Math Reading _ sept 25th_Understanding learning systemsx

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Abstract (summary)

Complexity science may be described as the science of learning systems, where learning is understood in terms of the
adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex
learning systems, we exp
lore some of the possible contributions of complexity science to discussions of the teaching
of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of
mathematical communities and the application

of principles of complexity to the teaching of mathematics. Through
the course of this writing, we attempt to highlight compatible and complementary discussions that are already
represented in the mathematics education literature.

Full Text

Complexity science may be described as the science

of learning systems, where learning is understood in terms of the
adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex
learning systems, we explore some of the possible contributions of complex
ity science to discussions of the teaching
of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of
mathematical communities and the application of principles of complexity to the teaching of ma
thematics. Through
the course of this writing, we attempt to highlight compatible and complementary discussions that are already
represented in the mathematics education literature.

Key Words: Classroom interaction; Cognitive theory; Knowledge; Learning; Teaching practice

Early in 2000, Stephen Hawking commented, "I think the next century will be the century of complexity" (Chui, 2000,
p. 29A). Hawking's remark was in specific refere
nce to the field of complexity science, which, as a coherent branch of
scientific inquiry, has only come together over the past 30 years.

It is not strictly correct to identify complexity science as a branch of inquiry. Unlike the analytic science of the
Enlightenment, complexity science is defined more in terms of its objects of study than its modes of investigation. It
first arose in the confluence of several areas of research
-
including cybernetics, systems theory, artificial intelligence,
and nonlinear
dynamics
-
many of which had begun to appear in the physical sciences in the mid
-
20th century. More
recently, certain research emphases in the social sciences have come to be included under the rubric of complexity.

In one popular account of the emergence o
f the field, Waldrop (1992) introduces the diverse interests and the diffuse
origins of complexity science through a list that includes such disparate events as the collapse of the Soviet Union,
trends in a stock market, the rise of life on Earth, the evol
ution of the eye, and the emergence of mind. Elsewhere the
list has been extended to include any phenomenon that might be described in terms of a living system
-
including, in
terms of immediate human interest, such levels of organization as the cell, bodily

organs, the person, various social
structures, an economy, a culture, and an ecosystem (Johnson, 2001). Two key qualities are used to identify such
complex phenomena. First, each of these phenomena is adaptive. That is, a complex system can change its own

structure and as such is better described in terms of Darwinian evolution than Newtonian mechanics. Second, a
complex phenomenon is emergent, meaning that it is composed of and arises in the co
-
implicated activities of
individual agents. In effect, a comp
lex system is not just the sum of its parts, but the product of the parts and their
interactions. Complexity scientists (or "complexivists") often describe such adaptive, self
-
organizing phenomena as
learning systems (Capra, 2002; Johnson, 2001), where lea
rning is understood in terms of ongoing, recursively
elaborative adaptations through which systems maintain their coherences within their dynamic circumstances.

Our intention in this article is to present some preliminary thoughts on the possible contribu
tions of complexity science
to discussions of mathematics learning and teaching. Our main thesis is that mathematics classes are adaptive and
self
-
organizing complex systems
-
a suggestion that we regard as complementary to several strands of current
discuss
ion in the mathematics education research literature. In particular, some key principles of complexity are
prominently represented within such theoretical perspectives as radical constructivism, situated learning, enactivism,
and some versions of social co
nstructivism.

Our discussion proceeds as follows: First, because it is not well represented in the current literature, we begin with a
brief introduction to complexity science. We use this account to frame a preliminary discussion of compatibilities and
c
orrespondences between research into complexity and research in mathematics education. We then discuss several
key conditions that must be present for a complex system to emerge. This section of the article is divided into two
parts. We begin with a descri
ption of the spontaneous emergence of a mathematical community among a group of
secondary school teachers and use this example to introduce and illustrate five necessary conditions for complex
systems. We follow this description with a report on a more del
iberate effort to enact principles of complexity science
within a mathematics classroom. Here we revisit each of the five principles in an examination of their possible
contributions to the project of school mathematics.

COMPLEXITY SCIENCE

One way to exp
lain what is meant by complexity within complexity science is to contrast the terms complex and not
-
complex. Weaver (1948) was among the first to draw such a distinction in an article that has come to be regarded by
many as seminal to the field. In rather
broad brushstrokes, he sketched out three categories of phenomena that are
of interest to modern science.

The first category in Weaver's analysis is simple systems. Such systems are determinate and tend to involve only a
few interacting objects or variabl
es. Examples include trajectories, orbits, and billiard ball collisions
-
in effect, the sorts
of phenomena that were of central interest to Galileo, Descartes, Newton, and other Enlightenment thinkers. The
analytic methods that they developed, in particular

Newtonian mechanics, continue to be the principal means to
examine and manipulate simple systems. In brief, for a simple system, actions and interactions of each part can be
characterized in detail and the behavior of the system can be predicted with grea
t precision.

As Newton himself recognized, however, the mathematical tools used to analyze the behaviors of, and interactions
within, simple systems can give rise to intractable calculations when the number of interacting components increases
only slightl
y. In the 19th century, as scholars met up with more and more such phenomena, new analytic methods
based on probability and statistics were developed. These methods were useful for the interpretation of instances of,
as Weaver (1948) dubbed them, disorgani
zed complex systems
-
situations such as astronomical phenomena,
magnetism, and weather that might involve millions of variables or parts. (As developed below, disorganized complex
systems are not complex in terms of current usage.)

Significantly, the development of statistical and probabilistic methods represented more a resignation than a shift in
thinking. That is, statistical methods did not arise from or prompt a change in the fundamental assumption that
phenomena are determinate

and reducible to the sums of their parts. The universe was still understood as pregiven
and determined by unchanging laws. The new tools were understood to provide only a veneer of interpretation, one
that humans were compelled to impose on a universe owi
ng to perceptual and conceptual limitations. Laplace (1951)
summed up the underlying sensibility at the close of the 18th century:

Given for one instant an intelligence which could comprehend all forces by which nature is animated and the
respective situa
tions of the beings which compose it
-
an intelligence sufficiently vast to submit these data to analyses
-
it would embrace in the same formula the movements of the greatest bodies and those of the lightest atom; for it,
nothing would be uncertain and the fut
ure as the past, would be present to its eyes. (p. 3)

The move to statistical methods, then, was an acknowledgement that no flesh
-
based intelligence was sufficiently vast.
The more intricate and complicated the phenomenon, the more one was compelled to re
ly on descriptions of gross
patterns rather than analyses of interacting factors.

Weaver noted that these two categories
-
simple systems and disorganized complex systems
-
do not cover the full
range of possibility. In both cases, the systems are mechanical
or, in the terms of cyberneticist von Foerster, "trivial"
(1981, p. 201). Simple systems can be understood in terms of their inputs and outputs, since their operations remain
constant. However, many events and systems emerge in the interactions of agents t
hat are themselves dynamic and
adaptive. These nontrivial systems change their own operations through operating. Such phenomena are not entirely
predictable, as they have capacities to respond in different ways to the same sorts of influences. More signifi
cantly,
they can team new responses. Candidates for such systems range from the level of microorganism through the
biosphere and include several layers of phenomena that are of particular relevance to discussions of formal education:
brain function, indivi
dual sense
-
making, mind, social collectives, and culture. Such phenomena do not readily submit
to analytic methods based in Newtonian mechanics or statistical regression. As Weaver (1948) noted,

The problems, as contrasted with the disorganized situation
with which statistics can cope, show the essential feature
of organization. We will therefore refer to this group of problems as those of organized complexity. (p. 66, emphasis
added)

Since Weaver's formulation, the terms disorganized complexity and organ
ized complexity have been replaced by the
terms complicated and complex. As Waldrop (1992) develops, the term complicated is used among the complexity
sciences to refer to mechanical events, and thus includes both simple and disorganized complex systems. T
he term
complex, in the contemporary frame, corresponds to what Weaver described as organized complexity. In terms more
recently adopted within the field, complexity science is interested in events of emergence1
-
that is, in those instances
when coherent co
llectives arise through the co
-
specifying activities of individuals. Discussions of emergence are often
accompanied by such illustrative examples as the flocking of sandpipers, the spread of ideas, or the unfolding of
cultural collectives. These sorts of s
elf
-
maintaining phenomena transcend their parts
-
that is, they present collective
possibilities that are not represented in any of the individual agents. Moreover, such self
-
maintenance can arise and
evolve without intentions, plans, or leaders. They are in
stances of order for free (Kauffman, 1995).

This quality of transcendent collectivity is useful for drawing a distinction between analytic science and complexity
science. Complexity is not just another category of phenomena, but an acknowledgement that so
me phenomena are
not deterministic and cannot be understood strictly through means of analysis (i.e., literally, by taking apart or cutting
up). A different attitude is required for their study, one that makes it possible to attend to their ever
-
shifting
c
haracters and that enables researchers to regard such systems, all at once, as coherent unities, as collections of
coherent unities, and (likely) as agents within grander unities.

Interest in complex phenomena is not new. It has, for example, been represe
nted in Charles Darwin's studies of the
evolution of species, Alan Turing's curiosity toward co
-
evolutions of life
-
forms and their contexts, Jane Jacobs'
characterizations of living (and dying) cities, Lynn Margulis's work on symbiotic relationships, Frede
rich Engel's
discussion of social collectives, Niklas Luhmann's studies of self
--

organizing and self
-
maintaining social systems, and
Humberto Maturana's and Francisco Varela's research into autopoesis (see Capra, 1996; Casti, 1994; Johnson 2001;
Kelly, 19
95; and Waldrop 1992 for discussions of these and other historical examples). In fact, as Dewey (1910)
noted, by the turn of the 20th century many researchers across the physical and social sciences had already
embraced Darwinian processes to make sense of

their objects of inquiry.

The processes of Darwinian evolution, however, are not sufficient to account for the emergence of complexity. A more
recent realization is that the phenomena of self
-
organization is also vital. For reasons that are not fully und
erstood,
under certain circumstances agents can spontaneously cohere into functional collectives
-
that is, they can come
together into unities that have integrities and potentialities that are not represented by the individual agents
themselves. These event
s of self
-
organization might be further described as "bottom
-
up," as emergent
macrobehaviors (i.e., collective characters, transcendent capacities, etc.) arise through localized rules and behaviors
of individual agents, not through the imposition of top
-
do
wn instructions.2 In this regard, complexity science is an
example of the very phenomenon it aims to understand: the emergence of more complex possibilities through some
sort of boot
-
strapping processes.

Through most of its history, complexity science has

been focused on efforts to better understand self
-
organization,
mainly through close observations of complex systems and through computer modeling. For the most part, this work
has been descriptive in nature, as researchers have attempted to identify qual
ities and conditions that are necessary
to self
-
organization. More recently, however, there has been an increased emphasis among complexivists on the
creation and maintenance of systems that are able to learn. We use this shift in emphasis later to structu
re our
discussions of two mathematical communities in a manner that reflects the recent evolution of complexity science,
first focusing on a description of the spontaneous emergence of one collective, then on a deliberate effort to occasion
emergence. Prio
r to that discussion, however, we highlight some important correspondences between complexity
science and recent research in mathematics education.

COMPLEXITY, COGNITION, AND MATHEMATICS EDUCATION RESEARCH

Over the past 30 years, the emergence of complex
ity science has paralleled the embrace of new theories of knowing
and knowledge among mathematics education researchers
-
in particular the development of nonrepresentationist
epistemologies that problematize the deep
-
seated cultural assumption that learning

is a matter of assembling an
internal model of an external reality (e.g., Bauersfeld, 1988, 1992; Cobb, Yackel, & Wood, 1992; Confrey, 1995;
Steffe & Kieren, 1994; von Glasersfeld, 1995). The two movements are compatible on several counts, including
share
d emphases on Darwinian evolutionary dynamics and a broader use of biology
-
based metaphors to describe
knowing, knowledge, and other phenomena (see Davis & Sumara, in press, and Varela, 1995, for discussions of these
compatibilities). Complexity science an
d contemporary theories of learning, however, are not coterminous. They differ
in terms of both their objects of study and the scopes of their assertions.

With regard to objects of study, contemporary nonrepresentationist theories of knowing and knowledge

tend to be
focused on particular phenomena, as opposed to the broad category of phenomena that are addressed under the
umbrella of complexity. For instance, radical constructivism follows Piaget in the examination of the individual's
construals of the wor
ld (von Glasersfeld, 1995), whereas social constructivist discourses are more likely concerned
with some collective phenomenon, such as the development of a disciplinary subject matter or the emergence of
cultural norms (cf. Ernest, 1991). Complexity scien
ce, on the other hand, is concerned with a range of nested learning
systems, which includes the co
-
implicated processes of individual sense
-
making and collective knowledge
-
generation
(among other layers of activity that extend at least from the subcellular

to the planetary).3 We might say that
complexity science is more a meta
-
discourse, useful for reading across theories that are concerned with different
levels or aspects of complex nested learning systems.

At the same time, complexity science explicitly
rejects any attempt to collapse such phenomena into instances,
variations, or elaborations of the same thing
-
as in, for example, attempts to reconcile radical and social constructivist
accounts of learning. This is not to say that such phenomena as individ
ual sense
-
making and collective knowledge
-
production must be held apart (indeed, they cannot be separated), but that observers must be attentive to the fact
that new possibilities arise and new, emergent rules apply in each case. To extend the example, the

dynamics of
individual sense
-
making and collective knowledge
-
production may be similar and co
-
implicated, but the phenomena
themselves cannot be collapsed into or mapped onto one another
-
or with any other complex phenomenon. It is thus
that a core tenet o
f the discourse is that complex unities must be studied at the levels of their emergence. In brief,
complexity science suggests that discourses concerned with different phenomena (such as radical or social
constructivism
-
or neurology, ecology, or biologica
l evolution) can be simultaneously incommensurate with one
another and appropriate to their particular research foci (see Rorty, 1981; Sfard, 1998).

The assertion that one must not collapse the phenomena that are of interest to neurology, to varied constr
uctivisms,
to ecology, and so on should not be taken to suggest that complexivists regard, say, events of individual cognition and
events of cultural evolution as well defined and distinct. On the contrary, such instances of complexity are understood
in te
rms of forms nested within forms, not in terms of well
-
defined regions. For example, the sorts of distinctions that
are necessary to study one learner's sense
-
making (versus learner
-
in
-
context, or learner
-
and
--

teacher, or learner
-
as
-
cultural
-
agent, etc.)
are understood as arbitrary.4

For these reasons, we join with other mathematics education researchers in the suggestion that the field must move
beyond efforts to bridge the phenomena of individual learning and social learning (e.g., Bowers & Nickerson, 2
001;
Cobb, 1999; Kieren, 2000); that indeed different frameworks are needed. Cobb's (1999) personal research narrative
speaks to the movement in the field from radical constructivism to trying to understand the emergent classroom
community:

In my own case
, for example, my colleagues and I initially intended to analyze students' mathematical reasoning in
purely psychological terms.... We treated social interaction and discourse as a catalyst for otherwise autonomous
mathematical development.... [Incidents].
.. led us to question our sole reliance on an individualistic, psychological
orientation.... Our interest in social norms did not arise as an end in itself. Instead it emerged within the context of
developmental research.... It was within this context that

we subsequently came to view one aspect of our analysis of
classroom norms as inadequate.... We attempted to address this limitation by shifting our focus to normative aspects
of students' activity that are specific to mathematics... [and now there is] th
e need for a theoretical construct that
allows us to talk explicitly about collective mathematical development. (pp. 7
-
9)

Along similar lines, Bowers and Nickerson (2001) comment:

The notion of a collective conceptual orientation can be viewed from a pra
gmatic level as well as a theoretical one.
Most instructors who have taught two sections of the same course can attest to the claim that each class tends to
develop its own unique characteristics or orientation over the course of a semester. (p. 3, emphasi
s added)

We have made similar observations, which have prompted us to turn to complexity science as a means to redescribe

a
classroom collective as a learning system. We suspect, that is, that the tendency to see classroom groupings as
unities with personalities is not simply a matter of figurative reference or anthropomorphism. Rather, for us, this
tendency reveals a capaci
ty to recognize events of emergence.

In research terms, this move toward understanding the collective as a cognizing agent (as opposed to a collection of
cognizing agents) presents some important advantages. Most obviously, the teacher or researcher can a
ctually
observe the "thinking" of this agent
-
that is, the interactions and prompts that trigger new possibilities and insights for
the collective. Examples of such analyses have been provided by Bauersfeld (1992), Cobb (1999), Kieran (2001),
Kieren (2000),

Sfard and Kieran (2001), and Zack and Graves (2001) in their discussions of the developments of
interaction patterns and the emergences of social norms within mathematics classrooms, albeit that these discussions
are not always framed in terms of the coll
ective cognition of a social grouping.

Of particular relevance here is a conceptual shift described by Cobb (1999), away from mathematics as content and
toward emergent terms. As he explains, the "content metaphor entails the notion that mathematics is pl
aced in the
container of curriculum, which then serves as the primary vehicle for making it accessible to students" (p. 31). By
contrast, in his emergent terms, mathematical ideas were "seen to emerge as the collective practices of the classroom
community
evolved" (p. 31). Cobb further highlights that this perspective is compatible with the view of mathematics
as a socially and culturally situated activity.

Such analyses and discussions, however, seem to stop short on the matter of the actual identificatio
ns of classroom
collectives or knowledge domains as learning systems. Social norms, interaction patterns, and mathematical ideas are
understood to emerge and evolve, but these phenomena tend to be described on the level of interacting agents, not
as proper
ties on an emergent unity. In terms of a key principle of complexity science, the notion of self
-
organization
tends not to be taken up in such discussions (Crawford's, 1999, examination of networked mathematical
communities
-
versus hierarchies
-
is a notable
exception). Pragmatically speaking, the addition of the idea of self
-
organization is critical in the move from descriptions of complex activity to efforts to affect the activities of complex
unifies
-
that is, in terms of this discussion, in the shift from d
escriptions of learning to recommendations for teaching.
On this count, complexity science seems to have moved from a focus on description to something more prescriptive
(or, perhaps more appropriately, proscriptive
-
this point is developed below). The curr
ent emphasis now reaches
beyond the question, "What's happening?" to include the question, "How can it be made to happen?" With this shift
toward a pragmatics of transformation, we believe that complexity science has become not just a valuable means to
int
erpret, but a source of practical advice to mathematics teachers.

To elaborate, much of current work in complexity science is concerned with the articulation of qualities that are
necessary to complex systems
-
that is, the conditions that must be met in or
der for agents to come together into
collectives that might supersede the possibilities of those agents. In other words, complexity science has become
more relevant to such deliberate social projects as schooling. In particular, complexity science has high
lighted that, by
attending to particular matters, a teacher can greatly increase the likelihood of complex transcendent possibilities
within the classroom. We address these sorts of practical issues in the next two sections.

CONDITIONS OF COMPLEXITY: THE
EMERGENCE OF A MATHEMATICAL COMMUNITY

We begin this section with a description of the emergence of a coherent social collective among a group of
mathematics teachers that might be characterized as a complex learning system. Through this example, we introd
uce
several conditions that were met in this emergent collective, conditions that we will revisit in the final section as we
report on a more deliberate effort to prompt the emergence of a mathematical community (i.e., a collective learning
system) within
a middle school classroom.

We came across this first case of complexity by accident, when a teacher asked Simmt, the second author of this
article, if she might be able to help with an assignment from a university mathematics course that he was taking. By

way of background, this teacher and many of his colleagues were taking one of six mathematics courses that the
school board requires of all secondary mathematics teachers who had few or no university mathematics courses as
part of their undergraduate degr
ees. Because almost all of the affected teachers have full
-
time positions, they are
limited to taking those few courses that are offered during the evening over the regular term and during compressed
summer programs. Each of these options presents problems
. The courses that are offered during the school year, for
example, often compete with the already heavy demands of teaching. Those that are offered during the summer are
often too compressed to support deep appreciations of the topics at hand.

With these pressures, many teacher
-
students
-
who were oriented by a common purpose, constrained by similar
educational histories, and linked by fax machines and other electronic technologies
-
organized themselves into study
groups. At the start of a recent t
erm, one group of teacher
-
students came together around weekly assignments to
share expertise, to collect ideas, and to distribute labor. An early assignment in a geometry course, in particular,
prompted many dozens of pages of correspondence among the mem
bers of this collective as they discussed patterns,
possible relationships among mathematical concepts, hunches, blind alleys and dead ends, and so on.

As we already mentioned, we became aware of the collective when Simmt was asked for her help with a geo
metry
question by a teacher acquaintance as he searched for a solution to a particular problem. In helping this teacher with
his weekly assignments, she in effect became a part of the collective and as a result began to learn about how this
particular stud
y group had emerged. When the teacher came to her with the question, he also brought records of his
attempts to solve the problem, as well as a number of different partial solutions that his classmates had faxed him.

The teacher shared his working papers
and class notes with us and talked to us about the assignments and how the
teachers worked together to meet the demands of the course. On the basis of these interactions, and an after
-
the fact
analysis of the artifacts, we noticed a few interesting and con
sistent qualities. One is the nature of the routes that the
group tended to take on its way to acceptable solutions. We might describe these paths as meandering or roving. It
appears that at moments, the communications among the collective were unoriented
and poorly connected; at other
moments, they were focused and coherent. At times, important contributions seemed to be bypassed. At other times,
minor points triggered major diversions. The engagements, that is, bore little resemblance to the well structur
ed
arguments that are popularly associated with mathematical thought and progress, tending much more toward the
syncopated structures of conversations (cf. Gordon Calvert, 2001).

As mathematics education researchers, we became interested in what we viewed

as the similarity of these teachers'
interactivities and the historical developments of collectives and movements among mathematicians. Recent popular
accounts of, for example, chaos theory (Gleick, 1987), complexity science (Johnson, 2001; Waldrop, 1992)
, and fuzzy
logic (MacNeill & Freiberger, 1994), offer descriptions of the unruly histories of a few strands of mathematics inquiry.
Such accounts illustrate the point that, at various times and in varied circumstances, intellectual movements can arise
spo
ntaneously and may quickly exceed the possibilities of any of their members
-
at the same time as they provide the
conditions for these members to advance their personal understandings and insights. The knowledge is a property of
the collective.

These sorts

of emergent events within mathematics have several common features, including differentiated
participations of contributing agents and uneven, sometimes erratic evolutions
-
qualities that were also represented in
the teachers' learning collective. Notably,

the members of these clusters were not only willing but found it necessary
to promote and to draw on differentiated areas of expertise, to work with and through several mathematical concepts
at the same time, to accept ambiguity and uncertainty, and to ma
ke use of analogy and metaphor as much as logical
proposition. On the surface, such engagements stand in stark contrast to some of the strategies used to organize
students in mathematics classes. For example, group work in which all students are expected t
o complete the same
sets of practice exercises or problems can become settings for individual work, not necessarily spaces for collective
action.

Significantly, the activities of the teachers' collective and the group
-
work activities in their classrooms a
re both
oriented by tasks that are selected and imposed by an instructor who is external to the group. The key difference,
then, is not who gets to pose the orienting question, but the conditions of engagement with that task
-
a matter that
we address presen
tly. Before delving into this analysis, however, we must note two important qualifications. First, the
suggestion that a new or transcendent unity can emerge from a group of previously disorganized agents is not a claim
about a superorganism, a superior co
nsciousness, or a metaphysical event. It is, rather, a statement about expanded
possibility that comes about when differentiated agents, who operate at a local level with local rules, come together in
manners that complement and amplify existent possibilit
ies while opening up others in the space of joint action. In
other words, we do not mean to erase or to minimize the activities or insights of individuals as we focus on the
activities and insights of the collective. In fact, we argue that an attendance to

the collective actually makes space for,
and supports the development of, individual students' ideas.

A second qualification is that emergent events cannot be caused, but they might be occasioned (Davis, Sumara, &
Kieren, 1996; Simmt & Kieren, 1999). A s
hift in interpretive focus is implicit here, away from what must or should
happen toward what might or could happen. Pragmatically speaking, decisions around planning are more about
setting boundaries and conditions for activity than about predetermining o
utcomes and means
-
proscription rather
than prescription. The proscriptive attitude might be stated as "This is what's forbidden; everything else is allowed,"
which represents a much more open stance than the prescriptive, "This is what's allowed; everythin
g else is
forbidden." Examples of proscriptive situations include most games, social interactions, business dealings, and artistic
endeavors. One example of a proscriptive system is the set of "Thou shalt not..." statements in the Ten
Commandments. These s
orts of examples are framed more by what one must not do than by what one must do. The
proscriptions set the bounds on acceptable behavior while they offer the conditions to expand the sphere of the
possible.

Despite the broad range of phenomena that are
captured under the umbrella of complexity, it appears that they all
have several features in common. In particular, certain necessary but insufficient conditions must be met in order for
systems to arise and maintain their fitness within dynamic contexts
-
t
hat is, to learn. We focus on five such conditions
here (adapted from Bloom, 2000; Casti, 1994; Johnson, 2001; Kelly, 1995; Lewin & Regine, 2000): (a) internal
diversity, (b) redundancy, (c) decentralized control, (d) organized randomness, and (e) neighbor

interactions.

Each of these interdependent conditions is simultaneously a reference to the global properties of a system and to the
local activities of agents within a complex system. Such conditions, of course, are not easily pried apart. Indeed, as
we
have attempted to do so in our considerations of various events, we have found ourselves caught up in tangled
cross
-
references, repetitions, and qualifications.

Internal Diversity

As we examine some of the artifacts from the communications within the tea
cher
-
students' collective, an obvious
quality is that members of the community contributed in very different ways. One person, for instance, interjected
with possible courses of action, but there is little evidence that she followed up on her own suggestio
ns. Another's
major contribution seemed to be testing out others' suggestions and marking rough spots and blind alleys. One group
member had clearly taken on the role of reformulating informally stated contributions in more formal mathematical
terms
-
that i
s, in terms suitable for submission to the course instructor. One teacher drew on his contacts outside the
group to collect suggestions and tips. Still another seemed to make few conceptual contributions but was very active
in ensuring that information was

relayed among participants. Out of such specialized manners of engagement,
acceptable responses to the assigned questions were crafted by the collective. In some cases, it was obvious who was
most responsible for particular solutions. In other cases, howe
ver, the route to a response was not so direct. In one
instance, for example, an eventual half
-
page proof required for a geometry problem was preceded by dozens of pages
of faxed and emailed communications among group members
-
correspondence that was filled

will with speculations,
cautions, questions, and so on, from all participants. The eventual insight belonged to the community, not to any
individual. The intelligent agent, that is, was the collective.

In complexity terms, intelligence is an aspect of le
arning, and is often defined in very similar terms. Johnson (2001),
for example, describes intelligence as the capacity of a system to respond not just appropriately, but innovatively to
novel circumstances. The extent of a system's intelligence is linked
to its range of possible innovations, which in turn
is rooted in the diversity represented among its agents. A system's range of possibilities
-
its intelligence
-
is thus
dependent on, but not necessarily determined by, the variation among and the mutability
of its parts.

This quality of internal diversity has been used to discuss the importance of, for example, the tremendous amount of
unexpressed "junk" DNA in the human genome, the range of vocational competencies among the residents of any
large city, the
biological diversity of the planet, or the array of specialized functions of different brain regions. In each
case, the internal diversity is seen as a source of possible responses to emergent circumstances. For instance, if a
pandemic were to strike human
ity, previously unexpressed DNA might enable the survival of a few people, and hence
the continued existence of the species
-
an intelligent response to a shift in circumstances. A more intelligent response
to the same circumstances (from the perspective of
the species, at least) might arise among the interactions of
researchers with expertise in such diverse domains as immunology, sociology, entomology, and meteorology. A
critical point here is that one cannot specify in advance what sorts of variation will
be necessary for appropriately
intelligent action, hence the need to ensure the presence of diversity.

Our main point here is that the teachers' mathematical collective was an intelligent system, enabled by the ranges of
interest and expertise among the t
eacherstudents. Significantly, the same manner of describing intelligence can be
applied at the level of the cognizing individual. This sort of statement may have a familiar ring to those acquainted
with radical constructivist discourse. For example, von G
lasersfeld's (1995) often
-
quoted assertion that "cognition
serves the subject's organization of the experiential world, not the discovery of an objective reality" (p. 51) might be
read as an alternative statement of intelligence as an agent
-
determined but
situation
-
conditioned phenomenon that is
enabled or disabled by the learner's range of experience and interpretative possibility. This similarity should not be
surprising, given radical constructivism's roots in both Piaget's genetic epistemology (structur
ed around biological
metaphors) and cybernetics (a major tributary to complexity science). (There is an important difference, as already
noted, in the fact that the radical constructivist discourses do not explicitly address the issue of transcendent
emerg
ence.)

Internal diversity is also seen as an important quality with situated learning, social constructivism, critical, cultural,
ecological, and enactivist theories.5 A common thematic to all these discourses is that the viability and adaptability of
wha
tever agent or body is under examination is contingent on that unity's internal diversity. (Even so, in most of
these cases, discussions tend to revolve around evolutionary dynamics and not the self
-
organizing emergence of new
orders of complexity.)

Of co
urse, such diversity is present in every collective organized around mathematical study, no matter how
homogeneously conceived, including school classrooms. But we suspect that relatively few such situations can be
legitimately described as complex unities

in terms of mathematical engagements. By contrast, most classrooms may
be appropriately understood as rich examples of complex unities when one considers, for instance, the continuous
projects of negotiating positions within the social corpus. Our point i
s that complexity is always and already present in
social groupings. Thrust together into classrooms or other settings, humans self
-
organize
-
that is, codes of acceptable
activity, group hierarchies, and so on emerge (see Rich Harris, 1998, for a review of
research into this phenomenon
among adolescents). Our concern thus becomes how the mathematics teacher might occasion the emergence of a
complex collective whose interactions and products are mathematical.

On this point, we are prompted to offer a critiqu
e of those group
-
based classroom strategies that are organized
around formal roles. One cannot impose diversity from the top down by naming one person a facilitator, another a
recorder, and so on. Diversity cannot be assigned or legislated, it must be assu
med
-
and it must be flexible. A further
problem with these formal group
-
based cooperative strategies is that they are often explicitly consensus
-
oriented,
rather than being framed by the fact that the collective might open up more diverse and sometimes nove
l possibilities.
In other words, although such structures can serve to educate individuals in matters of particular social roles, in
complexity terms, they may be no more effective in prompting the emergence of transcendent possibilities than
traditional,
teacher
-
centered classroom structures.

Redundancy

Redundancy is used to refer to duplications and excesses of the sorts of features that are necessary to particular
events. For a collective to build a house, for instance, it is not necessary that everyon
e know how to use a hammer.
Such redundancy among agents, however, will likely be very useful. Unfortunately, the term tends to be associated
with aspects that are unnecessary or superfluous and that contribute to inefficiency. We do not mean to invoke the
se
qualities here, as we frame our discussion in terms of adequate or appropriate activity, not optimality. The sorts of
redundancies we discuss are utterly necessary to the emergence of an intelligent collective.

When we first became aware of the emergen
ce of the teachers' collective, our attentions were focused on the
differences among participants. As we have learned more about the group, it has become clear that an important
quality underlying these events of emergence is the complement of variation: s
imilarity. In a phrase, the members of
the community were much more the same than different
-
culturally, professionally, educationally, and so on.
Sameness among agents
-
in background, purpose, and so on
-
is essential in triggering a transition from a collect
ion of
me's to a collective of us.

In more technical terms, a system's capacity to maintain coherence is tied to the redundancy expressed among its
agents. As demonstrated by the ways that some people's brains recover from strokes, some companies cope wit
h
employee disloyalty, and some ecosystems adapt to the loss or introduction of new species
-
and, for that matter, the
way that the teachers' collective dealt with being stumped
--

redundancy among agents is an important quality for
coping with stress, sudde
n injury, or other impairments. Redundancy thus plays two key roles. First, it enables
interactions among agents. Second, when necessary, it makes it possible for agents to compensate for others' failings.
It is in these senses that redundancy is a complem
ent to diversity. Whereas internal diversity is more outward
-
oriented in that it enables novel actions and possibilities in response to contextual dynamics, internal redundancy is
more inward
-
oriented, enabling the moment
-
to
-
moment interactivity of the age
nts.

The importance of redundancy in educational settings was underscored for use during a recent small
-
scale teaching
experiment conducted by Davis, the first author of this article. Five adolescent girls, each identified by a school district
as having learnin
g disabilities in mathematics, were engaged in an activity that had been designed to capitalize on
their particular interests and strengths. True to the pedagogical intention, each of the learners went off in a different
direction with the task. However, w
hen they were invited to present their insights to one another, the interactivity
came to a grinding halt. They simply could not effectively communicate with one another about what they had done.
The creativity of the collective was stifled
-
not in spite of

the internal diversity, but because of it. Inadequate attention
was given to such redundant systemic qualities as shared vocabularies, experiences, and expectations.

The upshot, perhaps obvious, is that educators who are interested in interactivity in th
e complex collective must
attend to the common ground of participants. In the mathematics classroom, such redundancy involves more than
shared vocabularies, symbol systems, and resources
-
all of which were present in the teachers' collective. It also
appear
s that certain commonalities of experience, expectation, and purpose are important. Put differently, it is not
enough that participants speak the same language, they must also be languaging the same or compatible
experiences. The interplay of such commonal
ity and internal diversity define what Lewin and Regine (2000) call the
"zone of creative adaptability" (p. 28), a notion that is reminiscent of Vygotsky's (1978) "zone of proximal
development." Both ideas are references to immediate possibilities for co
-
a
ctivity, as conditioned by a certain
redundancy among agents.

The balance of redundancy and diversity among agents
-
or, in systemic terms, stability and creativity
-
is not strictly
dictated by the system itself. In fact, it is more a function of the stabili
ty of the context. Minimal redundancy (i.e.,
high specialization) among agents is most valuable in relatively fixed settings, but it contributes to a loss of
robustness in the face of sudden change. (Wide
-
scale extinctions of species are often linked to th
eir overspecialization
and consequent inability to adapt to new conditions.) Maximum redundancy (i.e., highly interchangeable agents) is
more appropriate to more volatile situations. Increased redundancy does engender decreased diversity, however, and
take
n to an extreme, a reduction in internal diversity can diminish a system's capacity to adapt (see Cohen & Stewart,
1994).

The qualities of redundancy and diversity might be used to characterize an important difference between various
approaches to the tea
ching of mathematics. Traditional, whole
-
class strategies in which all students are provided the
same explanations and are expected to complete the same practice exercises in more or less the same way seem to
be focused on the establishment of redundancy.
A goal in such settings is to ensure that everyone achieves a certain
minimal competence within a prescribed set of topics and procedures. By contrast, more student
-
centered classroom
approaches seem to place priority on individual diversity. However, in b
oth cases, pedagogy is oriented by the
assumption that the individual is the locus of learning
-
an attention that is problematized by complexity science. That
is, popular enactments of both traditional and contemporary teaching might ignore the complex poss
ibilities of
collective engagements as they focus on the qualities of single subjects.

Decentralized Control

When we compare the interactive structures of the teachers' collective to structures of classrooms, one point of
contrast seems to stand out: No
one seemed to be in charge of the teachers' community. It organized itself
-
or, as
Varela writes (1999): The whole does behave as a unit and as if there were a coordinating agent present at its
center.... [A coherent global pattern] emerges from the activit
y of simple local components, which seems to be
centrally located, but is nowhere to be found, and yet is essential as a level of interaction for the behavior of the
whole. (p. 53)

Johnson (2001) speculates that the tendency of observers to suspect the existence of a coordinating agent may be
rooted in cultural habits of interpreting phenomena in cause
-
effect rather than complex emergent terms. As it turns
out, across studies of comp
lex learning systems, the former seems to be the less powerful of the two when it comes
to efforts to prompt or to interpret intelligent behavior, especially with regard to social and cultural collectives (see
Buchanan, 2000; Johnson, 2001; Varela, 1987).
To our observations, this is also true of the teachers' collective.
Simply put, there was no overseer, no director. The collective emerged and sustained itself through shared projects,
not through planning or other deliberate management strategies.

Anothe
r prominent example of this point is provided by current artificial intelligence research, where efforts to create
massive central processors and core databases have given way to projects developed around the parallel linking of
small, specialized subsyste
ms that are capable of learning (i.e., of adapting and self
-
organizing). Although we are still
far from achieving the sorts of super intelligence projected by scientists and science fiction writers of the 1950s,
considerably more progress has been made in
the past decade with this distributed, self
-
organizational approach than
was made in the previous half century with the massive central processor approach (Brooks, 2002; Clark, 1997).
Many other examples could be cited. In fact, Kelly (1995) develops his h
istory of complexity science around the slow
emergence of the realization that one must relinquish a certain amount of control if complexity is going to happen.

In terms of schooling, the notion of decentralized control should be interpreted neither as a
condemnation of the
teacher
-
centered classroom nor an endorsement of the student
-
centered classroom. Rather, it compels us to question
an assumption that underlies both teacher
-
centered and learner
-
centered arguments
-
namely, that the locus of
learning is t
he individual. Learning occurs on other levels as well, and to appreciate this point one must be clear on
the nature of the complex unities that might be desired in the mathematics classroom. For us, these complex unities
are shared mathematical ideas, ins
ights, concepts, and understandings that collectively constitute the body of
mathematical knowledge in the classroom. This point is critical to our discussion of the potential contributions of
complexity science to mathematics teaching.

This suggestion pr
ompts our educational attentions away from matters of teacher performance and into the
consensual domain of authority. Within a complex system, appropriate action can only be conditioned by external
authorities, not imposed. The system itself "decides" wha
t is and is not acceptable. Pragmatically speaking, with
regard to the emergence of shared mathematical insight, this means that the teacher cannot position herself or
himself (or a textbook or a curriculum document) as the final authority on matters of ap
propriateness or correctness.
Structures can and should be in place to allow students to participate in these decisions. For us, then, a key element
in effective teaching is not maintaining control over ideas and correctness, but the capacity to disperse c
ontrol.

This is not a new issue among mathematics educators. Several researchers and theorists have commented on the
matter, particularly those who work from critical or liberatory stances (e.g., Powell & Frankenstein, 1997; Skovsmose,
1994; Walkerdine, 1
988). Rejecting such dichotomies as child versus society, teacher versus learner, and knowledge
versus knower, these theorists have attempted to offer alternative interpretations of power and authority. For
example, instead of seeing these phenomena as ext
ernal and monological impositions, power and authority might be
redescribed in terms of discourses (Walkerdine, 1988); formatting (Skovsmose, 1994); or shared action, consensual
domains, cospecifications, and emergent choreographies (Davis, Sumara, & Kiere
n, 1996). Such shifts in vocabulary
are intended to problematize the terms of popular debate over the relative merits of teacher
-

and child
-
centered
instruction. Although presented as opposites, these emphases rely on similar assumptions about the organiza
tion of
social systems. To elaborate a point already mentioned, the individual tends to be seen as the locus of learning and
the fundamental particle of social action. Once the learner is cast in these terms, it follows that the classroom must be
structure
d either around the fiction of the "normal child" (see Walkerdine, 1988) or around the fiction of the isolated
human subject. In the extreme enactment of the first case, one must deliberately ignore diversity as one treats the
class as a teacher
-
led homoge
nized mass. In the extreme version of the second case, one must fragment the
classroom community into an accidental collection of fully independent agents whose needs must be addressed in a
student
-
centered manner.

We doubt that anyone regards mathematics

teaching in terms of either extreme. By the same token, it often seems
that many teachers regard their activity as being caught between these two poles. That this dichotomization does not
span the full range of educative possibility, however, is evident i
n the example of the emergence of the teachers'
mathematical collective. Notions of teacher
-
centeredness or student
-
centeredness are not very useful for making
sense of these collectives
--

in large part because the phenomenon at the center of each collecti
ve is not a teacher or a
student, but the collective phenomenon of a shared insight, similar to what Bowers and Nickerson (2001) call a
collective conceptual orientation. Shared is a key notion here, and we use it as a synonym for distributed
-
a move that
w
e recognize to be at odds with current usage in the mathematics education literature (and especially with the notion
of "taken
-
as
-
shared," see for example Cobb & Bauersfeld, 1995), but that we feel is consistent with popular usage. To
elaborate, just as re
sponsibilities, meals, opportunities, traits, and other objects or identifications might be shared or
distributed, so might mathematical insights. But the suggestion that ideas can be shared only makes sense if the
observer allows knowing and knowledge to
be spread across agents' actions in collective contexts.

Once again, if one turns from a focus on mathematical knowledge to the cultural norms and ongoing projects of social
positioning, the fact that a classroom community is already a complex self
-
organi
zing adaptive unity can be more
readily appreciated. These sorts of social activities are, for the most part, allowed to unfold on their own in the
classroom, subject only to occasional interventions by teachers and others who might seek to affect qualitie
s of
interactions among students. That is, such projects are shared or distributed among the participants. This is not often
the case for the mathematics, however. Far from being seen as a matter of decentralized control, mathematics is
more often regarded

as the embodiment of regimented, top
-
down, knower
-
independent knowledge.

Nevertheless, as demonstrated within the teachers' collective, it is evident that control or authority might be
redistributed
-
without getting caught in the trap of relativism. That
is, the distribution or sharing of the authority
across the classroom community need not be a matter of "anything goes." On the contrary, with the emergence of
any complex collective, standards of acceptable response and acceptable activity
-
of rightness an
d wrongness
-
inevitably arise.

Organized Randomness

At first hearing, organized randomness may sound like an oxymoron. It is, however, a notion that is critical to the
potential of complex emergence. It is a structural condition that helps to determine th
e balance between redundancy
and diversity among agents. Complex systems are rule
-
bound, but those rules determine only the boundaries of
activity, not the limits of possibility. The teachers' collective, for example, was organized by some rather severe
co
nstraints, including those that arose from course requirements, time restraints, available technologies, and
established mathematics. Yet, as limiting as these conditions were, they seem to have defined a territory rich in
possibility.

In effect, the idea

of organized randomness is a reiteration of the proscription versus the prescription notion. To
elaborate, the prescription
-
proscription distinction might be seen to map onto the extremes of current popular debate.
At one end of the spectrum, the caricatu
re of a strictly rote
-
based, examination
-
driven, teachercentered, traditional
classroom seems the epitome of a prescriptive situation because the focus seems to be on specification of what is
expected, and by consequence prohibiting everything else. On the

other end, an exploration
-
based, experience
-
oriented, student
-
centered, classroom appears to be proscriptive because there seems to be a commitment to open
exploration.

Such a mapping, however, is inappropriate. A difficulty arises in the extreme interpr
etation of the more exploratory
classroom in terms of an "anything goes" abandonment of rigid structures and imposed expectations. A proscriptive
situation does not represent an abandonment of constraints, but a shift in thinking about the sorts of constra
int that
are necessary for generative activity. As Johnson (2001) explains, complex emergence occurs in

rule
-
governed systems: their capacity for learning and growth and experimentation derives from their adherence to
low level rules.... Emergent behavior
s, like games, are all about living within boundaries defined by the rules, but also
using that space to create something greater than the sum of its parts. (p. 181) With regard to activities for the
mathematics classroom, Davis, Sumara, and Luce
--

Kapler
(2000) have used the term liberating constraints to draw a
distinction between tasks that are proscriptive and those that are prescriptive. Examples of tasks that fall into the
latter category range from those that are too narrow to invite much variety of
interpretation (e.g., "Turn to the
Multiplication of Fractions page and do the questions in the first column.") and those that are too open to foster
focused interpretations (e.g., "Write about the multiplication of fractions."). One is too organized and r
edundancy
-
oriented, the other is too unfocused and diversity
-
oriented.

The structures that define complex systems, in contrast, maintain a delicate balance between sufficient organization
to orient agents' actions and sufficient randomness to allow for fl
exible and varied response. Such situations are
matters of neither "everyone does the same thing" nor "everyone does their own thing" but of everyone participating
in a joint project. In our experience, minor modifications are sometimes all that is needed
to transform tasks that are
either too narrow or too open into liberating constraints. By way of example, prompts such as "Write down five things
that you know about the multiplication of fractions" (versus the sorts of prompts mentioned in the previous
pa
ragraph) have sponsored rich discussions in our middle school mathematics classes.

In our experience, it is not often that we manage to identify appropriate liberating constraints when planning for
teaching
-
that is, outside the classroom, it is impossible

to decide whether or not an activity will be a liberating
constraint. Most often, these parameters of activity must be refined and negotiated while teaching. Conditions
sometimes must be relaxed, other times made more rigid, and still other times abandone
d in favor of emergent
student interests. An elaborated example of such negotiations is presented by Kieren, Davis, and Mason (1996).

Neighbor Interactions

Agents within a complex system must be able to affect one another's activities. On the surface, it

might seem almost
unnecessary to make this point. What is not so obvious, however, is what might constitute a neighbor in the context
of a mathematical community.

In our ongoing efforts to interpret and to prompt complex activity around mathematics, it w
as only recently that we
realized that the "neighbors" in mathematical communities are not physical bodies or social groupings. That is,
personal and group interactions may not be as vital or useful as is commonly assumed. Rather, in mathematics, these
nei
ghbors that must "bump" against one another are ideas, hunches, queries, and other manners of representation.'
This is nicely illustrated in the classroom transcripts that Cobb (1999, pp. 14
-
15) uses to show how a mathematical
practice is developed. It is
the students' ideas, metaphors, and words that are made to interact with one another.
Rotman (2000) makes the point this way in his description of mathematics:

Mathematics is an activity, a practice. If one observes its participants, then it would 'be per
verse not to infer that for
large stretches of time they are engaged in a process of communicating with themselves and one another; an
inference prompted by the constant presence of standardly presented formal written texts (notes, textbooks,
blackboard le
ctures, articles, digests, reviews, and the like) being read, written, and exchanged, and of all informal
signifying activities that occur when they talk, gesticulate, expound, make guesses, disagree, draw pictures, and so
on. (pp. 7
-
8)

This point was esp
ecially obvious within the teachers' collective, where mathematical insights arose in the
juxtaposition of electronically transmitted representations. What was important was not the opportunity for direct
interaction (which was actually very limited), but
the possibility of making ideas collide with one another.

Within the context of the mathematics classroom, an implication here is that group work, pod seating, and class
projects may be no more effective at occasioning complex interactivity than tradition
al straight rows
-
if the focus is not
on the display and interpretation of diverse, emergent ideas. Concepts and understandings must be made to stumble
across one another. Without these neighboring interactions, the mathematics classroom cannot become a
mat
hematics community.

In effect, this point is similar to the assertion that the knitting of logical propositions is not the principal means by
which mathematical knowledge is generated. Rather, as Varela (1999) suggests, "the proper units of knowledge are
primarily concrete, embodied, incorporated, lived" (p. 7, emphasis in original)
-
and, hence, these embodied units of
knowledge must somehow be made to interact with one another within those spaces dedicated to collective
knowledge. Lakoff and Nunez (2000) f
urther develop this point through investigations of how mathematical
possibilities have been enabled through the blending of body
-
based images and metaphors.

A caveat here is that not only must there be neighbor interactions, there must be a sufficient de
nsity of such
interactions. It would seem that conceptual possibilities are rarely crowded enough in mathematics classrooms to give
rise to rich interpretive moments
-
a matter to which we now turn.

USING PRINCIPLES OF COMPLEXITY TO ORGANIZE A MATHEMATICS CLASSROOM

Our interest in complexity science extends beyond its utility for describing events such as individual and collective
learning, and in this section we attempt to make sense of what complex
ity science might have to offer by way of
"advice to teachers." This discussion is rooted in the conviction that complexity science has some direct applications to
classroom situations
-
although we qualify this claim by saying that its implicit proscriptive

sensibility is not always well
fitted to the predominantly prescriptive project of schooling.

We proceed here with a description of a classroom event that occurred several months into a yearlong teaching
experiment undertaken by Simmt in a seventhgrade c
lassroom. To fully appreciate her intentions and actions in this
event, it is helpful to know that her efforts were not principally focused on the development of individual
understandings or technical competencies, but on the emergence of conceptual and in
terpretive possibilities that
might not have been previously considered by any member of the group, including the teacher. The development of
individual insight was still seen as important, but as dependent on
-
and, hence, secondary to
-
establishing the
cond
itions necessary to an adaptive, self
-
organizing mathematical community.

We qualify this account by emphasizing at the outset that it is not offered as an event to be replicated or to be held up
as a model. It is offered as illustration, not exemplar. It
also bears mention that not all of the mathematics classes in
this project are as generative as the one in this account. On the contrary, the classes seem to oscillate between ones
that revolve around provocative mathematical engagements and ones in which
the students and teacher "do school."
The intention, then, is not to suggest some sort of reliable route to complex collective mathematical engagements.
Rather, we present and interpret an instance in which deliberate and conscious efforts to structure a l
earning activity
around the principles presented above seemed to occasion some provocative possibilities in a mathematics classroom.

This episode unfolded near the end of November in the school year, 2 weeks into a study of integers. The initial focus
of the unit was to foreground the sorts of applications and images that students brought to the topic
-
including number
lines, thermometer
s, profit and loss calculations, and altitude, as well as more primitive (and more implicit)
set/container schemas (i.e., use of metaphors of collecting and ordering objects to understand processes of counting
and other binary operations; see Lakoff & Nune
z, 2000). An effort was made to elaborate and link these sorts of
images through the introduction of two
-
sided chips, and these chips were made available to the students throughout
the unit.

In the session under discussion, students were asked to consider

the statement, "3 x
-
4 =?" They were paired off by
the teacher and each pair was given 10 minutes to agree on a product and to "Show how you know"
-
that is, to
prepare an explanation on chart paper for presentation to their classmates.

In our analysis, st
udents' responses fell into five different categories (see Sfard, 2001, for a complementary analysis
of similar sorts of prompts): * grouping of objects, whereby 3 x
-
4 =
-
12 was described and explained as three groups
of four negative chips;

* number line

movement, where 3 x
-
4 was interpreted as three hops of length four, starting from zero, moving in a
negative direction (which was not always presented as leftward or downward);

* repeated addition, where 3 x
-
4 was rewritten as
-
4 +
-
4 +
-
4, and the alr
eady established rules of integer addition
invoked;

* appeal to external authority, of either a previously learned rule or the answer generated with a calculator;

* extending a rule, in which one pair determined the magnitude of the product by ignoring t
he signs (i.e., 3 x 4 = 12),
and assigned the product a negative value "because the minus was on the bigger number."

A feature of these presentations, which took up the rest of the 45
-
minute block of class time, was that the students
closely attended to o
ne another's responses and explanations, without prompting or nagging from the teacher. Their
engagements with one another's work were notable around two explanations in particular: the case of extending a
rule and one version of repeated addition. Unfortu
nately, the case of extending the rule came too late in the session,
and there was no opportunity that day to pursue the challenge, raised by the teacher, that whereas this strategy gave
the same response as the others to 3 x
-
4, it would not for 4 x
-
3.

In the other noted case, one pair had written on their poster that "0 +
-
4 +
-
4 +
-
4 =
-
12." The zero had clearly been
squeezed into the left margin of the page, apparently added after the rest of the statement had been written. Because
this statement diff
ered from the other cases of repeated addition, the teacher asked about the zero. Pointing to a
classmate, the student responded, "He said we have to start somewhere." The "he" was a student who used a number
line in his explanation. In our interpretation,

the concept of multiplication as repeated addition was blended with the
concept of multiplication as movement along a number line.

The pair's response about their inclusion of a zero in the sum prompted some rumbling in the room, through which
another st
udent, Tim (a pseudonym), announced that "Actually it doesn't matter where you start from. It will be the
same." As neither the original explanation nor Tim's elaboration had yet been explicitly linked to the number line
image, the teacher asked Tim to say

more. In response, he offered, "Well, negative four, three times, is always
negative twelve, even if you start somewhere else on the number line." We focus our interpretations here on just a
few elements in this already extremely partial account
-
namely, o
n the case of the "extending a rule," on Tim's insight,
and on what these might have to say about the occasion of this classroom collective.

Internal Diversity

It must be reemphasized that this episode unfolded nearly 3 months into the school year and wa
s preceded by many
similar activities. Therefore, there was ample opportunity for appropriate norms of classroom engagement to have
been established. Prominent among these was the importance of diverse contribution.

With regard to the task that framed thi
s session, for instance, 3 x
-
4 = ? it was not viewed by most as a challenge to
close in on a rule or a definitive explanation, but as a place to articulate and compare different ideas. In fact, even the
pairs that seemed to frame their activities in terms

of rule
-
generation rather than reasoning, were very much a part of
this "diverse possibility" game. A member of the pair that extended a rule was pleased with the fact that their
response differed from all of the others. We interpret her expression of ple
asure as an indication that members of this
group did appreciate that divergent thought was an important aspect of mathematical engagements in this setting,
although the students seemed less aware that their response was lacking in terms of the sorts of ra
tional explanations
that were expected. The importance of diverse possibility was perhaps most evident in Tim's contribution, around the
fact that 3 x
-
4 is "always
-
12, even if you start somewhere else on the number line." In formal terms, we interpret
Ti
m's thinking to be consistent with a vector
-
based interpretation of multiplication, where direction and magnitude
matter but location does not.

We would highlight two points here. First, we regard Tim's insight as a mathematically significant one, but not

because it had any great impact on the individuals' understandings. (In fact, it did not.) Its importance lies in the fact
that, for the second time in the class, it was an instance of conceptual blending
-
or, perhaps more appropriately, a
reblending. (The

first blend was formally represented when the one group inserted a zero at the start of the addition
statement.) That is, although it was not made explicit, Tim "invented" a vector
-
like interpretation of multiplication that
was added to the already known
interpretations of set grouping, repeated addition, and number line hopping. As
Lakoff and Nunez (2000) demonstrate, these sorts of metaphoric elaborations have often been linked to moments of
significant advances in the history of mathematics.

Second, we

would highlight that Tim's thought was clearly hinged to ideas that had already been represented. His
comment that "negative four, three times, is always negative twelve, even if you start somewhere else on the number
line" accommodates the grouping of ob
jects, the number line movement, and the repeated addition schemas. The
comment also points to a new layer of meaning, one that frees the construct of multiplication from the assumption of
a starting place of zero. This more abstract, potentially more powe
rful possibility arose in the juxtaposition of diverse
interpretations
-
the bumping together of ideas. As such, the insight was a property of the collective, not of any specific
agent of the collective. It was a case of distributed intelligence. This new bi
t of knowledge
-
this emergence potential to
action
-
was rooted in the diversity represented among the students.

Redundancy

The redundancies that underlie a system's robustness can be difficult to interpret because they tend to serve as the
ground of activi
ty, not the figure. Aspects such as shared vocabularies, compatible experiences, and assumed social
roles are not often the subjects of conscious attention. They are, rather, the transparent backdrop that gives shape to
and enables activity in the mathemat
ics classroom.

Arguably, then, this part of our analysis should be the lengthiest
-
and it would be if we were to attempt a more full
-
bodied description of the setting. In the months preceding the reported event, Simmt had worked hard to establish
expectati
ons for classroom interactions, standards of acceptable explanation, and appreciations of divergent thought.
As well, she had placed heavy pedagogical emphases on the sorts of experiences, images, tools, and metaphors that
infuse interpretations of various

concepts.

The intention of these efforts was to establish a necessary redundancy among participants
-
that is, to ensure that
common understandings among agents were adequate to enable the emergence of collective understandings. Part of
this redundancy is
related to the sociomathematical norms of the setting (Cobb, 1999), but not all. Other forms of
redundancy included common artifacts, vocabulary, procedures, and mastery of facts deemed basic. Such
groundwork, although not the central theme of this analysi
s, is arguably the main work of the mathematics teacher,
as Cobb (1999) and Lampert (1990) have developed in more detail.

Decentralized Control

Was this lesson teacher
-
centered or learner
-
centered? As might be gleaned from the discussion thus far, we reg
ard
this particular dichotomy as a distraction. It reduces the complexity of the classroom as it prompts attentions toward
particular individuals rather than onto what we interpret to be shared and emergence understandings.

The knowledge in this setting,
as demonstrated in the range of interpretations, did not reside in any person in
particular. Nor was the authority exclusively lodged in one character, one argument, or one resource. The control of
the knowledge, that is, was decentralized, and that decent
ralization was necessary for some of the ideas to arise. This
was especially evident as the teacher
-
researcher observed this event. She had anticipated that the class would
progress in a certain way. However, the "3 x
-
4 = ?" prompt triggered possibilities

that she could never have
consciously or deliberately brought to her planning and teaching. In this setting, integer multiplication was an event of
complex knowing that was authored in the interactions of people, ideas, materials, displays, and other rela
tional
aspects
-
as opposed to being authorized by the teacher or a textbook.

This is not to say, however, that the teacher did not exercise a particular authority. Her role was more than facilitator.
In particular, the collective rules of engagement were v
ery much embodied in the manner in which she set the task
and in the accumulated history of her responses within this collective. And while such instruction could certainly be
analyzed in terms of power and authority, an interpretation based on such monolo
gical metaphors would have to
ignore the complex, co
-
specifying activities during the session. It would also have to focus on the activities of
individuals, rather than on the shared knowledge of the collective.

This distinction is similar to one develope
d by Bauersfeld (1992), as he contrasts the discovery approach to an
integrated (culture) approach. The former tends to be structured around prepared materials, preplanned steps, and
prestated learning outcomes. In the latter, "students are expected to sea
rch for patterns, to assume regularities, and
to relate developing or contrasting ideas, as well as to give reasons and arguments for the issue under discussion"
(1992, p. 23). So framed, the emergent interpretations and the ultimate learning outcomes cann
ot be prespecified
-
or,
in our terms, control is distributed.

Organized Randomness

Much in contrast to the manner in which the topic of multiplication of integers is most frequently presented, this
lesson involved just one item. On the surface, the decision to structure an entire class around the matter of 3 x
-
4
might seem a rather seve
re constraint. But, of course, the focus of the session was not really the rules of integer
multiplication, but the concept. The lesson was part of an ongoing investigation into the sorts of images and
metaphors that enframe discussions and applications of

integers and binary operations
-
and how those conceptual
tools might be worked to give insight into the topic at hand. (At least, this is how the teacher understood her activity.
It is not at all clear that any of the students regarded their engagements in

such terms
-
a point that underscores our
assertion that the authority or control in this classroom was distributed. Had the teacher insisted that learners
interpret their understandings in such terms, the session might have been more about uncovering corre
ct answers
and less about generating diverse possibilities.)

The imposed constraint of a single item for interpretation, coupled to requirements to provide reasoned statements
and to display their explanations, helped to organize the randomness. More desc
riptively, structures of organization
were operating on both tacit and explicit levels. As already suggested, on the tacit level, norms of appropriate activity
had already been established, and these served to frame student activities and interactivities.
Implicit rules of the
activity, for example, included the necessity of a convincing explanation and the requirement of attending to others.
On the more explicit level, specific to this session was the strategy of constructing posters, to present thoughts,
and
using the posters to organize ideas.

A key point is that, with regard to the notion of organized randomness, the challenge is not to prompt randomness. It
will always and already be present by virtue of the complex nature of human cognition. The challenge for the teacher
is to find means of u
tilizing the randomness that is rooted in individual interpretation within the collective space of the
classroom. The posters idea was developed in specific response to this matter, and maintained the balance between
sufficient organization to orient stude
nts' actions and sufficient randomness to allow for flexible and varied response.
Phrased differently, the structure of the activity served to mediate individual insights and the collective project of
learning mathematics.

For us, the deliberate attention

given to the mediation of individual and collective projects represents a rejection of
that approach to mathematics teaching that is structured around the sequenced presentation of isolated concepts. For
instance, introductions to integers are often handl
ed as organized movements through preparsed subconcepts in
which, for example, multiplication is assigned particular interpretations according to the topics that preceded it. By
contrast, in the lesson presented, the topic of integer multiplication was sim
ultaneously a discussion of other
operations, personal experiences, and possible applications. Although we believe it important for teachers to have well
developed understandings of how mathematical concepts fit together, complexity science prompts us to a
ssert that
such analytic understandings must be part of broader appreciations of concepts' experiential requisites, metaphoric
roots, and so on. Along with Ball and Bass (in press), we suspect that mathematics teachers need to be aware of the
sorts of idea
s and images that have been knitted together into the concepts and the abstractions that they are
required to teach (see also Sfard, 2001).

Neighbor Interactions

A driving assumption in this article has been that the collective dynamics involved in the p
roduction of knowledge
emerge not simply amid the juxtaposition of bodies, but amid the juxtaposition of interpretive possibilities
-
hence the
paired interactions, the poster presentations, and the surrounding discussions. Such was the condition that enable
d
the emergence of an idea not previously considered by anyone in the room, at least not in the terms presented: Tim's
merging of the location
-
specific suggestion that "we have to start somewhere" (as one multiplies by hopping along a
number line) with the

location
-
independent idea of multiplication as repeated addition. The consequent blending gave
rise to the vector
-
like notion that multiplication is a directed movement of specified magnitude.

This idea may never be taken much further in these students'
Grade 7 mathematics course. Use or elaboration of the
idea is not the point here, though. The critical issue is that the possibility of conceptual blends
-
that is, of melding
ideas and images with others
-
was present and explicit. This is important not just
because it presents the possibility of
surprising combinations, but because it foregrounds the place of interpretation. (Interpretation here is understood in
terms of the deliberate reading of two or more ideas against one another. So framed, an interpreta
tion is a sort of
emergent form.)

Pedagogically speaking, for us the idea of neighbor interactions prompts careful consideration of strategies for
representation of concepts and understandings. Our preliminary comparisons of the effects of verbalized and
written
representations, based on analyses of videotaped interactions and varied classroom products, suggest that physical
artifacts are a vital element in this regard. Not only do artifacts facilitate the juxtaposition of ideas, they help to
preserve poss
ibilities whose relevance may not be immediately apparent. In both examples of the teachers' collective
and the integers lesson, written representations made it possible to revisit, to compare, and to elaborate previous
insights. Further to this point, phy
sical artifacts can serve as a record of emergent ideas, means to offload some of the
demands on individual consciousness and memory (see Norretranders, 1998).

In the context of the reported episode, a benefit of the sorts of artifact
-
mediated neighbor in
teractions that were
present in this lesson is that, among such interactions, instances of "playing the wrong game" became very apparent.
In this session, it was clear (to the teacher at least) that three pairs
-
the ones who saw the task in terms of generat
ion
of rules rather than in terms of interpretation of concepts
-
had misunderstood what was required in this setting for an
argument to be accepted as mathematical. We suspect that such misunderstandings are commonplace in mathematics
classrooms, but opport
unities to problematize this rule
-
oriented sensibility are considerably less common.

COMPLEXITY SCIENCE AND MATHEMATICS TEACHING

Twenty years ago, when we were both beginning our mathematics teaching careers, it was not uncommon to hear
colleagues draw a

distinction between teaching mathematics and teaching children
-
as though, somehow, an emphasis
on one entailed a certain ignorance of the other. Neither of us had an adequate response at the time. We believe that
we do now, however. The manner in which co
mplexity science casts both children and mathematics
-
along with
classroom communities, conversational interactions, and similar phenomena
-
as adaptive and self
-
organizing learning
forms prompts us to think in terms of unities nested within unities, rather t
han in terms of discrete entities. So
framed, the imagined tension between learners' bodies and bodies of knowledge, along with the need to build bridges
between the two, vanishes. To teach children well, we argue, we must conceive of our activity in terms

of an active
participation in the body of mathematics knowledge by creating the conditions for the emergence of bottom
-
up, locally
controlled, collective learning systems.

Phrased in somewhat different terms, for us, the main attraction of complexity sci
ence is that it provides means of
reading across the concerns and contributions of radical, social, and critical constructivist discourses. At the same time
it speaks to the multileveled, deliberate, and practical concerns of formal education (Davis & Suma
ra, in press). In
particular, it prompts us to suggest that, in terms of the range of complex forms, the teacher's main attentions should
perhaps be focused on the establishment of a classroom collective
-
that is, on ensuring that conditions are met for the

possibility of a mathematical community. Such an emphasis is not meant to displace concern for individual
understanding. The suggestion, rather, is that the individual learner's mathematical understandings might be better
supported
-
not compromised
-
if the
teacher pays more attention to the grander learning system (Bauersfeld, 1992;
Bowers & Nickerson, 2001; Burton, 1999; Cobb, 1999; Crawford, 1999).

To that end, the five conditions discussed in this article are not merely useful tools for after
-
the
-
fact an
alyses of
classroom events. They can also be put to instrumental use to guide the structuring of teachers' engagements with
learners. For instance, the notion of internal diversity points to the need to develop in
-
class activities that can be
adapted by le
arners to their particular understandings and interpretations. Redundancy highlights the importance of
shared experiences and established standards of engagement. Decentralized control, among other things, points to a
reconsideration of lesson planning, le
ss in terms of prescriptions and itineraries and more as thought experiments.
Organized randomness is in some ways an elaboration of this point. It foregrounds the importance of setting
boundaries and rules
--

constraints that operate proscriptively rather
than prescriptively. Finally, neighbor interactions
prompt attention to the manners in which ideas might be represented and juxtaposed. Again, that such conditions are
met in a mathematics classroom is no assurance that complex possibilities will arise. Ho
wever, a neglect of such
conditions will provide a reasonable assurance that collective activity will never exceed the collection of individual
insights.

At the moment, we are engaged in different efforts to assess the utility of these ideas in the teachi
ng of mathematics
-
projects that involve considerable collaborative work with preservice and practicing teachers in various settings.
Across these projects, a consistent and somewhat resilient theme has arisen. As conditions for complex emergence
are descri
bed, illustrations provided, and engagements prompted, inevitably discussions turn to the phenomenon of
the teachable moment
-
a cohering of many bodies in an instance of shared purpose and insight.

It may be that what we are talking about in this article i
s just that sort of event. However, we offer that the
phenomenon of the teachable moment can be the rule of mathematics classrooms, not the exception. The key is a
willingness to understand the classroom community as an adaptive, self
-
organizing
-
complex
-
un
ity.

Footnote

The research reported in this article was supported by the Social Sciences and Humanities Research Council of Canada
(SSHRC) through Grants #410
-
2000
-
0224 and #410
-
2001
-
0500. We would like to express our appreciation to
Thomas E. Kieren, Ral
ph Mason, Susan E. B. Pirie, Jo Towers, and Lyndon Martin for the conversations that triggered
and helped to inform this writing.

Footnote

1 In this sentence (and in several others), we deliberately violate a rule of expression that is imposed on much of
academic discourse
-
namely, the avoidance of anthropomorphic expressions (see, e.g., The Publication Manual of the
American Psychological Association, 2001, pp. 38
-
39). Individual human qualities and intentions are not to be applied
or ascribed to nonhuman,

subhuman, or superhuman events. Such categories, however, are blurred in the nested
logics and organizations of complex phenomena. As the mathematician Paulos (1998) notes, in reference to the stock
market and other instances of complexity, "It might even

turn out that we're not always being absurdly
anthropomorphic when we tell stories about such systems and their moods" (p. 185).

Footnote

2 This suggestion has been a topic of considerable discussion and research among complexity scientists. Johnson
(200
1) provides a detailed report of the issue, along with several illustrations of phenomena that were once believed
to be the result of top
-
down processes, but which, under close scientific scrutiny, are revealed to be bottom
-
up
-
including slime molds, ant co
lonies, and many social movements.

Footnote

3 For a discussion of nested systems see Davis, Sumara, and Luce
-
Kapler (2000), especially pp. 172
-
173.

Footnote

4 On this count, complexity science might be interpreted to offer critique of an effort to understand mathematics
teaching that is not attentive to both the particular contextual conditions and the broader cultural circumstances that
give shape to the proj
ect of school mathematics. Further, complexity science renders problematic those discourses
that focus on peripheries, fringes, border spaces, novices, and other notions that suggest that complex forms might
have clear centers, boundaries, and origins. In
fact, considerable research has been undertaken to better understand
the vital roles played by those agents that are commonly imagined to exist on the edges of complex unities
-
including,
significantly for educators, persons who are popularly classed as nov
ices and initiates. Deacon (1977), for example,
provides a detailed discussion of the ongoing contributions of young children to the evolution of the complex form of
language. Kelly (1995) details several examples of "fringe groups" and "outliers" that hav
e played key roles in the
transformations of social groups, businesses, and cultures.

Footnote

5 Notably, situated learning, social constructivism, critical and cultural theorists, and ecological and enactivist theories
use different principal metaphors t
o frame their discussions
-
including apprenticeship, shared labor, power, and
knotted webs or networks, respectively. All of these are taken up in ways that project a sense of complex, co
-
implicative dynamic
-
albeit that they are applied to different bodies
(including the social, epistemic, politic, and
biological).

Footnote

6 It is interesting to note that debates around traditional and more progressive teaching approaches have grown more
intense as modern societies have been presented with the need to respond more flexibly to external conditions, such
as environmental degrad
ation, world politics, and global economics. According to complexity science, such contextual
circumstances would be expected to prompt shifts in educational emphases toward increased diversity among agents
in order to enable more flexible response. By con
trast, educational approaches that are oriented toward redundancy
would be better fitted to situations in which global circumstances are seen as more stable.

Footnote

7 We do not mean to invoke a knowledge as object metaphor in the suggestion that ideas m
ust be made to bump
against one another. There is, however, a physical aspect to the point. For ideas to bump against one another, they
must be represented in some way
-
for example, as oral expressions or written statements. The juxtaposition of various
rep
resentations might then trigger other interpretations, which when represented might trigger still other
interpretations or elaborations. To reemphasize, the representations are, not the knowledge (although, of course, they
are tied up in knowledge). Rather
, we regard knowledge as the potential for action, and so the principle that we are
attempting to develop here is more about the importance of actuating these potentials in the hope that they might
trigger other, more sophisticated possibilities. Unfortuna
tely, as is likely evident by the preceding sentence, an
appropriate English vocabulary for this manner of interpretation has not yet emerged
-
hence, our reliance on the
inaccurate, but useful, notions of bumping, colliding, and juxtaposition of ideas.

Ref
erences

REFERENCES