Ground Zero
Whereas young people become accomplished
in geometry and mathematics, and wise within these limits,
prudent young people do not seem to be found. The reason
is that prudence is concerned with particulars as well
as universals…
–Aristotle, 1142a
For Homer, the patterns of geometric
and archaic thought and large parts of fourth-century Athenian
common sense were not tied to individual cases; they contained
generalizations and used universals to give them substance.
But these universals did not supersede or constitute particulars,
they connected them. We might say that the universals of
Plato…are tyrants which “annihilate” particulars while
the universals of their rivals mediate between them while
leaving them unchanged.
–Feyerbend, 2001,
52
I observed six youths, all males,
in the distance some standing and sitting on the same
road [that I was traveling along]. About 10 p.m., as
I reached a dark area on the right where the youth were
hanging out, without a word, I was attacked by them.
Some held my hands and some my shirt and I was repeatedly
beaten with their hands striking me in my head and face…These
youths then attempted…by pulling and pushing me to try
to get me in an even darker area which leads to the beach…however
I struggled with them and repeatedly called for help…
The ordeal lasted about three minutes during which I
successfully made my escape…These young men that attacked
and robbed me all appeared to be teenagers between the
ages of 14 and 18 years old and they all wore black [and
white] scarves around their faces. I have no knowledge
of my attackers and would not be able to identify them
should I see them again.
–Statement of Witness,
Steven Khan, 17th January 2008.
Welcome.
I have had to revise this piece over and
over again as I try to connect and reconcile the desires
expressed in the call for papers with my own recent experiences
as a ‘student’ of teenage gang violence, a citizen
of a small twin-island Republic psychologically crippled
by a statistically incomprehensible level of brutality,
and as a young writer/creator within the academy in (mathematics)
teacher education. Two and a half years ago, when I first
began trying to connect mathematics, aesthetics (the arts)
and ethics, and at that time witnessing rising tides of
violence and suffering from outside of my region, I wrote
in a course paper: “Academic
disciplines that fail to establish clearly a link between
what is learnt and the responsibilities of individuals
to other individuals, to their societies and to other societies,
fall short of realizing the full potential of the educational
project. Mathematics and mathematics education must find
ways and means to link the domain to the most basic of
human desires as well as the most noble of human aspirations.” I
take this statement now as a marker, a testament, to an
urgent imperative in the work I am trying to perform as a mathemaesthethician examining
relationships between mathematics and art/aesthetics, while
exploring the possibilities for more responsible (global)
civil society amidst the tensions wrought by encounters
between overwhelming diversity and conflagrate, contradictory
values (Appiah, 2006, Matsuura, 2004). My work/performances
are oriented around a question of how I engage myself and
others as a (mathematics) teacher/educator in forming (and
performing) lifelong ethical relationships with and between different, diverse
peoples, places, things, and thoughts (Khan, 2006) within
proliferating, putrefying, and petrifying carrion structures,
cultures, and societies, where wuk seeks
to consume space, time, energy, and even desire for Work?
In my meanderings I have arrived at a
tentative conclusion that the telos of mathematical experiences
ought not only to be mathematical knowledge and power.
Rather, mathematics’ destination, like that of Art discussed
below is, I believe, elsewhere: in love, ethics, responsibility for the Other and, I want to suggest, perhaps, something
deeper. An invitation to perform mathematics and mathematics
education differently, which I have chosen to call a ‘mathemaesthethic disposition,’ reveals one’s responsibilities as
an ongoing commitment to democratic community formation through the
welcoming of difference by dialogic communication (Alrø
& Skovmose, 2003; Bakhtin, 1981; Khan,
2006b; Renshaw & van der Linden, 2004). This commitment
also entails the rendering of mathematics in ways
that are “evocatively resonant” through attempting
connections between meaning-full personal and aesthetic
experiences (Betts & McNaughton, 2003; Papert,
1978; Picard et al., 2004; Sinclair, 2006), and by
providing a means for understanding and addressing/redressing
our seemingly infinite capacity as individuals, communities
and corporations for the brutalities generated by
pathological asymmetries of power (Gutstein, 2005;
Sen, 2005), by learning how to be for an Other, i.e.
ethics. Such a disposition is not the exclusive purview
of mathematicians, artists, or educators, but of
any individual or group committed to justice, compassion,
and ethics through education
viewed as a synaesthetic (Root-Bernstein, 2002) transcendent,
and immanent project.
I have also come to believe that mathematics
education is a universal tyrant in social and cultural
iatrogenesis, i.e., it makes people, culture, society and
social institutions sick and disrupts their agentive capacities
for recuperative intervention by denying them access, power,
and opportunities for creative transformation of their
own lives. Indeed, the legacy of mathematics education
globally is overwhelmingly that of massive failure and
systemic suffering on an incomprehensible scale (Higginson,
1999). Fractally amplified across economic, political, cultural,
technological, environmental and personal landscapes, stories
of poverty, inequity, failure, suffering, greed, corruption,
violence, and disaster speak of pervasive and pernicious
pathologies in which mathematics education, and her siren
sisters, science and technology, facilitators among discourses
of power, exclusion and oppression, are conspicuously implicated.
This pathology includes mathematics and mathematics education’s
complicity in the creation, rationalization, and maintenance
of an increasingly anachronistic, inhuman, industrialized,
corporatized, and militarized conception of schooling (Giroux,
2007), the globalization and ghettoising (Skovsmose, 2006)
of curriculum and culture, and the entrenchment of hypercompetitive,
exclusionary, and individually and socially harmful “winner
takes all” markets (Frank & Cook, 1995) which threaten
all of our liberties, physical and mental health, and undermine
democracy’s foundations worldwide.
Mathematics education’s complicity in
this ‘glocal’ pathology, its unrepentant sins of commission
and omission, however, provides direct access to the infrastructure
necessary for its transformation and healing. A mathemaesthethical
disposition provides (me) a means for effecting such transformations by re-conceiving
the answer to the question “who and what is mathematics
education for?” from
the present dominant and unquestioned corporate, military,
engineering, economic and technological pursuits of a privileged
few, to be for ethical relationships with an unknowable Other. I want to argue that
perhaps the human desire “to make beautiful” art can serve
as a necessary mediator between the tyrannical tendencies
of mathematical universals and the urgent need to develop
practical wisdom, prudence or phronesis in education and society by helping us to responsibly
connect the most powerful universals with the most familiar,
meaningful, and resonant particulars of individual lives.
Perhaps art can also help us to reorient our selves away
from brutal, egoistic and nationalistic concerns around
employment and economic competitiveness and too narrow
views of mathematical power emphasized in authoritative
international curriculum documents, towards re-conceived visions of the possibilities of education taken as a whole. Perhaps in this
way we might as mathematics educators recognize our complicity
in re-producing and re-inscribing suffering on violated
bodies around the world, including our own. Perhaps…
There are few specific treatments of mathematics
and ethics, and many more regarding the contested relation
between mathematics and art/aesthetics. The recent literature
of the last 30 years in mathematics education, however,
is encouraging as there is an increasing concern with ethical
issues, though usually covered under other terms like access,
equity, discrimination, oppression, liberation, emancipatory
discourse, culturally relevant, critical, inclusion, and
social justice. Many mathematics educators have begun to
respond positively to D’Ambrosio’s (2001) challenge and
critique that, “particularly in mathematics, there is an
assumption that we are fulfilling our responsibilities
if we do our mathematics well, thus instilling attitudes
of rigor, precision and correctness in our students…But
this is not enough” (331). Indeed, in facing the lengthening
shadows cast by violence, terrorism, war, corruption, growing
disparities between nations and among individuals and groups
within nations that create new and more permanent and insidious
forms of slavery and suffering, and facing the prospects
of global ecological Holocausts, we would be failing to
meet our responsibilities by merely “doing our mathematics
[or mathematics teaching] well.”
My work is oriented by a Levinasian view of taking ethics as an alternative
starting point for philosophy, relationships and sociality
(Levinas, 1969, 1981) and by Deleuzeian ideas regarding
philosophy’s task as the creation of concepts that enable
thinking which does not identify but rather ‘palpates’ difference,
and in so doing makes it ‘audible,’ thus allowing difference
to ‘speak’ before its visible identity makes it known.
I attempt to “trace a path between impossibilities” (Deleuze,
1995, 133) by trying to think the difference one never
thinks one can ever think, through posing Why and What if? questions (Fleener, 2004). Following Deleuze again, I experiment with
the ethical possibilities around the question: What might living/performing mathematics and mathematics education differently/ethically consist in?
(See May, 2005 for commentary on Deleuze’s question of
‘living’). These starting positions, I believe, reveal
but do not fully disclose or prescribe the situated principles,
practices, and processes, which render the possibilities of (more) sustainable and ethical democratic relationships
more likely.
In this paper I argue that Art, and mathematics
education when performed as an Art, can provide us with
a means to come face-to-face with and learn to welcome
Others, which reveals our responsibilities to be for Others. This revelation, which is a teaching, then
challenges us to accept and act on these responsibilities
in future encounters in a particular way, i.e. ethically,
and so opens up possibilities for new forms of social organization
and sociability, a necessary condition for reconstituting
relationships, including democratic ones. It is necessary
though to begin with a discussion of ethics.
First Iterations: Beginning with Ethics
The irreducible and ultimate experience
of relationship appears in fact to be elsewhere, not in
synthesis, but in the face to face of humans…morality comes
not as a secondary layer… morality has an independent and
preliminary range. First philosophy is an ethics. (Levinas,
1985, 77)
One can see this nostalgia for totality
everywhere in Western philosophy, where the spiritual and
the reasonable always reside in knowledge. (Levinas, 1985,
76).
The experiment motivating this section
and my recent thinking is “What if we
began thinking about mathematics education starting with
ethics?” Such an experiment finds resonance with recent
work such as Skovsmose (2006), Eisenberg (2008) and Stemhagen
(2008), the latter arguing for “a recognition, even an
embracing of the human and hence, of the ethical dimension
of mathematics” (60). In tracing the contours of the relationship
between mathematics, aesthetics and ethics, I have been
drawn to the writings of and about French theorist Emmanel
Levinas. All of Levinas’ major works are written “in the
shadow of the Holocaust…[and] bears a distinctive weight,
indeed a heaviness…There is present…an exigency to attend
to suffering, injustice and violence, and his account of
the ethical becomes a labour born not only of philosophical
interest, but human necessity” (Todd, 2003, 2). Such exigency
is what is necessary in responding ethically to the present
pathologies in which mathematics is implicated.
For Levinas, Western philosophy typically
approaches its phenomena of interest from an ontological
perspective, i.e. it attempts to determine ‘what is,’ to
establish identities and determine essences,
to demarcate between that which is and which is not (Pinchevski,
2005). From this position ethics is seen as subsequent
to ontology and emerges as a series of ‘oughts,’ prescriptions on being, based on what is. Levinas questions this obsession
of (Western) philosophy with ‘grasping’ the nature and identity of the Other as a prelude to diminishing or
collapsing the Other’s otherness to sameness. Ethics for
Levinas is first philosophy, i.e. it is pre-ontological. It precedes ontology. For Levinas, the meaning of ethics does not lie in identities and essences
that lead to prescriptions of what one ought to do, but rather:
Ethics does not have an essence,
its “essence,” so to speak, is precisely not to have an
essence, to unsettle essences. Its “identity” is precisely
not to have an identity. Its “being” is not to be but to
be better than being.
Ethics is precisely ethics by disturbing the complacency
of being…Ethics occurs as an an-archy, the compassion of
being. Its priority is affirmed without recourse to principles,
without vision, in the irrecuperable shock of being-for-the-other-person
before being-for-oneself, or being-with-others, or being-in-the-world…
(Levinas, 1985, 10)
This is a very different and difficult place
to begin thinking about mathematics education. The comfort
of content and easily defined and measurable objectives
vanishes, and with it attempts to define and delimit what
students ought to
learn. Such prescriptions, codified in curriculum, syllabus
and assessment objectives by their totality, violently
reduces everything and everyone to a singularity of sameness,
a mean, that precludes the very conditions under which
truly ethical responses to difference might emerge—the
experience of relationship to the Other. But what
if we were to be for
the Other, even before being for mathematics? What
if we were to teach/perform mathematics (education) in
this way? Perhaps, by adopting the position (or disposition)
of being for the Other we might begin to reorient our selves
and open possibilities for imagining a mathematics education
that is Otherwise (Säfström, 2003). But we must first come
into a presence, that of recognition of the difference
of the Other in the traumatic and open encounter with the
vulnerable face of the stranger.
The face is a central concept in Levinas’
writings. It is, he says, “that which stays most naked,
most destitute, though with a decent nudity…there is an
essential poverty in the face. (…) The face is exposed,
menaced, as if inviting us to an act of violence. At the
same time, the face is what forbids us to kill” (Levinas,
1985, 86). In mathematics education many faces are as yet
still conspicuously absent (Skovsmose, 2006) or their differences
are rendered within sterile academic tropes that marginalize,
pathologize or otherwise position and construct such difference
as deficiency, deviancy or disability. Many students do
not see their own faces, nor the faces of any Other in
the way(s) mathematics is presented. The emergence of the
ethical relation and the infinite responsibility of the
ethical subject, as Levinas conceives him, depend on appreciating
the phenomenological approach and address of the Other
manifested as face. The nakedness of the face, its vulnerability
and fragility, is a call that addresses me directly and
calls me forth to be responsible and answerable to the
Other who “orders and ordains me” (Levinas, 1985, 97) in
the very moment of encounter. Indeed, the face “speaks,
it is in this that it renders possible and begins all discourse”
(Levinas, 1985, 87). The presence of the Other manifested
as face, as Pinchevski (2005) notes, ruptures my egoist
concerns by “speaking, addressing, questioning, demanding…[it]
causes me not to be at-home with myself”(218). The face
requires response.
Our first relation then for Levinas is
one of responsibility for the
Other which emerges out of the meeting of face-to-face,
an intersubjective encounter with unknowable difference
that does not seek unity or synthesis, but which calls
me forth out of my self in his demand for a response. Such
response is a witnessing, “a revelation which is not a knowledge” (Levinas,
1985, 106). The witnessing however is as Zembylas (2005)
describes, “a witnessing to what the Other accomplishes in
me (i.e. the Other creates me as a responsible person)” (italics added, 148). In
another radical departure from the familiar (mathematical
and aesthetic) values of balance and symmetry which underlie
modern notions of equity, fairness, and justice, Levinas
proposes that the ethical relation, “is a non-symmetrical
relation...I am responsible for the Other without waiting
for reciprocity…Reciprocity is his affair” (Levinas, 1985,
98). And since his is a non-prescriptive or totalizing
ethics one can never be certain of having fulfilled one’s
debt to the Other who addresses you and so one is always
open, already responsible, ready to receive the call of
the other to responsibility. Indeed as Pinchevski (2005)
notes, “[r]esponsibility here means exceeding rather than
following social norms” (216). Thus our responsibility
for the other, rather than becoming an obligation, becomes a Desire. Such
Desire, as Levinas (1985) explains is “the relation to
the Infinite [which] is not a knowledge…Desire [unlike
need] cannot be satisfied…Desire… nourishes itself on its
own hungers…Desire is like a thought which thinks more
than it thinks, or more than what it thinks” (86). Such
is our responsibility for the Other, a relation to the Infinite that is not
a knowledge, and which overflows our capacity to think
it. How might we help students to come to desire by performing
mathematics as a means of and for responsible action?
For many students of mathematics, at all
levels, mathematical activity is an obligation, something
that must be done. It is often presented as a (largely)
faceless edifice to be scaled. In my recent third (final)
year B.Ed course, Mathematics Around Us, after some discussions and readings around ethnomathematics
I invited students to choose and interview two individuals
from different career tracks, one where mathematics was
clearly evident and the other where one could not easily
see mathematics being applied. This was to be an opportunity
for them to encounter difference face-to-face as well as
learn about how mathematics is actually used. Tricia, a
primary school classroom teacher, chose her school’s janitor.
She was quite moved by Mrs. Lewis’ response to being interviewed
and wrote in her journal:
On the day of presentation I was
all excited to tell of my interviewee, and not about
what I had learnt, but about how excited she was after
learning so many things about her career that she never
saw prior to this interview. My interviewee saw herself
as having some worth after she realized how much mathematics
there was in her daily cleaning of the school. This made
me reflect on how much we as teachers can use such strategies
to teach our children how valuable even the jobs that
we ‘look down on’ really are.
Tricia speaks not about what she has learnt,
but rather what Mrs. Lewis has accomplished, how her encounter
with this Other has created her, and now calls her to a
new orientation in her teaching. Tricia’s presentation/performance
also drew a significant response from her colleagues. Rinnelle,
for example wrote in her reflection on the assignment and
class presentations:
Though each presentation made by
fellow classmates was informative and provided ‘food’
for thought, the one that made the greatest impact on
me was the presentation about the interview conducted
with the school cleaner…Initially, as is the case with
many individuals who engage in activities that are not
explicitly mathematical, Mrs. Lewis could see no mathematics
in her job. However, it was seen that through her explanations
of how she conducted her duties, that there were many
mathematical concepts involved. When she did realize
this, as Tricia reported, she was surprised and proudly
assented to the fact that mathematics was involved in
what she did. This really touched my heart, as too often
cleaners like Mrs. Lewis are disrespected by teachers
and children because of their duties, but here it is
this cleaner gained some respect for herself and what
she did because something as ‘important’ as mathematics
was found in her job.
Through the interview and Tricia’s witnessing/performance
in the classroom, Rinnelle’s ethical sensitivity to the
brutal effects of social hierarchies is heightened and
she too testifies not to what she has learnt, but what
this encounter, through Tricia’s mediation/meditation, has accomplished in her.
In watching the recording of the interview
and reading Tricia’s report, what most stood out for me
was the moment when Mrs. Lewis’ countenance changed, her
face, somewhat fatigued throughout, at one point, lights
up with the joy and pride of recognizing that “something
as important as mathematics was found in her job.” In the classroom it is the joy and
profound respect with which Tricia’s face and words speak
of her encounter with difference, and connecting with their
own experiences that so moves the other members to a new
and deeper awareness of their relationship both with mathematics
and with Others, especially those perceived to be the least
members in the society. The desire that emerges is to help
their students to likewise encounter and value difference
and to respond respectfully and responsibly to this gift.
They begin to speak, to write difference, to imagine, mathematical
practices that might be Otherwise. They and I bear witness
to what these encounters accomplished in them: what and
how the presence of an Other teaches when welcomed.
Second Iteration: Welcoming the Other
- Art’s Revelation of Ethics
The presence of the Other is a presence
that teaches (Levinas, “The Transcendence of Words” Outside
the Subject, 148).
The Other is manifested in a Mastery
that does not conquer, but teaches. Teaching is not a species
of a genus called domination, a hegemony at work within
a totality, but is the presence of the infinite breaking
the closed circle of totality (Levinas, 1969, 171).
The emergence of ethical subjectivity,
for Levinas, requires a disposition, to welcome, the unknowable
Other, the stranger, who approaches and addresses me, whose
face invites me to respond, whose presence unsettles me
and teaches me. Welcoming the Other is the first response
in the revelation of witnessing to the emergence of an
intersubjective relation of responsibility for the Other. How the
self welcomes the Other, receives the Other and whether
(or not) he allows himself to be taught by the Other is
at the heart of what Todd (2006) identifies as a “theory of
learning” in Levinas’s philosophy of welcoming which is
less concerned with how a subject learns content than with
how a subject learns from the Other through the orientation or disposition to
welcome. The Other who is separate(d) from me, comes to
me as Master, teacher, a stranger to my self, who offers
gifts of difference, difference about what I do not yet know and difference from whom I cannot know completely, but to whom I am already in relation, and for whom I am already called to be responsible. The presence
of the Other teaches me “otherness itself” (Todd, 2006,
4). Thus the welcoming of the Other is a necessary disposition
for being taught. Such welcoming, as Todd (2006) explains,
is:
not a gesture that seeks to reduce
the independent nature of the Other’s existence through
domination, identification, understanding or even care;
it seeks not to “envelop” or to protect. Rather it stands
as an affirmation of the Other’s strangeness… Welcome is
an ethical testament to the separation between me and the
teacher…welcome invites as it receives the Other…(8).
The excerpts from Tricia and Rinelle’s
encounter above speak to this affirmation of strangeness.
Generosity and hospitality underscore
the orientation to welcome as “an extravagant response”
(Levinas qtd. in Todd, 2006, 11) to what the Other offers
and are thus appropriate manifestations of the acknowledgement
of the gift that the presence of the other bequeaths. Hospitality
(hospitalité) for Levinas however is inseparable from the dwelling
(home). As he says: “I welcome the Other who presents himself
in my home by opening my home to him” (1969, 171). Thus
as Gauthier (2007) argues, the home “achieves its full
dignity when the Other is welcomed into it, thereby transforming
it…”(160). The home is unique in that it is the site from
which the self, by being safely housed, by being itself
welcomed as a guest into the home, can “recollect itself
in earnest” (Gauthier, 2007, 161) so that it can extend
hospitality to the Other as host. This sense of the self
as both guest and host is emphasized by the French word hôte, which Derrida (qtd. in Todd, 2006) uses to explain
the Levinasian welcome as:
The hôte who receives (the host), the one who welcomes the
invited or received hôte (the
guest), the welcoming hôte…is
in truth a hôte received
in his own home. He receives the hospitality that he
offers in his own home; he receives it from his own home…The hôte as
host is a guest (9).
In welcoming and inviting the Other to
dwell with me in my home not as an extension of myself, both
dwelling and dweller are transformed. As Gauthier (2007)
describes,
When the Other is welcomed into
the home, the latter ceases resembling a ‘root’ that the
self puts into the ground as a means of isolating itself
from its fellows: ‘The chosen home is the very opposite
of a root’ (Levinas, T&I, 172). Instead, the home attains
the status of a chosen place because the presence of the
Other graces it with the presence of the Infinite (164).
The ethical implications for education
and mathematics education in particular are clear and challenging.
They require us to ask questions regarding how we welcome
Others into the homes into which we have been welcomed,
and whether such dwellings are “instruments of ethical
compassion…or…little more than miserable domiciles…?” (Gauthier,
2007, 165). These questions also provoke concerns when
applied at other scales and are especially relevant to
current concerns about trans-national migrations and the
welcoming of foreigners and strangers.
Few students find a home in mathematics
education as presently practiced. It is not often a chosen
place. More often than not, the encounter with mathematics
is not welcoming and many of our classrooms are not places
where we invite students to dwell well with us through mathematics, but rather act as sites of
colonization, conversion, occupation, and dehumanization.
They are ‘miserable domiciles’ for teaching domestic docility.
However, perhaps by learning to welcome, and by inviting our students to dwell well
with us, each other, Others, and with the mathematical
ideas necessary to participate and create responsibly in
professional and civic mathematical discourses, we might
open possibilities for classroom practices and lives that
are “instruments of ethical compassion” (Gauthier, 2007,
165) and social transformation. But how is it possible
to learn to welcome such unsettling difference joyfully
in mathematics education?
The argument I am hoping to make, following
Dewey, and others, is that through art we can come to an
enlarged understanding of love, which reveals our awesome
responsibilities for Others, and which opens possibilities
for imagining and creating new, more just forms of social
relationships at different levels. Dissanayake (1992) for
example, argues for a central role of the arts in human
evolution (both biological and social) and the aesthetic
experience for her is filled with ‘evocative resonance,’
i.e. the sense that there is something deep and meaningful
in the experience, which provides a ‘satisfying fullness.’
When such experiences are incorporated into social activities
such as rituals and festivals she argues, they make tangible
the possibility of social cohesion and transformation.
Kerdeman (2005) also argues that aesthetic experiences
are not only integrative of mind, body and emotion but
offer therapeutic and ameliorative roles in the lives of
individuals and perhaps societies. I also draw heavily
on the work of Didier Maleuvre (2005) who explains his
view of art as:
…a form of sanctifying the human conversation,
… [having] a destination beyond itself….The aim of art
is not art… And the vehicle by which art travels into reality
is not just skill, insight, knowledge or intelligence.
It is love. (77)
Maleuvre makes
a very convincing argument for viewing Art as a “teaching
of Love” which resonates with me. For him, “Art is less
concerned with delivering information about the world than
teaching us about how to stand in relation to it, how to find our place in it, and live with it through art we do not
seek to master the world so much as become its denizens…”
(78-79). In this way he connects with Levinas’ ideas of
welcoming, hospitality and responsibility. This type of
education is radically different from the other sort that
typically occurs in schools. There is an intensely physical
aspect to this love that calls us to remember that there
is something special in being with other people. It is a teaching of Love. And of Love
he says:
Love, is a kind
of falling… To fall is to experience the pull of physical
reality… Now, love is a kind of falling because it calls
us back to the tangible. Love is a connection with the
particular and the unique, a face, a person, a body, a
moment, a gesture…Now, inasmuch as love is the state of
being in which the unique absolutely matters, it connects
us to the physical. It is spirit falling into the flesh…
Love is physical and sensuous, it needs to touch and embrace…Love
is the joyful realization of being made of flesh. (82)
This recognition
of the embodiedness and embededness of Love present in
art points to an intrinsic connectedness that unites us
with other beings. Thus, he says, “Love is the disposition
of being related, of being present to others. It is inaccurate
to say that love connects what is separate. Love is the
realization that nothing is separate” (85). These thoughts
have deep resonances not only with D’Ambrosio’s challenge
regarding what it might mean to “do mathematics well” and
how mathematics might be related to our human survival
with dignity, but also with embodied perspectives in mathematics
education (Lakoff & Nunez, 2000) which, by drawing
attention to the important role of the body in mathematical
cognition, re-establishes a sensuality of mathematics that
“cold” cognitive approaches eschew, and calls us to attend
seriously to questions of whose bodies are represented
or not represented in mathematical discourse, how they
are represented and how such presence or absence is felt
and experienced by real bodies. These perspectives thus
connect with the opening quote by Paul Feyerbend regarding
universal mediators, like Art and Mathematics, which have
the potential to mediate difference rather than annihilate
it.
Teaching at
times suffers greatly under the weight of an instrumental
view of teachers, students and curriculum, and becomes
merely a practical and increasingly economic arrangement;
a means to ill-defined and often conflicting ends. Teaching
though, understood from the perspective of Art as a teaching
of love, ‘opens us up’ to more hopeful possibilities. For
Maleuvre this radical opening up is exemplified by the
process of portraiture. He says:
The portrait is
an achievement of human sympathy, of opening one’s loving
sensibility to the experience of another person. To open
up in this manner requires time, patience, and dedication.
It happens through labor and effort…The artist is penetrated
by his subject until it becomes more central and immediate
to his own self than his own stream of thought. In effect,
the artist becomes a witness. He does not observe; rather,
he dedicates his own existence to testify to another person’s
life. It is less technical achievement than a gift: a labor
of moral generosity. (88-89)
This
idea resonates with the Levinasian encounter with the
Other, first experienced in the face-to-face and exemplified
in Tricia, Rinnelle’s and my own response to their faces
and biographical portraits. Art, as a teaching of Love, reveals the very possibility of ethics. Teaching
understood in this way, like the portrait artist, looks
and feels radically different. There is the acknowledgement
of Eros in the interpenetration and ‘intervulnerability’
of artist and subject, the caring dedication and effort
put into exposing the relation between beings, the task
that becomes not work alone but vocation that bears witness
to love of self, one’s work and the other. What if we
tried to teach/perform mathematics in this way as a process
of witnessing?
The
encounter with Art also teaches us how to be attuned
to the violence that permeates modern life and educational
systems. Maleuvre (2005) also proposes a role for art
in acting as an ethical indicator and as a vehicle for
addressing moral issues. He says:
…art is especially suited to decry the
mutilation of human life… The violence that lacerates beings
by the same token lacerates art…Violence maims, mangles,
humiliates, and reduces persons to meat. It denies the
victims language and self-expression. It punctures the
human conversation that holds us above animality…[works
of art] teach us to take care; to pause; to heed; to orient
our attention away from egotist concerns; to attend to
the other; to enter into a relation; to participate; to
see as also we are seen. They are moral lessons, lessons
in gentleness and sensitivity, in compassion and listening
(91-92).
Art teaches us how to be for an Other who is not like
our self without trying to make her like our self, how
to hold off violently collapsing difference to the sterile
singularity of sameness. Art teaches us how to be attuned
to the violence in our lives and the lives of others
when vulnerability becomes a source of humiliation or
oppression. It gives us courage and hope and a springboard
for action. Again the students’ responses to each other’s
interviews and the interview subjects, their reports—literary
and audio-visual portraits—awaken in them, a sensitivity
to the ways in which mathematics education, when practiced
as an art is able to speak to issues of gate-keeping,
exclusion, reduced opportunities for self-determination,
social stratification, and impaired conceptions of human
dignity and the dignity of all work.
In my own experiences as a neophyte teacher I have been
to the brink of depression and rage as a consequence
of the violence and humiliation that one sometimes finds
in school. But it has been Art: writing, drawing, photography,
conversation and Love that have taught me how to find
my way back, and which gives me the courage now to embrace
the challenge of beginning to speak to those atrocities
and to respond to more recent violations. Art has much
to teach us, but not in the sense of facts about the
world, but rather how to learn, how to feel, and how
we might be differently in the world. Art offers us its self
as a means to help us understand what it is that we do
when we teach and what/who we might do
it for. It offers us an opportunity to reconnect with
the most fundamental aspect of modern schooling, of being
with others in conversation, in Love, one that is romantic,
erotic, communal and at times painfully aware of its
own shortcomings.
In
remembering, reading, and witnessing the contempt and violence
with which we have treated (and continue to treat) our
artists and their works one finds parallels and resonances
with the contempt, violence and humiliation with which
we treat each other and each other’s work in education
and elsewhere. The vulnerability of Art, like the naked
face of the Other, offers us an opportunity to learn, an
invitation to become an hôte and to extend hospitality from the dwellings, including
mathematics, where we have been welcomed. The challenge
is to begin to perform ourselves differently.
Third Iteration: Performing oneself
differently
In the same course described above I introduced
students to ethnomathematical ideas and practices. My choice
of articles and activities were based primarily on aesthetic
functions such as those proposed by Sinclair (2006) of
evaluation, motivation and generation, but also oriented
by the cultural diversity of Trinidad and Tobago where
I have taught and lived, and a belief in the potential
of learning about ethics through artistic creation and
engagement with others. Students read about line drawings
in different cultures and explored kolam patterns (Ascher,
2002) in class. They were then challenged to go to the
beach and create their own patterns for inclusion in their
portfolio. Their reflections provide some insight into
what might happen when one attempts to perform mathematical
activities, and oneself, differently in non-traditional
spaces.
When I was doing my tracings in the
sand, many individuals stopped and looked on but no one
asked any questions. At first I was a bit embarrassed
and I was not getting the pieces right, but I persevered
and got a few. (Sabrina)
I can recall going on the beach one
Sunday afternoon to do some line drawings and everyone
watching me like if I am about to perform a ritual in
a public place. This created a kind of anger within me
because I don’t like to be embarrassed. As a result I
moved to a private beach out of the sight of individuals
to get my drawings done…The line drawings were very difficult
at times and required a lot of thought before attempting
to complete them in the sand, especially when trying
not to raise my hand out of the sand. I enjoyed this
aspect of the exercise because it challenged me and gave
me the opportunity to work exceedingly hard at times
to figure out how to complete the drawing…(Deon)
Figure
1 a,b,c,d: Student kolams and Malekula nitus (reproduced
with students’ permission)
Having to make my own line drawings
in sand turned out to be a more interesting activity
than I had anticipated. I discovered that the drawings
I appreciated most on paper were not the ones I liked
when they were drawn in sand. I had never thought that
the medium used could affect the aesthetics so much. (Rhoda)
In performing
oneself differently, one opens one’s self up to scrutiny,
isolation and embarrassment. In asking students to step
outside the safety of the classroom space for the performance
of an aesthetically motivated mathematical activity, they
were placed in a position of being seen as strangers in
their own land. Through it they were given an opportunity
to become aware of their own vulnerabilities before others.
What, though, did they learn from this risky undertaking?
Deon found satisfaction in an activity that was intrinsically
rewarding, aesthetically motivating, yet sufficiently challenging,
while Rhoda learnt how the medium of representation, sand
versus paper, can affect the aesthetic qualities of the
product, an insight that also has relevance for how we
choose to present mathematics in classrooms. In all cases,
the students came to care for their drawings, not as mathematical
objects, but as aesthetic creations linked to a sense of
their own self-worth, frustrations and accomplishment.
Their performances also raised questions about performing
‘unnatural’ rituals, such as line drawings, or even formal
mathematics, in public spaces with their rigid social expectations
and norms. In performing this unnatural ritual, mathematics,
in a public space like a beach, students engaged not only
in a mathematical and aesthetic activity but a political
and deeply personal one as well.
Attention
to the relationship between the aesthetic and the political
is discussed by Linker (2003), who provides a way into
discussions about the practice or performance of politics
in classrooms, schools and wider society. He writes:
…all politics has
an aesthetic nature…culture is a self-organizing, non-equilibrium
system in which there is only the
aesthetic and the political. Understood in this way, politics
and aesthetics are not discrete areas of practice. The
political, in this sense, represents the entire field of
human relations and production. The aesthetic is the performativity of those relations…Though they may be examined in
isolation, they are irrevocably bound; the aesthetic shapes
the political, is the source of its power, and the political
provides context for the aesthetic. (italics added, 16)
By applying
this performative model to education, he argues that the
fundamental necessary condition for a revised pedagogy
is a revised conception of knowledge as a means to pursue
purposes that are not prescribed outcomes, but rather an
entering into ongoing processes that are always partial
and incomplete. Such an education, he suggests, is fundamental
to preparing citizens “for responsible participation in
the societas, and
education in the arts demonstrates the opening of liminal
spaces required to facilitate such participation” (105).
In stepping into this liminal public space, a beach (between
sea and land), students’ aesthetic activity took on a political
dimension. In performing the activity, they were required
to be courageous and face fears and anxieties imposed by
social conditioning as they mediated the psychological
liminality related to expectations around performances
in different spaces. In performing themselves differently,
artistically and politically, they discovered something
about their selves.
Of interest is the way in which changes
in mathematical thought are related to changes in aesthetic
considerations, and perhaps changes in patterns of human
behavior (ethics). Whitehead (1941/1951) alludes to this
relation between mathematics and patterns of social organization
and behavior by casting art as the study of pattern. He
argues:
…the cohesion of social systems depends
on the maintenance of patterns of behavior; and advances
in civilization depend on the fortunate modification of
such behavior patterns. Thus the infusion of pattern into
natural occurrences, and the stability of such patterns,
and the modification of such patterns, is the necessary
condition for the realization of the Good….Mathematics
is the most powerful technique for the understanding of
pattern and for the analysis of the relationships of patterns.
(677-678)
Thus for Whitehead, the approach to the
good via mathematics depends on a sensitivity to pattern,
the cultivation of which occurs in both mathematics and
aesthetics. The history of (Western) aesthetic thought
however is dominated by values derived from Euclidean geometry,
structures that still linger, underlie and influence our
conceptions of school, teaching, curriculum and social
organization (Davis & Sumara, 2005). Plotinsky (1998)
however argues that non-Euclidean geometries, such as those
present in many non-European cultures, as well as research
into the foundations of mathematics, Godel’s incompleteness
theorems, as well as more recent work in topology, algebra,
chaos and complexity theories, are challenging key aspects
of classical aesthetics, as well as the epistemology and
ontology of mathematical knowledge. Relying on Nietzche
and Derrida he states:
…we may be forever hampered by the absence
of any fundamental absolute center…The resulting aesthetics
and epistemology may be seen as the aesthetics and epistemology
of networks that are both radically decentered and radically
oblique …the radical aesthetics of mathematics and science
that may emerge…may no longer be fully mathematical…[rather]
the new aesthetics is the aesthetics of multiple interconnections…The
conjunction of the known and the unknown, the knowable
and the unknowable, may well be… the most sublime feature
of mathematics and science. (197-198)
This analysis of the co-implicated, co-specifying
philosophies of aesthetics and mathematics, suggests the
possibility for conceiving of mathematical (educative)
projects that are radically decentered and radically oblique
to the traditional aims
of both mathematicians and mathematics educators. The de-centeredness
conjunction of the known and the unknown, in the Levinasian
frame, suggests not only the need to perform mathematics
differently, but to do so with a Desire for the Other.
In
my practice as a mathemaesthethician I am aware that my
performances, courses, assignments and invited talks, the
way I welcome students and visitors, what and how I write
in public fora are always political and aesthetic, and
are attempts to open spaces within the academy, where difference
might dwell well. Nobel Laureate in literature V. S. Naipaul,
who exists in a somewhat strained relationship with this
homeland, has criticized Trinidad in particular by saying
that ‘nothing’ has been produced here. I agree with him
in the sense that if ‘thing’ is akin to a well-defined
object (physical or intellectual) that is reproducible,
commodifiable and easily exportable to other contexts,
then no ‘mere thing’ has been produced here, rather what has and is continually
produced is not as simple as a thing. No, what is produced
here is performance. And if we wish to say that our culture and cultural
products, including our mathematics, arts and contested
democracy, are per-formative (for formation),
then the criteria for evaluating such performances are
vastly different from the criteria for things and objects.
Thus, I have come to believe, taking culture as an autopoietic
system, that a carrion culture of domination, disempowerment,
disenfranchisement and death in mathematics education,
and education more generally, can only be dynamically transformed
by performing itself differently,
and in so doing, transform itself, its context, and its
knowledge about the relations between itself and its context,
and through this performance come to know itself
better and know its
better selves.
The end of education, taken as a living
artistic practice, is Love. And indeed I would like to
offer that even this Love has to be attuned to an even
greater end. Perhaps by performing mathematics education
differently we (me) and our (my) students might come to
know ourselves better and indeed know and perform our better
selves? This concern with the performative dimension thus
provokes for the educator the question of a standard for the performance. I want to propose that such a standard is to be found in Levinas’ conception
of the Holy.
Fourth Iteration: Towards Holiness
as an end for education
This is the fourth iteration of the L-system
introduced in the first section. The image is ‘evocatively
resonant’ for me in that it is at once hauntingly familiar
yet somewhat ambiguous. The image summarizes the identity
I hope to perform as a mathemaesthethician. It re-presents
my attempts at exploring non-standard, ab-normal mathematical/artistic performances
and experiences with Others. Before proceeding to finish
the reading of this paper, spend a few moments reflecting
on what it means for you. For me, in the outstretched arms
and wire-like frame I see Trinidad’s indigenous art form,
the mas, incarnated
once per year in the Carnival, a two-day street theatre,
rooted in practices that are slowly being extinguished
by the forces of globalization, consumer capitalism and
education. I see a human being performing him/herself.
The head bent forward, an indication of the weight of responsibility,
as well as the joy of work that is also play, and the satisfying
fatigue that one feels when one is rewarded by one’s work.
In the image, generated mathematically,
there is celebration as well as suffering. The image is
also an invitation to a difficult conversation about the
ends of education in general, and mathematics education
in particular. Following Amartya Sen’s (1999) provocative
and important work in rethinking economics, Development
as Freedom, in which individual human freedoms and their expansion form a necessary
prerequisite for sustainable economic development, and
drawing on Emmanuel Levinas and his commentators, I have
begun to ask What if we take Holiness as an end for (mathematics) Education? What might
it mean to perform Holiness
as a mathematics educator?
For Levinas there is a distinction between
the sacred (le sacré)
and the idea of Holiness (la sainteté).
The sacred refers to religious experiences that encourage
a loss of one’s sense of self and capacity for rational
engagement, a phenomenon that perhaps also occurs to some
degree in mathematics education. It represents the desire
to merge with, to become one with, or take on the characteristics
of, what is believed to be the divine realm or the attributes
associated with the supernatural, such as power or omniscience.
Holiness for Levinas, however, means ethical separation, as
before the Holy one comes into an experience of ‘presence,’
an aesthetic experience like the one has with great Art,
where upon hesitation and lingering, one is welcomed and
becomes increasingly aware (learns) of oneself as separate,
a unique and distinct being with unique responsibilities.
The recognition of this distance, the absolute and irreducible
difference of the other to whom we are responsible, invites
us to welcome and venerate the Other’s difference without
seeking a reduction to the singularity of sameness. It is this that marks the encounter with difference
as Holy. To be morally whole, for Levinas then, is “to
accept the authority…of the gaze that questions our self-absorption
and that makes us aware of our capacity to be cruel. Only
this gaze can cut through the hardened shell of the ego”
(Caruana, 2006, 578). Indeed as Caruana goes on to note,
for Levinas the true sign of integrity “…is the ability
to affirm one’s bad conscience, or…to refuse to make compromises
with the moral indifference of existence” and thus, “the
true divide [or] fundamental split for Levinas is not between
believer and non-believer, but rather between those who
“are shaken by their own potential for brutality and those
who are oblivious to it” (Finkielkraut, 1997, in Caruana,
2006, 579).
These words in particular have deep resonance
with me as a mathematics educator and researcher. I am
continually having to confront my own immense ‘capacity
for brutality’ that comes from being part of systems of
power and privilege, which at times appear to be ‘faceless’
or attempt erasures and silencing of identities, which
allow a slippage, dilution, or evasion of personal responsibility.
As a Trinidadian, I am compelled to reflect on what is
a transnational concern, perhaps even a human universal,
namely, the seeming inability/incapacity of many citizens,
including children, to be shaken by their escalating capacity
for brutality in every sphere of life, both public and
private, and while it is becoming more and more difficult
to avoid the consequences of our past brutalities, I fear
that we are close to a tipping point in which we may choose
not to acknowledge
the existence far less the authority “…of the gaze that
questions [our] self-absorption and that makes [us] aware
of our capacity to be cruel” (Caruana, 2006, 578).
I am only just beginning to articulate
responses to questions such as, “What might our
roles as teachers be if we
accept Holiness as a worthy end for education?” How
might the practice of mathematics education be more welcoming
to difference and help us to recognize our capacity for
brutality? In reading Levinas alongside Caribbean philosophers
of art I see resonances as in the example below, with my
annotations in parentheses: [].
Meditative Meanderings
Ethics
is forceful not because it opposes power with more power,…
with a bigger army, more guns…but rather because it opposes
power with what appears to be weakness and vulnerability
but is responsibility and sincerity (Levinas, 1985, 13).
We have to see creation as tracing a
path between impossibilities…Creation takes place in choked
passages…A creator who isn’t grabbed around by the throat
by a set of impossibilities is no creator. (Deleuze, 1995,
133)
Cruel…curling fingers, squeezing…bruising…impairing
speech.
Follow, hollow eyes direction…compliantly,
silently, into darkness.
Thunderstorm…a violent reign of blows,
from children’s empty hands,
Enough!
There is a transformative potential in
(some) traumatic events and I am trying to feed the incident
described in my Witness Statement at the beginning of this
paper, and even more recent witnessing to my own complicity
in violence, positively into my emerging practice.
Questions arise. How might this incident be related to
my purposes for being in that particular place at that
time? How might this be related to the professional mathematics
education practices I was engaged in during the day as
the Chair of a panel revising/updating a regional curriculum
(syllabus) document? In what ways might ‘mathematics’ have
contributed to these young men’s desire for illicit enrichment
and personal humiliation? Reading the incident report again,
I wonder, surely some, perhaps many, of our students, feel
that they are being held and led by the throat, menacingly
‘invited’ to silently follow to a dark place, where they
may not wish to go? They too are creators.
The teaching/learning autobiographies
I read from my final year B.Ed students, as well as my
own, all speak to a long history and a more than passing
acquaintance with violence in mathematics at all levels
of their educational experience. Ingrid’s biographical
statement, for example, relating her earliest memories
of mathematics: “My first introduction to arithmetic and
the first resources I interacted with were a copybook,
a pencil and a ruler. The ruler was not for measuring,”
is stark and clear in its understated brutality. The others
recount similar brutalities, even at the tertiary level.
It is amazing that they have survived, bruised, battered
but hopeful, the result of a chance encounter with a wonderful
and inspiring teacher, an artist, someone who reached out
and transformed their relationships with mathematics. I
hope I can do the same in mathematics education. Others
though do not make it. Why are their narratives filled
with stories of contempt, attempts to dominate and humiliate?
Why do we allow teachers, including ourselves to wield
such mathematical power without making them aware of their/our
overwhelming capacities for brutality?
Members of our committee worked late and
finished our assignment on time. This part of the curriculum
renewal process completed in the four days assigned to
it. Rationale and aims being the last sections to be worked
on, the word propaganda delicately excised from the document
and replaced by critical interpretation. Good work team.
We don’t question the economic justifications that led
to us being there, to take on a task that rightly requires
more time and perhaps more heads and hands. We never question
it. We just get on with doing the job, our wuk. Mathematics
teachers have been selected for precision, persistence
and passivity. We are Empire’s somnambulant heirs. Recommendations
for changes based on responses from those in the field;
20 odd teachers respond to questionnaires. A ‘representative’
sample and responsible advocate for tens of thousands of
students? Can we get rid of the multiple-choice paper?
No, that’s not possible. We bind ourselves to little boxes. The structure of the examination
remains relatively intact. There will be little change
in teachers’ practices in the classroom. I predict. We’ll
be doing the same thing in a couple year’s time, same problems,
same frustrations, same solutions proposed. Much like the
examination paper itself! I wonder though, on the final
day, whether what we did would change anything, whether
it would make any difference to those boys, my assailants,
my teachers, and so many others like them? I look at the
document, printer ink still drying, and think not.
Coming home, safely, my wife reads the
story of my learning on my body, traces the welts and scratches
on my neck, imagines teenage fingers squeezing; sees the
black and blue bruise above my lip; kisses the bump on
my head that is slowly going down but still sore. My injuries
are, thankfully, not serious. They are nothing in comparison
to the scars and wounds carried by some of our teachers
and students. Our bodies are palimpsests, written on, over
and over again, by so many scribes, poets, lovers, armies;
we always bear their traces, they ought never to be written
off, written out or forgotten in history’s exiles. That
is a path that leads to genocide. We must read and write,
witness and testify, honor and hallow these markers.
What can we read from the bodies of the
students in our mathematics classes and mathematics education
courses? What stories of violent encounters are already
being written over and over again? Are we the scribes?
Dare we write differently and teach them to write their/our
bodies responsibly? Who will dress and soothe their wounds
when they come broken and hurt, humiliated and afraid?
Anushka, a teacher-student, stands in front of me, trembling,
with eyes and heart like levees overflowing, a state no
one ever wants to be seen in. She has been robbed at gunpoint
by unmasked men in broad daylight. I offer her a gentle
touch and a re-assuring squeeze. I feel like a miser. Lucia
and Marketa, strangers, offer me smiles and conversation.
I am overwhelmed by such generosity. Savitri deviates from
her assigned task, describing, testifying instead, to perceived
age, caste, ability and gender biases as a young Indian
female mathematics teacher. I am outraged, with her. Listening, we honor her meander.
Mathematics education is iatrogenic but
perhaps as Ed Dolittle (2007) suggests it might also be
medicinal. But it cannot do this alone. It needs Art and
it needs Ethics. I am coming to another tentative conclusion
that perhaps Mathematics education needs an ethical psychoanalytic
hermeneutics—a practice of an archaeology of interpretation
for healing and responsible action—as it works through
and towards learning/living to welcome Love and Holiness.
I have come to realize that as long as
I choose to teach ‘mathematics’ or the even less well defined
‘mathematics education,’ I have already lost the war no
matter how many battles are won. The challenge however
is to win the peace. How do I do this? Eighteen months
later I finally have an answer for Bill, who asked at my
thesis defence, Where’s the math? The
math will always be there Bill, our students’ hearts, minds
and bodies may not be for one reason or another. This is the
reality in the violent societies we live in. But it doesn’t
always have to be so. Instead, I elect to ‘teach’ my
students humanity and holiness. To remind them, that they
teach for humanity, and challenge them to teach humanly, how
to be Holy. Perhaps I can do this through mathematics and art and ethics. “Perhaps”, as Solzenhitsyn (1972) hopes:
the
old trinity of Truth, Goodness and Beauty is not simply
the dressed up worn-out formula we thought it in our
presumptuous, materialistic youth? If the crowns of these
three trees meet…and if the too obvious, too straight
sprouts of Truth and Goodness have been knocked down,
cut off, not let grow, perhaps the whimsical, unpredictable,
unexpected branches of Beauty will work their way through…and
thus complete the work of all three?...Then what Dostoevsky
wrote –“Beauty will save the world”—is not a slip of
the tongue but a prophecy… (7-8)
What comes after the thunderstorm?
Acknowledgements
I would
like to express gratitude to my wife Shalini for her
assistance and patience during the preparation of the
paper; Bill Higginson, for directing my early inquiries
into mathematics, aesthetics and ethics; Dalene Swanson
and an anonymous reviewer for their thoughtful reading,
provocative comments and encouraging critique; and to
the students named herein for their generosity and permission
to share parts of their stories.
The title is, unapologetically, a play on Hardy’s (1940)
biography, “A Mathematician’s Apology” and seeks to
orient the discussion away from an apologia to
an ethics.
Todd
(2006) explains that she is “using theory rather loosely
to indicate a constellation of ideas that reflect a certain
consistency with respect to how a subject is taught”
rather than “in the technical sense of a well-developed
explanatory system.” (13)
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