/dissonant harmony, like the way in which a duo’s talk-refrain creates a harmonious third between the two of them, whilst in relation to another child (C2) a dissonance is created that marks how the duo is enacted as disturbing.  

Music helps explain how affect has no absolute value that can be measured by universal metrics. Effects of affect have no point zero. Discord is not one specific pitch in isolation. Affect is the consonance or dissonance of the sounding harmony resulting from the immanent relations between actualized intervals. Harmony as a thinking tool helps illustrate how the seemingly insignificant addition of one singular body in a classroom, such as an entering classroom assistant or a child leaving, can alter the harmony of an entire assemblage.    

So, when Deleuze’s Spinoza ([1981]1988) speaks about ‘joyous and sad affects’, such affective changes can be conceptualized as relational architecting where acts (here refrains) in certain material configurations augment or diminish the various and varying affective capacities of bodies. Let us once more take the example of the duo’s talk-refrain; talking diminishes another child’s (C2) working capacity. It is thus not talking per se that is ‘bad’, but talking can become a ‘sad affect’ under certain circumstances. In the score (see Illustration 4), therefore, it is a child’s (C2) act of resistance that creates dissonance between bodies in the moment when becoming-teacher is enacted as too lenient and thus the two classmates too talkative[16].

Put differently, there is no mode of existence that is universally ‘Good’/’Evil’ in itself, just like there is no note that is dissonant/consonant; rather, a note’s/act’s capacity to affect, and the acted upon’ s capacity to become affected, has to do with immanent relationality. This is why this kind of inquiry into the different modes of existence belongs to Ethics:  

Ethics, which is to say, a typology of immanent modes of existence, replaces Morality, which always refers existence to transcendent values. Morality is the judgment of God, the system of Judgment. But Ethics overthrows the system of judgement. The opposition of values (Good-Evil) is supplanted by the qualitative difference of modes of existence (good-bad). (Deleuze [1981]1988, p. 23)

Enacted refrains are therefore the ways in which bodies attempt to create livable order with other non-/human bodies. One might not know beforehand what augments/decreases (a) body/-ies’ capacity/-ies, but in the course of eighty minutes of ‘math’ there are ample opportunities to experiment with how one might make such space-times livable.  

Punctual System

A brief recap of the presented base lines of musical notation before adding the last elements necessary for an exploration of the full score of eighty minutes of (air)plane-math.

First, there was melody (horizontal axis) which enables presenting the inexorability of the present as the change and movement of each body/instrument over time. Then, there was harmony (vertical axis) which enables presenting the immanent relationality of living, and how refrains entail a continuous relational architecting. Musical notation is accordingly not an ideal system, it too entails grammaticality. Rather, musical notation is what Deleuze and Guattari speak of as a punctual system[17].

Illustration 5: The two base lines of musical notation taken as a 'punctual system'

The ‘transversal tie-in’ (Deleuze & Guattari, [1980]1987) of musical notation and language is what provides necessary tools to examine molecular alliances and molar tendencies[18]. Put succinctly, we know each encounter is unique, but how do refrain pulsations work over time – and how do refrains affect teacher-becoming during eighty minutes of ‘math’? Before introducing the entire score, we turn to the last elements of musical notation, namely duration and velocity.   

Duration and Velocity as Rhythm

Each (non/human) body and refrain have durations and are at the same time tessellations of different velocities. In this mosaic and in the context of music, durations and varying velocities are bodies proposed as rhythms[19].Let us be reminded of how Deleuze’s Spinoza describes bodies:  

The important thing is to understand life, each living individuality, not as a form, or a development of form, but as a complex relation between differential velocities, between deceleration and acceleration of particles. A composition of speeds and slownesses on a plane of immanence. In the same way, a musical form will depend on a complex relation between speeds and slownesses of sound particles. It is not just a matter of music but of how to live: it is by speed and slowness that one slips in among things, that one connects with something else. One never commences; one never has a tabula rasa; one slips in, enters in the middle; one takes up or lays down rhythms. ([italics in original] Deleuze, [1981]1988, p. 123)

All (non-/human) bodies form different duration(s) that come with different speeds and slownesses[20]. In encounters with other bodies, we must therefore calibrate our velocities to those of others and the matter we engage with. The beforementioned exchange (see Illustration 3, [bar 8-12]) where a challenge-refrain can be followed alternating between becoming-teacher and a child (C2), illustrates how the stasis of an assignment sheet participates in speeding up speech and pulse in a becoming-teacher trying to get an agitated child (C2) to do what is conceived as futile exercises – all whilst the rapidness of the exchange arrests physical movement.

Effects of affect in encounters therefore also involve the changes in a body’s time spent on a refrain, and/or the speed with which a refrain is enacted – and the duration and speed of a body doing something. These temporal matters can be shown through musical notation as differences in a body’s duration (note value[21]) and the velocity of doing (tempo) in relation to other bodies (rhythm and syncopation)[22].

Note Value   

A last return to bar 17 where a child trying to finish a stack of assignments (C2) is asking becoming-teacher for permission to go sit elsewhere due to the incessant talking of two classmates (C3 and C4):

 
Illustration 6: Refrain and note value [bar 17]

“Relations of motion and rest, of speeds and slownesses”, together with how “a body affects […] or is affected by other bodies”, are told to “define[…] a body in its individuality” (Deleuze, [1981]1988, p. 123), which is why I elsewhere have tried to keep the rhythmicity of living at the fore by presenting bodies as rhythm bodies (see Deleuzian Ontology in Education). A collectively shared event such as bar 17 (Illustration 6), therefore, shows how it is the situation and our shifting ways of acting upon it that brings out our individuality – not the other way around. We are not readymade I’s clashing with the world, but rhythm-I’s teased out through events. Child 1 not actualizingchild 3 and 4’s talk-refrain as unsettling (which does not exclude that C1 feels disturbed), can in musical notation be shown as calm[23]. Becoming-teacher, similarly, is shown as not acting on talk before child 2 points it out[24].

Hence, this line of reasoning suggests that it is the specifics of eighty minutes of ‘math’ in this classroom that brings about a becoming-teacher that proposes dancing as solution to the problem of not knowing how to do fractions whilst two children consistently talk to make time pass. All the while another child sends a mayday through never-to-be-finished calculations on a whiteboard, and a fourth leaves to sit elsewhere.   

(Air)plane-Math

Now the full Score where eighty minutes of ‘math’ has been transposed into ninety-four bars in keeping with the aforementioned rational. Then the Music.

A reminder: Melody provides the movement and variation within enacted refrains in relation to each instrument (horizontal axis), and harmony presents the immanent relationality where bodies sharing a space-time come together through refrains that have bodies engage in relational architecting (vertical axis), whilst rhythm presents the different and varying durations and velocities of bodies trying to calibrate to one another, to lesson time and to materials. In the score, all three dimensions come together and present how bodies enact refrains that serve to make chaos livable. All refrains have been marked and color coded in keeping the overview provided below the score (full-sized score is appended).  

Score

 

Becoming (Air)plane-Math 

The score provides an overview of the production of refrains in terms of their frequency, order, and distribution. At a quick glance the many talk-refrains come into view, the ways in which challenge-refrains move between parts, how bodies keep entering and leaving the classroom (EE-ref.), and the many attempts at territorializing math through math-refrains. Some refrains go together with specific bodies[25], whereas challenge-refrains, whiteboard-refrains, math-refrains, and music-refrains are borrowed and move contagiously between instruments.

The score also makes visible patterns, such as how becoming-teacher repeatedly follows up discipline-refrains with math incentive-refrains, as if trying to smooth over reprimands with suggestions on something fun to do instead. A closeup into the contents of incentive-refrains show how refrains morph over time; from incentives to do math to incentives to do other things than math[26]. As a folded airplane lands in front of becoming-teacher, the initial response is a discipline-refrain. But then the frowned-upon airplane becomes reterritorialized as an incentive-refrain[27] as becoming-teacher says, For each paper you finish, you may fold an airplane (C, p. 2). Eighty minutes of ‘math’ thus becomes The Music of (Air)plane-Math, which is a morphing that can be visualized through musical notation.

Relay Work

Bars 7-16, 26-39, and 48-73 illustrate the kind of relay work that goes on between instruments during eighty minutes of ‘math’. Relay work consists of becoming-teacher enacting refrains that children respond to, or children enacting refrains that becoming-teacher responds to. However, while the children can be seen taking turns in actualizing challenge-refrains, whiteboard-refrains, help-refrains, or shoot off airplane-refrains – often in combination with ongoing talk-refrains – becoming-teacher is alone in responding to children’s refrains. The children can rest and let someone else take the lead for a while before making a new attempt at organizing chaos. Becoming-teacher, on the other hand, is affected and affects other bodies in every turn with no chance to rest. In fact, it seems that the only opportunity to rest is by leaving the shared space-time by taking a stroll to the staffroom [bar 45-48] and reframing the situation as a case of children not knowing how to do fractions, or when children are mimicking a twerking reindeer on the projector [bar 76-82].

Fragile Math-Refrains

Most refrains have in common that they seek to territorialize math content in a space-time that provides limited resources to augment one’s fraction-solving capacities. In fact, it is not the absurd institutional framing that affects teacher and pupil becomings; as part of schooling they must stay in this space called ‘classroom’ for eighty minutes as part of a ‘lesson’ to pretend they learn ‘math’. Had it been a ‘let’s hang around here for a while and come up with something fun to do’-event, then the same situation would most likely instill a sense of joy and accomplishment. But bodies keep trying to do ‘math’, which is why talking, refusing, proposing that they dance, or fold airplanes, suddenly seem reasonable solutions to create order in a world that has none. But the stakes are high for all involved, and endurance painful when having to stay for another hour in a space where a child has asked its supposed teacher, Can’t you teach us something instead of doing this fart?[28] (C, p. 2).

Vulnerable Lead

Upon the return of the second violin [C2, bar 56], the child hands over a finished stack of assignments to becoming-teacher. Becoming-teacher begins reading whilst suggesting that the group could dance before doing more math. No one wants to dance. With the help of the child’s finished assignments, becoming-teacher puts everything on the line by taking back the front stage position from the whiteboard-child [bar 68], writes a fraction and then asks the rhetorical question How do you solve this? [29](C, p. 3). The score illustrates the relay work becoming-teacher and a child (C3) engage in as they explore fractions through math-refrains [bar 68-72]. But the child’s answer puzzles becoming-teacher who asks the child to explain, How do you arrive at ‘a half’? (C, p. 4). Rhetorical questions have turned into earnest interrogation where a becoming-teacher asks a child for help. So, the child explains. There is a vulnerability to the unfolding scene where the attempt to take the lead fails. After this, becoming-teacher abandons the idea of doing more fractions.

More dancing, another airplane, bodies entering, and incessant talking, the frenzy escalates before settling in the moment becoming-teacher goes about territorializing the teacher-function in a way that has math itself morph: No more dancing, there’s ten minutes left of class, work on your laptops or play math games [bar 82, 83]. ‘Work’ is not ‘dancing’, it is working on their laptops or playing math games, a shift musically expressed by having math-refrains change rhythmically from eight and quarter notes into quarter-note triplets [bar 83-93] (see appended full-sized score). What the remaining time entails is playing chess, or rather, talking about the program where one enters to play chess and the opening proverbs that greet entering players.

Music

Listen. Hear flux. However, it is not a Crumbian experimentation[30], but an orderly and in many ways ordinary piece of music where instruments dutifully arrange refrains by abiding the institutional rules of (air)plane-math. Ordinariness is what makes refrain-making so potent, so hazardous; for you may not notice distortion unfolding in the open. Standing on the threshold of eighty minutes of ‘math’, and you would see nothing but familiar materials (math assignments, whiteboards, markers, desks, chairs, laptops, children, and an adult referred to as ‘teacher’) doing familiar things (trying to finish an assignment sheet, writing fractions on a whiteboard, children chatting, someone seated in the adjacent group room working, working on laptops) – even children twerking to a reindeer may pass you by when you notice a ‘teacher’ contentedly watching. The music of (Air)plane-Math is equally deceptive.

Listen to the quintet work through melodic lines on their expedition through ever-changing harmonies. Each measure is an aurally crafted duration, a present past eagerly reaching towards future beginnings. Intervals shifting through sonic frequencies by gliding and leaping in pitch through rhythms that sooth, hesitate, exhaust and limp. Just like the groping practices of life in classrooms.

Music makes the many-ness of few felt. Said differently, the intensity and force of as few as five bodies can be expressed through music as repeatedly climbing fluctuations between consonance and dissonance. In fact, following individual melodic lines and examining how they travel between and into other instruments is possible thanks to this fewness[31]. Yet, few-ness must not be confused with effortlessness. Instead, it is humbling to consider the complexity that goes into the melodic calibration between a becoming-teacher and even one child. 

Music evades naming, transcends category, and leaps towards the future by suturing presents saturated with pastness. Refrains as the mundane acts performed to create transient horizontal order in the vertical flux of relationality, where bodies without time to ponder, pause, nor plan, grabs hold of what is in the hope of finding math-treasures in unfolding presents. Nothing less than everything is at stake in each moment.

But it seems as if the most blatant aspect of all processes of becoming is so obvious that it has been forgotten, passed over, or simply ignored; that is, the way in which becoming is at the mercy of the inexorable present pushing bodies through the capriciousness of presentness. It is thus the ontological force of the present itself that most profoundly affects all, teacher becoming included. The best way to do the inexorability and capriciousness of the present – make it felt again – is through music on play. Just like the present, visitors are offered no chance to pause the music, only to stop and leave all together. What refrains do you create to keep chaos at bay?

Epilogue

12.13, the staffroom. Nine colleagues. Becoming-teacher talks about how math class went. Tells a colleague about a child who stated that they “wanted to learn something” but that it turned out that the children didn’t know how to do what they were supposed to when working with the assignments. 12.19, becoming-teacher talks about how ‘the teacher calling’ was challenged last week when there were so many on sick leave and becoming-teacher had to fill out for absent colleagues alone in class. […] 12.32, becoming-teacher by the computer with a coffee and chocolate. The supervisor hands becoming-teacher a stack of printed sheets for English class. Becoming-teacher looks at the stack of papers and turns to the accompanying teacher’s manual.  (C, pp. 4-5) 

The encounter with impromptu substituting is one of the things that diminish not only a becoming-teacher’s, but also children’s capacity to act. It is up to a becoming-teacher and children to invent ways of handling and enduring the inexorability and capriciousness of the present during eighty minutes of ‘math’ in a space-time with scarce resources to augment one’s math capacities. Through the production of refrains, bodies organize chaos and in so doing create glimpses of increase in one’s capacity to act – if nothing else, from the joy of resisting farce.

Covering for sick colleagues is a routine practice in this school assemblage and part of becoming-teacher’s workweek. Having to face children and enact ‘lessons’ in unfamiliar subject areas repeatedly is a setup that has becoming-teacher question ‘the teacher calling’. So, in the absence of actual success stories, becoming-teacher invents stories where only the children are uneducated. Actualized classroom events, meanwhile, remain a well-kept secret.  

Seventy minutes after (air)plane-math, becoming-teacher is handed a new stack of assignment sheets, this time by the supervisor to use in the upcoming English class. The support to decipher what the handouts cover and what to do with them is for becoming-teacher to find out from the appended teacher’s manual.  

References

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