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Susan Gerofsky:
Performance Space & Time
A symposium discussion paper for “Digital Mathematical Performance: A Fields Institute Symposium” Faculty of Education, University of Western Ontario 9 June – 11 June 2006.
Précis: I would like to propose a view of digital mathematical performance that creates a liminal space – a passageway between worlds where boundaries are deliberately blurry: boundaries including disciplinary boundaries (math vs. applied skills, music, art or drama, for example), distinctions between “teacher as knower” and “student as listener” (already challenged by the use of computer networks in schools), the differentiation between performers and audience, boundaries among the virtual, imagined and real, and mind/ body boundaries that have long played a central role in shaping the image of mathematics.
“Performance” as a generic category has its own new academic discipline, Performance Studies, that came into being in the 1980’s and 1990’s. Performance studies draws together notions of performance from such diverse sources as theatre, literature, visual arts, anthropology, sociology, linguistics, artificial intelligence and cultural studies. The kinds of activities included as “performances” are tremendously varied:
The term performance incorporates a whole field of human activity. It embraces a verbal act in everyday life or a staged play, a rite of invective played in urban streets, a performance in the Western traditions of high art, or a work of performance art. It includes cultural performances, such as the personal narrative or folk and fairy tales, or more communal forms of ceremony – the National Democratic Convention, an evensong vigil march for people with AIDS, Mardi Gras, or a bullfight. It also includes literary performance, the celebration of individual genius, and conformity to Western definitions of art. In all cases a performance act, interactional in nature and involving symbolic forms and live bodies , provides a way to constitute meaning and to affirm individual and cultural values. (Stern & Henderson, 1993, p. 3; italics in the original.)
In recent years, theorists of performance have expanded the notion of performance to include digitally-mediated performances, especially since these can now have a strong interactive, “live” component rather than simply acting as an archive of an ephemeral live performances. Post-structuralist theories of the performative have been readily taken up by performance studies:
Performance will be to the 20th^ and 21st^ centuries what discipline was to the 18th and 19 th, that is, an onto-historical formation of power and knowledge […] Hyphenated identities, trangendered bodies, digital avatars, the Human Genome Project – these suggest that the performative subject is constructed as fragmented rather than unified, decentered rather than centered, virtual as well as actual. […] While disciplinary
institutions and mechanisms forged Western Europe’s industrial revolution and its system of colonial empires, those of performance are programming the circuits of our postindustrial, postcolonial world. […] Research and teaching machines once ruled strictly and linearly by the book are being retooled by a multimedia, hypertextual metatechnology, that of the computer. (McKenzie, 2001, p. 18)
It is unusual (and energizing) to link mathematics and math education with performance, in no small part because many of the things that make a performance distinctive and interesting go squarely against many of the long-held traditions of mathematics. I will explore the human necessity of performance in mathematics, and suggest some of the varieties and features of performance that might be offered to mathematics education from other fields.
Later in this paper, I will consider the varieties of “live” performance included in performance theory and some of the ramifications of digital performance in mathematics education. First, however, some ideas about the necessity of mathematical performance in space and time.
Platonic vs. performative mathematics: “all-at–one-point” vs. space & time Kurt Vonnegut, in his novel Slaughterhouse Five (Vonnegut, 1968), describes novels on the (fictional) planet Tralfamadore which exist “all-at-a-point” and are taken in all-at- once:
…Each clump of symbols is a brief, urgent message – describing a situation, a scene. We Tralfamadorians read them all at once, not one after the other. There isn’t any particular relationship between all the messages, except that the author has chosen them carefully, so that, when seen all at once, they produce an image of life that is beautiful and surprising and deep. There is no beginning, no middle, no end, no suspense, no moral, no causes, no effects. What we love in our books are the depths of many marvelous moments seen all at one time. (Vonnegut, 1968, 88)
In my own experience as a mathematics learner, I often experienced frustration that I could not “read” theorems and proofs like a Tralfamadorian novel – all-at-once, rather than over time. For although both theorems and proofs are unfolded over time and space to facilitate communication and learning, there is as sense in which the culture of mathematics insists that the whole story truly exists as a single, elegant point, which in itself contains all its possible ramifications, whether we inadequate humans can perceive them in that form or not.
Mathematical proofs and solutions have an element of the tautological – the conclusions are logically contained in the assumptions. In some sense all mathematical proofs and algebraic calculations are identities , in that each line of the work is logically identical to every other line.
What could be meant by the opposite of performance in mathematics education? It would not be simply the introverted, quiet, private working out of solutions that has been associated so often with mathematical work, since this is still a matter of performance (albeit a physically restrained one, performed for an audience comprising only oneself). ‘The opposite of performance’ could mean an instantaneous, intuitive grasping of the mathematical implications inherent in a situation, an epiphany, all-at-a-point. These ecstatic, intuitive moments do come to mathematicians, often after long stretches of hard performative work on a problem, and even these require a more public performance in the verbal “unpacking” that enables others to judge the soundness of the intuition.
Varieties of performance Traditionally, and particularly in the secondary and postsecondary math classes that I have observed, students’ mathematical performance has been limited to a very quiet, private performance using head and hands, pencil, paper and calculator, with the rest of the body held still and quiet, and an audience of one or two (the learner and perhaps the teacher). The teacher’s traditional performance is somewhat more active but just as rigidly circumscribed – lecturing, writing on a board or overhead projector, pacing, asking for occasional short answers from the audience.
Performance studies offers a much broader range of activities that have potential as mathematical performances in education. I will explore three of these, which overlap with one another: ritual, play and improvised theatre.
Ritual in mathematics education The study of ritual comes to performance studies from anthropology. Ritual performances range broadly in context and purpose, from the liturgical order of religious rituals, to partially codified, partially improvised secular rituals like courtroom trials or the christening and launching of a ship, to private rituals and routines in everyday life which may be considered habitual or even compulsive. All these rituals share the importance of a fixed sequence of behaviour, often using language, rhythm and gesture. Rituals involve repetition and codified actions, which must be performed correctly when called for by tradition or circumstances. (Schechner, 2002, p. 163).
The function of ritual is often a repairing or remaking of the world, through “restored behaviour” (Schechner, 2002). The familiarity of ritual is a reassurance that, come what may, a culture has continuity over time. Even as these “restored behaviours” are repeated and recontextualized in changing circumstances, the core elements are preserved through oral tradition and memory. Rituals serve as links across generations and the vaguaries of life to reaffirm what is important, mysterious, and central to our cultural knowledge and beliefs. In this way, they fulfil many of the same functions as any formal education system.
In Finnegans Wake , James Joyce invoked “history as she is harped, rite words in rote order”, and his pun points out the close relationship between ritual and rote learning, the rightness of certain words in the order they are written or chanted in ritual performances (as in the singing of a culture’s oral histories to the accompaniment of a harp).
Mathematics education, for all its emphasis on logic and rationality, encompasses many rote and ritual elements that lend themselves to performance which may be memorable, playful, even trancelike and hypnotic.
For example, counting itself is first of all a ritual and rote activity. Learning to count initially involves a rhythmic chant of intrinsically meaningless sounds, which are associated first as a sequence (like the names of the letters of the alphabet), and only later with particular quantities or values. Counting initially consists of pointing at objects in a sequence and reciting the elements of the counting chant.
Children often learn to count by twos and by fives playground skipping and clapping games, which involve social interactions, tunes, and rhythms marked by kinesthetic movement:
Cinderella, dressed in yella How many times did she kiss her fella? Two, four, six, eight, ten, twelve, Fourteen, sixteen, eighteen, TWENTY Two, four, six, eight, THIRTY Two, four, six, eight, FORTY…
Bluebells, cockle shells Eavy, ivey, over Five, ten, fifteen, TWENTY Twenty-five, THIRTY Thirty-five, FORTY Forty-five, FIFTY…
Children’s culture is full of number chants that take on the nature of rote learning and ritual. Children engage with these with enthusiasm in playful settings, learning number patterns and the history of counting in the course of play.
Since children engage in counting rituals in their own play, could mathematics educators bring playful rituals to classroom learning? There is a danger in trying to deliberately invent folklore – the result may be stilted or too-precious, and it may be better to research and provide variations on existing ritual traditions. Nonetheless, it can be very interesting to tap into ritual elements for that rote work which is worth “having off by heart”. In order to make a ritual memorable, it must incorporate ritual features of rhythm, gesture, bodily movement, chant, and perhaps rhyme. I still remember the twelve times table off by heart because our Grade Six teacher made us walk across the classroom chanting a number for every step, We practiced this ritual day after day till everyone had got it.
To make a mathematical classroom ritual more memorable, teachers would do well to learn the elements of theatre to add multisensory associations to the ritual performance. Dorothy Heathcote writes about the essential oppositions used to create dramatic effect in all theatrical performances: light versus darkness, movement versus stillness, sound
Provocative ideas on configuring time and space from performance studies: (1) Liminal spaces Performance traditions of all kinds reconfigure our perceptions of time and space, creating places for play and transformation and disturbing the conventional order of things. Within performance studies, one of the most distinctive qualities of performance is seen to be the liminal , protocultural space it claims and creates.
The term “liminal” was introduced by Victor Turner, who came to performance theory from anthropology (Turner, 1982). Turner was looking at anthropological accounts of rites of passage within a variety of cultures, and saw these transitional rituals in the life cycle as occurring in an “in-between” or liminal space, between sites of more conventional cultural activity.
Richard Schechner, a key performance theorist with a background in both anthropology and theatre, develops the idea of liminal cultural space as a quality of the actual physical spaces where performances are enacted:
A limen is a threshold or sill, a thin strip neither inside nor outside a building or room linking one space to another, a passageway between places rather than a place in itself. In ritual and aesthetic performances, the thin space of the limen is expanded into a wide space both actually and conceptually. What usually is just a “go between” becomes the site of the action. And yet this action remains, to use Turner’s phrase, “betwixt and between”. It is enlarged in time and space yet retains its peculiar quality of passageway or temporariness […] An empty theatre space is liminal, open to all kinds of possibilities – that space by means of performing could become anywhere […] The spaces of film, television, and computer monitors […] apparently full of real things and people, are actually empty screens, populated by shadows or pixels. (Schechner, 2002, 58 – 61)
A classroom can be a liminal space – a space of possibility, a passageway, an expanded marginal space with room for play. Classrooms are designed to allow for flexible spatial arrangements; if we are willing to work in the space of the culturally liminal, a classroom can be as mutable as a theatre space.
Brian Sutton-Smith elaborates on the uses of liminal spaces as the source of innovation in a culture. He writes that liminal spaces provide a place to experiment and create new structures which may later be adopted by mainstream culture:
[…] The “antistructure” represents the latent system of potential alternatives from which novelty will arise when contingencies in the normative system require it. We might more correctly call this second system the protocultural system because it is the precursor of innovative normative forms. It is the source of new culture. (Sutton-Smith, 1972, quoted in Carlson, 1996, p. 23)
In considering mathematics education as performance, we are opening up the concept of schooling as a space for protocultural experimentation and invention, rather than (or as well as) cultural transmission, and rediscovery and exploration of what has gone before. Elementary and secondary schools are rarely viewed as hotbeds of new cultural forms and ideas. Is it possible to treat schools as the sites for culture-creating performance? Certainly the concept of digital performance, which I will discuss at the end of this paper, holds promise for cultural innovations initiated by young people.
(2) Sacred spaces In all kinds of performances (including sports, religious rituals, theatre and ceremonies), there is a demarcation of a special kind of space. A specially marked performance space begins to take on sacred or magical qualities. There are restrictions about who may enter the space and in what manner, what preparations consecrate the space for its special uses, and how a person must behave in and towards that space. There are performance spaces where one must bow or remove one’s shoes or cover one’s head on entering (for instance, a karate dojo, or a temple or church). Some performance spaces require that special words be spoken on entering, or that a person enters silently. Sometimes only the elect may enter the space (athletes on a field, priests at an altar, actors on a stage). The geometric shape of the performance space may have symbolic as well as practical significance.
It is rare that we consider the aspects of sacred space or consciously play with these parameters in the mathematics classroom, even though every class does establish tacit conventions and rituals around space.
Richard Schechner writes about the ways he establishes the sense of a demarcated sacred performance space in theatre workshops:
Because rituals take place in special, often sequestered places, the very act of entering the “sacred space” has an impact on participants. In such spaces, special behaviour is required […] Ordinary secular spaces can be made temporarily special by means of ritual action […] When I lead a performance workshop I insist that participants wear no street clothes, shoes, watches, or jewelry for the duration of the workshop. No one has a watch, so time is defined by our mutual experience. Each session begins with a careful sweeping and mopping of the floor. Such simple actions as sweeping and mopping in silence transport the workshoppers to a different place mentally and emotionally. These ritualized procedures help create a feeling of communitas even before the exercises begin. (Schechner, 2002, 63)
Simple rituals for marking off a performance space are available to mathematics educators if we want to engage in mathematical performances. If we are willing to set aside a Platonic disdain for physical, sensory modes of knowing, we can use tools borrowed from theatre to create an atmosphere of anticipation and set-apartness that facilitates affective performance.
(already challenged by the use of computer networks in schools), the differentiation between performers and audience, boundaries among the virtual, imagined and real, and mind/ body boundaries that have long played a central role in shaping the image of mathematics.
We live in a world where these and other fixed borderlines are dissolving in the new environments created by rapid technocultural change. Young people in particular are accustomed to moving easily, and sometimes living simultaneously in physically present embodied worlds, virtually embodied online worlds, and abstract conceptual worlds. They are very comfortable moving among websites, computers games, digital music and photography, cell phone calls, text messages, email, video filmmaking, engaging in many of these virtual worlds at the same time as they are playing a pickup game of basketball, walking or skateboarding with friends, riding on a bus, eating – or going to class. Multitasking and bricolage are dominant features of our culture. Schooling must meet kids where they live if education is to be a living dialogue.
The digital math performances I imagine would not be solely digital, but would move easily between online, onscreen experiences, physically-present, kinesthetic embodied experiences with other people in a variety of spaces, and quietly conceptual individual imagining, working and thinking. Students and teachers would use resources to address purposeful design problems – not exercise problems with ready answers, but complex, open-ended design questions that require skills, research, imagination, humour, aesthetics and judgement. If mathematical concepts and skills were at the heart of these performances, their uses would be immediately apparent as “digital performers” used them as elements of the design of (for example) songs, games, videos, three-dimensional sculptures, robots, paintings, animations, inventions, furniture, gardens, playground equipment… The liminal spaces of computer processors, interfaces and networks would flow seamlessly to and from other kinds of performance spaces. In the interplay of these spaces, schooling could become a source of new culture.
References Carlson, M. (1996) Performance: A critical introduction. London: Routledge. Huxley, M. & Witts, N. (Eds.) (1996) The twentieth century performance reader. London: Routledge. McKenzie, J. (2001). Perform or else: From discipline to performance. New York: Routledge. Quoted in Schechner (2002), p. 20. Schechner, R. (2002) Performance studies: An introduction. London: Routledge. Schechner, R. (2003) Performance theory. London: Routledge. Stern, C. S. & Henderson, B. (1993) Performance texts and contexts. New York: Longman. Quoted in Schechner (2002), p. 16.
Sutton-Smith, B. (1972) “Games of order and disorder”. Paper presented to the symposium “Forms of Symbolic Inversion” at the American Anthropological Association, Toronto, December 1972, pp. 17 – 19. Quoted in Carlson (1996), p. 23. Turner, V. (1982) From ritual to theatre: The human seriousness of play. New York: PAJ Publications.
Vonnegut, K. (1968). Slaughterhouse Five. New York: Bantam. Wagner, B. J. (1989). Dorothy Heathcote: Drama as a learning medium. London: Hutchinson.
