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QIX18710.1177/10778004
v
The Classroom as Rhizome: New Strategies for Diagramming
Knotted Interactions
Qualitative Inquiry
18(7) 588­–601
© The Author(s) 2012
Reprints and permission:
sagepub.com/journalsPermissions.nav
DOI: 10.1177/1077800412450155
http://qix.sagepub.com
Elizabeth de Freitas1
Abstract
This article calls attention to the unexamined role of diagrams in educational research and offers examples of alternative
diagramming practices or tools that shed light on classroom interaction as a rhizomatic process. Drawing extensively on
the work of Latour, Deleuze and Guattari, and Châtelet, this article explores the power of diagramming as a creative force
in research rather than a reductive one. The concepts of rhizome, assemblage, and knot are developed and applied to the
study of classroom interaction.The author then shows how these concepts and their application to classroom interaction
can be studied through topological knot diagrams. The author discusses the specific qualities of knot diagrams that make
them suitable tools for the study of rhizomatic processes and offers some examples of such diagrams. The author offers
these knot diagrams as tools that actually undermine the usual conventions of graphic representation in our field, not
simply to disrupt for the sake of disruption, but to invite speculation about how one might develop different diagramming
habits that better capture the entanglement of interaction.
Keywords
assemblage, Deleuze, diagram, classroom interaction, entanglement, knot
Introduction
A rhizome is an acentric nonhierarchical network of
entangled and knotted loops, folding and growing through
multiple sites of exit and entry. Rhizomatic growth is contrasted to arboreal or tree growth in that the latter is hierarchical and bound to one trunk, whereas the rhizome is a
network of proliferating roots and offshoots. Recent work
in educational philosophy has drawn on the work of
Deleuze and Guattari (1987, 1994) to study social interaction in educational contexts as a complex rhizomatic process.1 According to this view, educational systems and
subjectivities emerge and interact by way of discontinuity,
rupture, and multiplicity in a vast interleaving rhizomatic
assemblage. Such an approach demands both a radically
new ontology of the social and a radically different methodology for studying social interaction. Methods for
coding and diagramming research data need to be entirely rethought so as to better grasp the dynamism of
this rhizomatic process. Jackson and Mazzei (2009),
Maclure (2011), and St. Pierre (2004) have leveraged the
work of Deleuze and Guattari to criticize the reductive
arboreal coding habits of most educational research,
whereby linear models of growth are imposed on complex
rhizomatic processes. Such models usually deploy various
kinds of diagrams as representations of the interaction
under study.
The primary aim of this article is to call attention to the
unexamined role of such diagrams in educational research,
to argue for a radical rethinking of the function of diagrams, and to offer examples of alternative diagramming
practices or tools that shed light on classroom interaction
as a rhizomatic process. This article explores the power of
diagramming as a creative force rather than a reductive
one. I am not proposing an alternative model of diagramming that might be universally applied to all interaction (an
act of symbolic violence), but an inventive diagramming
experiment, a map that disrupts the very idea of image and
representation, a creative act of proliferation and rupture.
My aim is to show how the diagram might be rescued from
the stodgy state-sanctioned schematics of so much educational research, and be set free along rhyzomatic lines of
1
Adelphi University, Garden City, NY, USA
Corresponding Author:
Elizabeth de Freitas, PhD, Adelphi University, Harvey Hall 130, Adelphi
University, 1 South Avenue, Garden City, NY, USA
Email: defreitas@adelphi.edu
de Freitas
flight, allowing it to punch through the surface of semiotics
and push back at the regimes of signification that curtail
creative inquiry in our field. My aim is thus to do philosophy (and educational research) in the experimental spirit
and reclaim diagramming as artful abstraction. Deleuze
and Guattari use diagramming in just this way2 and argue
for a new kind of noneductive decoding of interaction—a
kind of diagramming that amplifies and ramifies and multiplies that which it engages.
I first discuss issues with current diagramming practices
in educational research, drawing on the work of Latour
(1990) and Lynch (1991), and elaborate on the concepts of
rhizome and assemblage as tools for studying interaction,
further developing these through the work of Bennett
(2010). I then discuss Deleuze and Guattari’s use of the
term diagram and extend that discussion through the work
of Gilles Châtelet (2000), who has argued that diagrams
should be considered physicomathematical entities.
Finally, I put forward the proposal that topological knot
diagrams, which were developed in the late 19th century,
offer educational researchers a creative method for diagramming the entanglement of classroom interaction. I discuss the specific qualities of knot diagrams that make them
suitable tools for the study of rhizomatic processes, and I
offer a few examples of classroom diagrams. Following
Lynch, I suggest that these diagrams might function as
tools for deconstructing our current diagramming practices, where “deconstruction displaces (and, if taken far
enough, dissolves entirely) the Flatland of pictorial rationality” (Lynch, 1991, p. 1). Thus, I offer these knot diagrams as tools that actually undermine the usual conventions
of graphic representation in our field, not simply to disrupt
for the sake of disruption but to invite speculation about
how we might develop different diagramming habits. The
knot diagram is meant to function as a “breaching experiment” (p. 15), inviting the reader to break with the usual
diagram conventions and imagine a new diagramming
practice that might better address the irregular and asymmetric tangles of interaction.
The Power of Diagrams
The diagram has a long illustrious history in the social sciences as a tool for representing social networks of various
kinds. In educational research, diagrams are used regularly
to convey the “essential” components and relationships
involved in teaching and learning. The diagram, however,
functions all too often as a crude tool for reducing the
complexity of these situations to a set of inadequate tags
that often entirely misrecognize the actors and actions
involved. Lynch (1991) showed how diagrams in sociology journals used a limited repertoire of iconic, geometric,
and semiotic elements to reduce complex phenomena to
“bounded labels, quasi-causal vectors, and spatial symmetries
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and equivalences” (p. 11). Many of these were “gratuitous
textual gestures” (p. 6) that were meant to function rhetorically as marks of quantitative and mathematical sophistication. Lynch argues, however, that diagrams do not have
to function this way, and he suggests that Deleuze and
Guattari show us how the diagram might be reconceived as
a powerful tool for thinking creatively.
Similarly, Latour (1990) suggests that we attend to the
way that our diagramming practices actually constitute and
control what is taken to be visible (and invisible) and that
such practices are indeed part of capitalist, imperialist, and
commercial interests. He notes how social and physical
scientists gain status and leverage when they mobilize their
preferred inscriptions and gather the gaze of others to these
inscriptions: “Scientists start seeing something once they
stop looking at nature and look exclusively and obsessively
at prints and flat inscriptions.” The flatness of the diagram
is also crucial in invoking and mobilizing mastery; one can
dominate a flat surface where there are no hidden convolutions or shadows. Whenever one needs to master a subject,
says Latour, look for the flat surfaces that enable that
mastery—a map, a list, a file, a census, a diagram.
These concerns speak directly to researchers whose aim
is to advance social justice through inquiry. If we attend only
to the diagrams without placing them within the turmoil of
political vying for power and status pursued by various subjects (or actants/ quasi-subjects, in Latourian terms), then
we will either be “mystical” about semiotic material—by
fetishizing it’s free and emergent power—or we will imagine that there is some a priori logic to why some inscriptions
work and others don’t. Latour identifies nine aspects of the
power of diagrams in furthering a cause, mustering allies
and mobilizing people (and things) in support of particular
agendas. These nine aspects of the power of diagrams are
mobility, immutability, flatness, reproducibility, changes in
scale, potential for being superimposed or recombined with
text and other inscriptions, and lastly, occupying the plane
where axioms of geometry and measure can be applied. One
can see immediately how qualities such as flatness, immutability, and rigid measure lend themselves to reductive arboreal coding habits whereby linear models of growth are
imposed on complex rhizomatic processes.
Through these particular drawing practices, diagrams
come to dominate that which they claim to reference or
depict. In effect, Latour suggests that this is the “papering of
things by inscriptions.” Optical consistency allows for “a
two-way path of access to the thing and back . . . making the
thing something that was invariant from one place to another.”
In effect, the diagram emerges as a model that carries with it
a rigid Cartesian metric to be imposed on all the contexts to
which it will be applied in the future. In the field of educational research, we rarely acknowledge the power of diagrams in structuring and confining our understanding of
classrooms. If our diagrams mobilize particular ways of
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Qualitative Inquiry 18(7)
Figure 1. Technology mediated student-teacher interaction
Source: http://wiki.laptop.org/images/8/8c/Edu_Toolkit_Use_Case1.png
thinking about classrooms and impose a flatness and
Cartesian measure onto classroom interaction, we need to
recognize the ways in which they also muster support for particular curricular and instructional agendas. To summarize,
Latour identifies the particular diagramming habits that have
come to dominate the linear arboreal models of growth and
interaction we take for granted in educational research.
A survey of all education articles from 1950 to 2010
available in the database JSTOR under the search terms
“classroom” and “interaction” reveal more than 1,500 with
diagrams of some kind. Setting those aside that are charts or
tables and those that represent the physical arrangement of
the classroom, the ones that truly aim to capture interaction
often impose simplistic reductive categories onto the lived
experience of students and teachers. The tradition of diagramming in educational research often involves a schematic rendering of material practices evidenced in the
classroom (writing, speaking) as well as presupposed but
invisible cognitive constructs (problem solving, reflection),
the two categories often bound to each other through arrows
or overlapping circles invoking quasi-causal relationships.
These tree diagrams and Venn diagrams tend to enforce hierarchical models and metaphors of cognitive individualism
and causality. In other words, they represent causality as unitary and linear, and they represent the individual student in
isolation and fail to capture the dynamic and collective
nature of interaction. Consider, for instance, Figure 1, where
student and teacher interact through digital platforms.
Diagrams that do a somewhat better job of mapping interaction often identify the spatial coordinates of students and
teacher and show connections between these points (lines)
as traces of communication (Figures 2 and 3). These are
more powerful in that they begin to map the physical residue
of classroom interaction and the perceptual coordinates of
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de Freitas
Figure 2. Frequency and source of verbal interaction
Source: Nathan & Knuth, 2003.
the students and teacher, but the diagrams of this sort found
in the sample focus only on interaction through spoken word
between human agents, neglecting the complex ways in
which interaction involves engaging material and nonhuman
agents as well. For instance, the two examples in figure 2
and 3 trace the verbal interaction between the teacher and
students using the arrow to indicate who is speaking to
whom, the thicker lines indicating higher frequencies of
interaction. Such graphical representations, however, evoke
the concepts of linear and direct transmission and reception
and fail to capture the entanglement of interaction.
These diagrams remain reductive in that they trace or
represent links between isolated speakers without attending to how nonverbal links might function on occasion as
disruptive lines of flight that actually rupture the presumed
dimensionality and flatness of the diagram. Like other diagrams in the field, these ones function as highly restricted
and ossified representations of linear causality, rather than
as creative forces of imagining interaction as entanglement. In the next section I discuss Deleuze and Guattari’s
concept of the rhizome and describe how this concept can
help us think about entanglement in classrooms.
The Rhizome
Although their terminology is sometimes difficult, and their
analogies always playful, Deleuze and Guattari (1987, 1994)
offer up a set of creative philosophical tools for tackling the
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Qualitative Inquiry 18(7)
Figure 3. Frequency and source of verbal interaction
Source: Hudson & Bruckman, 2004.
complex social dynamics of classrooms. They provoke us to
consider a radically new ontology of the social, where subjectivity is formed by way of discontinuity, rupture, and
multiplicity in a vast interleaving rhizomatic assemblage.
This is a subject with distributive agency, whose capacity to
act evolves through a process of repetition, variation, proliferation, and differentiation in an acentric network of dispersed affect (Grosz, 2008; Sermijn, Davlieger, & Loots,
2008). The rhizome is contrasted with the arboreal model or
tree diagram, which has dominated humanist theories of
sociality and subjectivity. The arboreal is hierarchical and
linear and grounded in a unitary trunk or cause. Rhyzomes,
unlike trees, are such that any node can be connected to any
other—there is no strict hierarchical structure that confines
contact. A rhizome can be broken or cut, but it starts up again
elsewhere on one of its old lines or it starts new lines. Every
rhizome contains “lines of segmentarity” by which it is
stratified and territorialized and doused in signification, but
they also contain lines of flight or lines of deterritorialization,
where a rupture breaks with the entity. These lines of flight
tie back into the rhizome: “These lines always tie back to one
another” (Deleuze & Guattari, 1987, p. 9). The looping back
is not simply a mimetic form of recognition or replication but
a creative act of capturing. Looping back is a “veritable
becoming . . . not an imitation at all but a capture of code,
surplus value of code, an increase in valence, a veritable
de Freitas
becoming, a becoming-wasp of the orchid, and a becomingorchid of the wasp” (Deleuze & Guattari, 1987, p. 10). It is
through these lines of flight and capture that rhyzomatic
assemblages stretch outside of code in some material way.
New kinds of agents become part of the loops, and the rhizome itself incorporates these new agents. Although the
looping back is often the result of control and/or oppressive
forces, a rhizome involves sites of “asignifying rupture,”
which work against the “oversignifying breaks” that segment
and separate and cut across the structure (Deleuze &Guattari,
1987, p. 9). A differentiated line suddenly splits off and
erupts into another potential for differentiation, disrupting
the patterns of growth that may have become entrenched. As
Braidotti (2006) suggests, Deleuze is challenging us to think
past linguistic mediation and identity, toward a “non-unitary,
radically materialist and dynamic structure of subjectivity”
(p. 2). The rhizome becomes more and more appropriate as a
way to think the socius and the subject, as each and together
are a multiplicity without unity or essence. Indeed, the classroom as rhizome helps us think about the event-structure of
classroom interaction. To study the classroom as rhizomatic
assemblages is to study the moments of rupture, to identify
and follow the lines of flight and differentiation: “Always
follow the rhizome by rupture; lengthen, prolong, and relay
the line of flight; make it vary, until you have produced the
most abstract and tortuous of lines of n dimensions and broken directions” (Deleuze & Guattari, 1987, p. 11). It is precisely this proliferation of rupture and capture—this acentric
and nonlinear growth—that allows a rhizome to thrive. We
are always in the midst or the milieu of a rhizome, always
located at one of the many middles that constitute a rhizome.
The classroom as rhizomatic assemblage includes many
kinds of agents or nodes aside from human or biological persons. The blackboards, the projectors, the furniture, even the
announcements over the intercom, all factor into the assemblage. We need a theory of rhizomatic processes that recognizes these diverse kinds of interactants. Latour (1993)
argues that we must resist the modernist tendency to divide
the world into two spheres: the human sphere (a sphere of
originary freedom, will, intention, agency, diversity) and the
sphere of nature or the external world (a sphere of limited to
no agency, acting with mechanical precision). It is precisely
this division, he argues, that has forced us into the confining
theoretical boxes that forever hobble our attempts at studying social interaction. He rails against Modernity’s claim for
an ontological divide between human thought and that which
is outside of thought and demands that we reconsider agency
as distributed across a heterogeneous ontology.
Consider, he suggests, a world composed of “actants” or
quasi-objects that all partake in some degree of agency, as
though a network or assemblage of catalysts or mediators.
An actant is a source of action: “something that acts or to
which activity is granted by others. It implies no special
motivation of human individual actors, nor of humans in
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general” (Bennett, 2010, p. 9). An actant may appear “sometimes as a thing, sometimes as a narrative, sometimes as a
social bond, without ever being reduced to a mere being”
(Latour, 1993, p. 89). This emphasis on “thingness” aims to
set things in motion, conceiving of all entities as events or
trajectories enduring through space and time. Even inanimate “objects” are seen as active mediators in a social material network. A quasi-object is “as much force as entity, as
much energy as matter, as much intensity as extension”
(Bennett, 2010, p. 20). This is not to suggest that there are
no ways of distinguishing between human actants and nonhuman actants. Bennett argues for a “vital materialism” that
might pursue a new kind of political theory following
Latourian metaphysics. She taps into Spinoza’s concept of
“conatus” to explore the embodied nature of assemblages.
Conatus is the “active impulsion” of a body, the tendency to
persist or persevere in “its own being” (Bennett, 2010, p. 2).
For more complex assemblages, conatus is the “effort
required to maintain the specific relation of ‘movement and
rest’ that obtains between its parts, a relation that defines the
mode as what it is” (p. 22). In this sense, conatus refers to
the dynamic entwinement of force and matter, quite literally
the relationship of movement and rest between various
quasi-subjects, be they human or other.
The theory of assemblage allows us to reimagine agency
as distributed across the surface of these heterogeneous rhizomatic alliances. Indeed, the theory of assemblage suggests that the power of an alliance is mostly enhanced by the
heterogeneity of it. Bennett gleans from Latour and Spinoza
a method for studying human-nonhuman interaction:
Bodies enhance their power in or as a heterogeneous
assemblage. What this suggests for the concept of
agency is that the efficacy or effectivity to which that
term has traditionally referred becomes distributed
across an ontologically heterogeneous field, rather
than being a capacity localized in a human body or in
a collective produced (only) by human efforts.
(Bennett, 2010, p. 23)
Assemblages are composed of diverse elements and
vibrant materials of all kinds. The classroom assemblage is
composed of humans, writing implements, writing surfaces,
texts, desks, doors, as well as disciplinary forces whose
power and agency are elicited through various routines
(singing the anthem) and references (“In algebra, we always
do this . . .”). Power is not distributed evenly across the surface of an assemblage, since there are joints or nodes where
there is more traffic and affect than at others. Assemblages
have “uneven topographies” and possess emergent properties (Bennett, 2010, p. 24). Certain individuals and objects in
a classroom, often being those who speak up or take action
or function as sites for visibility (i.e., smartboards), leverage
this power differential. Mapping the movement of power
594
across the classroom involves attending to the way that
affect or feelings emerge and are mobilized and blocked.
Affectivity is associated with students and other agents, as
well as with the assemblage itself, which is more than the
sum of its elemental affects. It is important to conceive of the
agency of the assemblage in terms that do not simply reduce
it to a static structure imposing fixity on the active agents
within it. The power of the assemblage is not merely negative as a constraint or passive as an enabler. The assemblage
is a fluid folding agent as well. Bennett (2010, p. 30) cites
Coole’s (2005) recent revisioning of agency in terms of
“agentic capacities” by which one might escape from the
discrete individualism assumed in most approaches to the
study of classroom interaction. Coole describes a spectrum
of agentic capacities housed sometimes in persons but
sometimes in physiological processes and sometimes in
transpersonal intersubjective processes. Bennett wants to
extend these capacities beyond the human realm and into
the realm of human-nonhuman assemblages and powers
that might not always be associated with human agency.
There is always a swarm of intensities at work when there
is agency: “The task becomes to identify the contours of
the swarm and the kind of relations that obtain between its
bits” (Bennett, 2010, p. 32). In the context of the classroom, there is a proliferation of surfaces (whiteboards,
blackboards, projectors, the “verbal” plane of speech) in
relation to which a distributed agency emerges. As Means
(2010) suggests, Deleuzian concepts allow us to map the
“tactical navigation of perceptual forces” and the “dynamics of affect within acts of dissensus” so that we might
understand how subjects leverage and navigate the flow of
affect and “become ethically recognizable as visible and
audible subjects of equality” (p. 8).
Rethinking the Diagram
Can we diagram classroom interaction as though it were
rhizomatic? Can we consider the classroom as though it
were “some couchgrass or some of a rhizome” (Deleuze &
Guattari, 1987, p. 9)? Can we find lines of flight erupting,
differentiating, proliferating, and being brought back into
the fold? Are there creative moments when a rupture in the
seam is allowed to flourish? The problem with modeling
classroom interaction using the rhizome is that the rhizome,
by definition, “is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or
deep structure” (Deleuze & Guattari, 1987, p. 12). It might
seem at this juncture that diagrams—as Latour has described
them—are inappropriate for the study of rhizomes. If there
is no “generative model” in a rhizome, then none of our
all-too-familiar diagramming practices can capture its
operation. Although Deleuze and Guattari ask that we abandon the search for deep structure and genetic axis, they are,
I believe, deeply committed to a manner of diagramming
Qualitative Inquiry 18(7)
that experiments with rhizomatic assemblages. They offer
up the rhizome less as a tracing of the social (since a tracing, they argue, encapsulates the tree logic of representation) and more as a map. A map, unlike a tracing, “is
entirely oriented toward an experimentation with the real”
(p. 12). The map does not represent the territory—the map
constructs the territory. The map is itself part of the rhizome. A map, like the rhizome, has multiple entry points
and can be opened up for additional connections in all of its
dimensions. In this sense, mapping subjectivity across the
classroom would be less about identifying stages in a
genetic axis or positions in a deep structure, or about representation of interaction, but would involve following the
affects and percepts in their twisting, braiding, and knotting
emergence. Along these lines, we could create a rhizomatic
“group map” for the classroom, by which we might capture
or evoke the conflicting processes of dispersal and massification. We might draw the lines that survive the folding
back of state-sanctioned curricula, and map their potential
growth into new dimensions that rupture the map itself, as
these lines continue to proliferate, “if only underground,
continuing to make a rhizome in the shadows” (Deleuze &
Guattari, 1987, p. 14). This group map might function as
the abstract machine or diagram of the classroom, not as a
reductive encoding of classroom interaction but as an
engine itself for creative imagining.
For Deleuze and Guattari (1987), diagrams do not have
to be reductive. They can function as pragmatic exercises of
finding out how something works. Reduction, on the other
hand, is a form of encoding, as though a priestly unveiling
of meaning. As Buchanan (1997) argues “of abstraction,
Deleuze and Guattari say they can never get enough; of
reduction, they say they can never get too little” (p. 83).
Thus, a diagram can be used to find out how a rhyzomatic
network thrives rather than unveiling what it means. Such
diagrams would be about realizing abstractions rather than
identifying the referent that is being stood-in-for. Buchanan
suggests that “to abstract means to isolate the triggers that
will produce new Universes of reference—to use Guattari’s
felicitous phrase—whereas to reduce means to isolate the
triggers that would enable the restoration of a Universal referent” (p. 84). Thus, the function of the diagram, suggests
Frichot, is actually to step outside of illustration, figuration,
and representation. A diagram is an experiment, inventing
lines of flight at the threshold between the actual and the
virtual. As a “gestural deployment of material” (Frichot,
2005, p. 73), the diagram operates through potentiality and
possibility. In Deleuze’s terms, “The essential thing about
the diagram is that it is made in order for something to
emerge from it, and if nothing emerges from it, it fails”
(Deleuze, 1994, p. 102, quoted in Frichot, 2005, p. 73).
The diagram must be seen to construct a new real that is
yet to come, a plane of creation pushing back at the regimes
of signification and sundering preexisting forms of content
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de Freitas
and expression. The diagram, in this sense, should not be
seen as a tracing or reductive model. According to Deleuze
and Guattari (1987), the inventive diagram knows nothing
of the distinction between the real and the not real, since it
is neither substance nor form, but rather pure function and
abstract matter. The inventive diagram knows only potentiality, lines of flight, “particles-signs” (p. 142). Such a diagram is absolute positive deterritorializing, an abstract
machine, operating by function (not form) and by matter
(not substance). Regimes of representation—and in particular the semiotic triple of index, symbol, and icon—impose
an axiomatics of confinement on the diagrammatic. To
honor the creative impetus of our diagrams, we must see
them as actants, agents, or events that carve up matter in
new ways.
Châtelet (2000) pursues this line of thinking, arguing
that diagrams are material experiments, cutting up space,
folding surfaces, and multiplying dimensions. The diagram
is not simply a representation, or an illustration or code—
there is no algorithm or rule for determining it. Reducing a
diagram to a representation “ignores the corporeality, the
physical materiality” of diagramming as a creative activity.
Diagrams are allusive and allegorical, elastic, and never
exhausted. For Châtelet, diagrams act as interference or
intervention in that they are potentially creative events,
conjuring and shaping the sensible in sensible matter. The
diagram invites an erasure, a redrawing, a “refiguring”
(Knoespel, 2000, p. xvi). He suggests that innovative diagramming techniques have historically pushed through
confining axiomatics and state-sanctioned practices to
allow for new forms of doing mathematics and science.
Like Deleuze, he sees the diagram as a potential that is
never entirely actualized, since it stands somehow outside
of representation. “Diagrams are more than depictions or
pictures or metaphors, more than representations of existing knowledge; they are kinematic capturing devices,
mechanisms for direct sampling that cut up space and
allude to new dimensions and new structures” (de Freitas
& Sinclair, 2012, p. 12). According to Châtelet, diagrams—
and abstractions of other kinds—cannot be extracted from
sensible matter through an act of reduction or subtraction
because diagrams are a kind of capture technology, a
machine for grasping, trapping, contracting, folding, and
twisting matter.
Knot Diagramming
The development of knot diagramming in the 19th century
is an excellent example of an inventive practice that would
“smash the classical relationship between letter and image”
(Châtelet, 2000, p. 184). Knot diagramming formally
began in the 1880s when physicists speculated that each
atom in the enveloping ether was a contorted knot. In the
19th century, the term ether referred to an elastic quasi-rigid
Figure 4.A mathematical knot and corresponding diagram
inert medium purportedly filling up universal space. Each
fundamental atom was considered a unique knot in the
ether. Scientists using these diagrams recognized how they
were not simple illustrations in that they broke the rules of
the Cartesian plane.3
Mathematicians—unlike sailors—were less interested
in how tension made a knotted rope taught and more interested in the distinctness of knots, and so they joined the
ends of the rope after tying the knot, essentially capturing
the knot in a closed curve. Such contained knots could
then be studied, compared, and sorted. Mathematical knot
diagrams are topological, elastic, and deformable creatures. One can stretch or distort any part of a knot diagram,
twist it and fold it, and make it appear entirely different in
a new diagram. Figure 4 shows a knot and its corresponding knot diagram.
If a knot can be unraveled using straightforward unfolding techniques, termed “the reidemeister moves,” so that
it’s minimal number of crossings (termed “the crossing
number”) is equivalent to one of the “prime knots,” then
one has identified a knot masquerading as a more complex
knot. For instance, the trefoil knot (Figure 5) is a prime
knot with a crossing number of 3. The knot in Figure 6
remains equivalent to the trefoil knot, even though it’s
three leaves have been twisted. The apparent crossings in
the knots in the figures below create a multidimensional
effect, suggesting a layering precisely where Cartesian
geometry would have imposed an intersection.
One can obtain the trefoil from Knot 6 by simply unfolding these twists and contracting the elastic rope. Note that
neither of these knots can be deformed into the circle
(Figure 7), which is termed the “unknot” and has crossing
number 0. And yet the seemingly complex knot in Figure 8
can be shown to be equivalent to the unknot after a series of
simple unraveling moves.
Topological diagramming, like that used for knots, introduces depth into the plane, conjuring a virtual dimension
within the two-dimensional surface. Topology shifts our
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Qualitative Inquiry 18(7)
Figure 5. The trefoil knot
Figure 6. The trefoil knot with twisted loops
Figure 7. The unknot
Figure 8. A distorted unknot
attention away from concepts of measure and rigid transformation (like linear causal links) and focuses on the stretching and distortion of continuously connected lines and
regions. Topologists and knot theorists study boundary relationships and connectedness. Knots are themselves composed of curves or curvilinear lines that are embedded in
dimensional spaces, but the curve in n-space is an unruly
and nonlinear creature. Depending on the space—whether it
be three dimensional or four dimensional or other—different
structural relationships emerge, while other relationships
will suddenly dissolve. For instance, some knots in threedimensional space will suddenly become unknotted in fourdimensional space. We are accustomed to seeing curves on
the flat page tracking their methodical extension across the
two-dimensional space, confined to the page, regardless of
how curly or fractal. In contrast, the curve or line in n-space
is a multidimensional entity, suddenly possessing perspective
and depth, moving out toward us and away from us. One can
relax the knot and loosen its crossings, and then imagine
crawling along the rope, following its path into the depth of
the page. The knot has no interior or exterior. It is all line, or
all outside. Thus, the knot behaves rhizomatically in pursuing it’s proliferating lines of flight—we are always in the
milieu of the knot, along its paths and expanding or contracting loops, and never positioned at a fixed point, neither a
beginning nor an end. The knot is like a rhizomatic assemblage, which is “precisely this increase in the dimensions of
a multiplicity that necessarily changes in nature as it expands
its connections. There are no points or positions in a rhizome, such as those found in a structure, tree, or root. There
are only lines” (Deleuze & Guattari, 1987, p. 8). A knot follows a one-dimensional path through three-dimensional
space (or greater dimensional spaces), connecting elements
in all these different planes. “It is necessary to imagine foldings, invaginations, exquisitely complex situations that generalize the practice and the idea of the knot to all imaginable
dimensions” (Serres, 2008, p. 78).
Cutting is the only way to introduce new knotted segments into a knot diagram. Through a cut, one is able to
create new lines of flight, new crossings and folds, ruptures
and new segments. Cuts or “Lines of flight” or positive
nomadic paths of (de)territorialization mark the emergence
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Figure 9. Cutting and combining knots
Figure 10. Links composed of distinct knots
of a singularity within the system. If we cut the trefoil
(Figure 5), and the line of flight folds and twists and loops
back, and we then stretch and retie it with two more knots
(Figure 9), we have an entirely new knot and a new network, with new sites of intensity and new contractions.
The knotted assemblage is always growing, cutting and
chasing the proliferations of lines, while also assembling or
joining with other knots. At any site of rupture, two knots
can be joined to form compositions or more complex
assemblages, like that found in Figure 10 where three components fold across each other but never join.4 Knot diagrams play havoc with three of the essential qualities of
diagrams that Latour pointed out. First, they are not immutable. They can be stretched and deformed until they barely
resemble the original. Second, because of their over- and
under-crossings, they break with flatness. And third, they
don’t occupy the plane in the traditional sense, and so they
resist the usual confining metric of Euclidean geometry.
For these reasons, knot diagrams are an inventive way of
studying affect and interaction in the classroom. In the next
section I explore these ideas in relation to classrooms and
institutional settings.
empirical methodology, knot diagramming can be used during classroom observation or video analysis. Instead of simply tracking the straight linear path between two students or
teacher and student and then applying a Euclidean measure
of the content exchanged, I suggest that we follow the meandering tangle and conceive of interaction as a genuine mixture. The diagrams in this article were generated using
knotplots, a dynamic graphing software package that allows
one to generate knotted lines and surfaces and to then cut,
fold, twist, stretch, and join various components. In the classroom assemblage, the looping, folding, and segmenting of
the dynamic knot captures the flow of information, affect,
and movement. Certain crossings in a knot might reduce the
flow and proliferation of these material encounters and
inhibit movement. Such crossings mark the ossification of
particular relationships among people and objects. When
passionate thresholds are reached in the classroom assemblage and the knot of relationships is stretched beyond some
presumed material capacity, the rope is suddenly cut; the
thread torn, and a new node or singularity opens up; a line of
flight erupts and twists and folds its way into new dimensions. These lines are almost always recaptured and tied back
onto the knot, looping back to infuse new energy into the
network, while always subjected to code and containment.
The lines are diverse and link across the ontological heterogeneity of the assemblage. In other words, the linkages are
between people, machine, door, language, and so on. The
triggers for rupture can be minor and seemingly inconsequential—a broken overhead projector, an intercom
announcement, the accidental pairing of two excited students. Or they can actively involve the introduction of
actants with high degrees of agency and the simultaneous
proliferation of surfaces to work on, when the knotted network is joined to another or a new component is linked
in—imagine the folding that occurs when a valued classroom
visitor comes or when the students are introduced for the first
time to clay or oil paint or when the teacher suddenly
explores the “what if not” alternative of a closed mathematics problem. Figure 11 offers an example of a knot composed
of two components, one with a rupture and the other without.
If you pick a line and follow it, you can uncover the two
distinct components.
Experiments With Knot Diagrams
and Classrooms
A focus on the knotted nature of interaction points to the
folded, segmented, and multidimensional aspects of activity,
while directing our attention to the topological relationships
between agents (borders, connectedness) rather than metrical
(measure) or linear relations. We mix topologically, through
cuts and joints that affix the one knot to another, bound by
proximity and connectedness rather than Cartesian concepts
of discrete distance and separation. One can imagine these
cuts and singularities in terms of subjectivity and affect,
where a singularity or rupture offers the space of the potential, of the new, a site of liberation and recombination of
affective forces (Deleuze, 1993). In this new network,
subjectivity is dispersed across the old and new crossings
and recaptured by the assemblage in different ways. In this
section I suggest that knot diagrams can be used to capture
the complex entanglement of classrooms. As a concrete
598
Figure 11. Multiple components (either closed or ruptured)
Note.
A. Component 1
B. Component 2
C. Kink or narrow passage where two previous knots were
joined.
D. Space contained by the two entangled components
E. Line of flight
F. Rupture or singularity
Figure 12. Classroom interaction (closed)
Note.
A. State curriculum mandates (a separate but entangled
component)
B. Teacher folding into curriculum
C. Student compliance
D. Teacher folding under state mandates
E. Potential line of flight or rupture
F. Student leaving classroom
G. Announcements of future tests (tightening)
In what follows, I focus on the ways in which knot diagrams might capture the competing forces at work in shaping classroom interaction. There are five facets of these
diagrams that seem to support this: (a) multidimensionality, (b) joining and entangling of multiple components, (c)
capacity for distortion, (d) lines of flight,5 and (e) crossing
and unknotting numbers. Every closed knot has an
“unknotting number,” which is the number of crossings
Qualitative Inquiry 18(7)
that need to be changed (from over to under, or the reverse)
in order to turn the knot into the unknot (Figure 7). The
unknot is a closed curve without any folds, segments, or
breaks, indicating an absence of interaction. When classroom interaction pays too much tribute to language mastery and linguistic communication, without adequate
attention to other material encounters, the folds of the knot
are mere twists, and the knot itself masquerades as something more multidimensional than it is. In such cases, student participation is enlisted primarily in the service of
state standards, where learning involves merely the reproduction of linguistic capital. The crossings of such a knot,
where the lines twist without creating complex entanglements, are manufactured and segmented by the institution
and then saturated with signification and code. The
uncrossing number for these knots is always minimal so
that control and obedience can be quickly instilled in the
classroom if need be. In Deleuze and Guattari’s terms,
positive nomadic paths of deterritorialization (where
divergent thinking and activity is expressed) and subterranean molecular trajectories (where affect traverses the
tangle of interaction) are subjected to the molar massification of the social assemblage.
Since subjectivity is always already emergent within
the assemblage, within the milieu of the rhizome, since
subject and system are reciprocally presupposed, the
challenge for teachers and students alike is to grasp how
the state tightens the knot and twists the rope. How do we
map the affect and power across the assemblage, map the
segmentation of the state while also following the emergent ruptures? If affect circulates across the psychic network of the classroom assemblage, dispersing intensity
on body and other surface, then we need to imagine interaction in terms of apersonal and nonindividuated agency,
as though energy were flowing across the surface or
through the rhizomatic assemblages of linked singularities (Massumi, 2002). In Figure 12, the classroom interaction is highly controlled, without line of flight or rupture.
The legends under Figures 11 and 12 point to the actions
of different kinds of agents (teacher, curriculum, tests)
and various forms of folding whereby compliance and
resistance occur. The knot consists of two components,
and I have indicated that these have directionality (arrows)
whereby the flow is forced along the line, and the sites
where the two cross will generate friction.
In contrast, Figure 13 is composed of one component
with ruptures. At the far right the knot is highly symmetric, which might capture cooperation or coercion, depending on how one values symmetry. The diagram was created
by joining three distinct knots, one with 10 crossings (far
right), 5 crossings (middle), and 3 crossings (left). Joining
the knots meant reducing the number of crossings by one
in each case and then distorting the left portion introduced new crossings. Since this knot diagram is not a
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de Freitas
Figure 13. Classroom interaction (with rupture)
Note.
A. Distraction (dispersal of affect)
B. Site of conflict between two students
C. Teacher occupies the immovable space within the symmetric
arrangement
D. Group of students excited by student conflict
E. Students on cell phones
F. Contraction (slows flow of affect)
closed curve, it can easily unravel. As a diagram of classroom interaction, it captures the delicate nature of interaction and points to the tensions between order or symmetry
(managed behavior) and more divergent thinking or
actions.
These diagrams and their legends are offered as experiments rather than signifying codes. Granted, I am tapping
into knot theory as a source of inspiration, but I am using
these diagrams as a “countersignifying semiotic” rather
than a model or system that blocks lines of flight through
axiomatization.6 The diagrams are meant to act as transformations of the current regime of diagramming, transformations that aim to disrupt or even “blow apart” the codes of
interpretation that dominate our field (Deleuze & Guattari,
1987, p. 136). Of course, even an attempt at an asignifying
diagram “harbors knots of coincidence” that seem to be
“waiting” for code (p. 138). And indeed my labeling is precisely an act of coding, an act of affixing signifiers and
signifieds, but my hope is that this lettering of the knots
resists closure in being deconstructive of more dominant
practices.7 I invite the reader to rearrange the letters to different locations on the knot, to deform the knots and stretch
them in unexpected ways, and to experiment with the act of
lettering. My hope is that these diagrams might trigger
both a critique of our current practices and an exploration
into new strategies for studying interaction. The diagrammatic, as an abstract machine, constructs a real, yet-tocome, and “a new type of reality” (p. 142). This new reality
breaks apart the linear models of positivist social science
research and shows us how to advance educational inquiry
so that the complexities of classroom interaction might be
better studied.
Concluding Comments
The aim of this article was to take up Latour’s call to
interrogate assumptions built into our graphic representations and to explore diagramming practices that might
engage with the classroom as a rhizomatic assemblage.
Knot diagrams lend themselves to the study of complex
linked networks because they break with Cartesian measure and honor the topological connectedness of the network. In addition, the over/under knot crossings punch
through the page and point to the multidimensionality of
interaction, whereas other characteristics—like the crossing and unknotting number—point to the degree of the
tangle and the complexity of the interaction. Because of
their topological nature—their focus on deformation—
they capture (without signifying) the ontogenetic nature
of classroom interaction. Knots evoke the stretching,
twisting, and folding processes that characterize becoming. A knot is an event, a potentiality. I offer these diagrams less as prescription or mirror and more as experiment
or mould. They are not meant to signify but rather to
express the emergent potential of interaction (Massumi,
2002). The legends of these diagrams—like the strange
taxonomy of animals in Borges’ apocryphal Chinese
Encyclopaedia—breach standard labeling conventions
and break ontological rules. Yet the absurdity they underscore is not altogether dismissive, since they function as
compelling devices for studying entanglement. My hope
is that these diagrams disrupt the usual conventions of
graphic representation in our field, while pushing us
toward new diagramming habits that better engage with
the classroom as rhizome.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
Notes
1. See for instance, Braidotti, 2006; Clough & Halley, 2007;
Gale, 2010; Gilbert, 2004; Goodley, 2007; Gough, 2004,
2005, 2007; Maclure, 2010; Ringrose, 2011; St. Pierre, 2004;
Semetsky, 2006; Tamboukou, 2008; Webb, 2008, Wyatt,
Gale, Gannon & Davies, 2010).
2. See, for instance, the diagrams on pp. 135, 137, 146, 185,
218. See also the beautiful diagramming of chapters from a
Thousand Plateaus by Marc Ngui, available at http://www.
bumblenut.com/drawing/art/plateaus/index.shtml
3. Theories of knots and knot diagramming continued to evolve
in the 20th century, even though the concept of ether was
abandoned.
4. In forming assemblages, a small piece of the rope is removed
from each knot, and the ends are tied to the other’s ends. Other
kinds of assemblages called “links” can be formed using multiple closed knots that become entwined without joining (Figure
10). Knots that involve two or more entwined components (two
twisted closed lines in space) can be studied and contrasted
by examining the nature of the crossings in each component.
600
Distinct components are identified and studied when each knot
is given an orientation (pick a direction to travel around each
of the closed curves) and then each crossing that involves both
components is labeled as either under or over.
5. Canonical mathematical knot diagrams don’t have breaks or
lines of flight. I am introducing these lines of flight as an
important facet for the study of interaction.
6. It is important to note that I am breaking with the rules of
formal mathematical knot diagramming by allowing ruptures
and lost threads to remain untied.
7. Massumi (2002) suggests that Deleuze and Guattari are not
interested in postmodern absurdity, parody, or ironic subversion because these tactics actually conserve “the true” (p. 5).
According to this view, the postmodern deconstructive gesture might be seen to actually foreground the speaking/
writing subject as though nostalgic for the “masterful presence” that it claimed (p. 5). Instead, Deleuze and Guattari
present their diagrams as genuine experiments and creations
of the new. My reading, however, is that the diagrams in
Thousand Plateaus, for instance, function more powerfully
if they are taken up as both creation and deconstruction.
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Bios
Elizabeth de Freitas is an associate professor at Adelphi
University. Her research interests include mathematics education, research ethics and methodology, as well as philosophy
and cultural studies. She has published articles in Educational
philosophy and theory; Educational Studies in Mathematics;
Qualitative Inquiry; Race, Ethnicity and Education;
Mathematics Teacher Education; The International Journal of
Education and the Arts; Teaching Education; Language and
Literacy; Gender and Education.