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Curriculum and Complex Systems Theory
Curriculum and Complex Systems Theory W-M Roth and J Thom, University of Victoria, Victoria, BC, Canada ã 2010 Elsevier Ltd. All rights reserved.
I am inclined to think . . . that unlike the basic question of computability itself, the issues of complexity theory are not quite the central ones in relation to mental phenomena. (Penrose, 1989: 145).
Educators are increasingly engaging in a discourse that includes terms such as chaos, complexity, emergence, and fractals to understand, describe, and provoke teaching and learning situations (e.g., Davis, 2004). These terms are founded in scientific and mathematical fields where modeling phenomena such as weather, the shape of ice crystals, galaxies, or mountain ranges has contributed to the development of, among other things, film animation and video games (Mandelbrot, 1983). However, actual mathematical modeling of curriculum development, curriculum as a process, or curriculum as praxis rarely, if ever, occurs in terms of the complex systems theory, although the mathematics of nonlinear dynamical systems may be the most appropriate tool. (At the time of this writing, the ISI Web of Science database did not list any educational publication under this search term.) There may be two main reasons for this. First, sociocultural phenomena such as real classrooms are so complex that they exceed the capabilities of the mathematical models and computers that generate them. In addition, if classrooms are more complex than any other type of natural system or ecology (Costanza, 2003), then the developmental speeds of sociocultural phenomena exceed those of natural systems by orders of magnitude. Second, to make sense of and to subsequently model even the simplest of complex systems necessitates not only the conceptual mathematical competency of the educator, but also his or her procedural mathematical competency. Given the enormous complexity of social systems, some may argue that concepts from the complex systems (complexity and chaos) theory be used primarily for analogical purposes, though the apparent simplicity underlying the models of complex systems appear to make it suitable for social systems as well (Guespin-Michel, 2005). However, as research on analogical reasoning shows, the user still has to understand the structures of both the source system and the target systems for any analogy to exist and be productive. For instance, if in the context of a fourth-grade lesson on electricity, the analogy linking a continuous train on the track to the electrons in the wire only proves conceptually effective when all the corresponding structural elements are precisely matched (track/train/people in station pushing/resistance and wire/current/battery/light bulb). Alternatively, one might
assume that the terms from the complex systems theory are to be used for metaphorical purposes, thereby drawing on the power of metaphors in understanding (Lakoff and Johnson, 1980). Even if this is so – and often, this is the case – using such terms for descriptive purposes does not eschew the need for assessing how these metaphors mediate one’s conceptions and enactions of curricula. Furthermore, if the meaning of a metaphor is not communicated and comprehended appropriately, it does nothing but hide or obscure, rather than explicate the structural relations among sets of concepts. For instance, second-grade students learning about geometrical objects will find it difficult to understand and even memorize the word cylinder if they do not know that it comes from the ancient Greek term for roller. Having such knowledge would allow them to relate the term to their experience of rolling cylinders down an inclined plane or rolling it between their two hands. The upshot of these considerations is that the use of concepts from the complex systems theory makes sense only if the speakers and their audiences know the source phenomena (the models related to them) and the target phenomenon (curriculum development and enacted curriculum). If this is not the case, one can use any imagery and does not have to refer to the complex systems theory at all. For example, it is possible to use the imagery of a branching tree and not have to refer to fractal geometry (e.g., Davis and Sumara, 2000) that allows producing a family resemblance between computer-generated images of trees and trees that can touched and climbed. If we do not understand the underlying mathematical processes at least conceptually, it does not make sense to refer to the complex systems theory, fractals, or chaos, or to choose its associated images as metaphors. Thus, the addition of fractal does not make the metaphor anymore effective.
Modeling the Complexity of Social (Classroom) Interaction Curriculum as a process (praxis) and curriculum development continue to be thought largely in terms of linear theories, whereby teachers translate a planned (written) curriculum into their classroom actions with the result of achieving specific learning outcomes in students. This is the case although every educator has had experiences where the planned curriculum and the curriculum that is actually lived out or enacted are not the same thing;
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where plans or instructions – for example, recipe, assembly instructions, or modes of operation – led to very different sets and trajectories of action. Failing to implement one’s own plans for making a dinner, or for operating a new gadget, points to the complexity of life. That said, one can only imagine how much more complex social actions are, such as a teacher bringing forth and carrying off a lesson in the company of a classroom of students. The linearity that underpins many accepted planning models does little to reveal the complexity of nature generally and, specifically, the many orders of greater complexity that is human life. This linearity, as should be clear from this article, underlies the fundamental inappropriateness that teachers experience in university classrooms or the irrelevance of the lesson plans they make to actual classroom events. Complexity is or at least it should be assumed inherent in the term curriculum itself: one might be talking about a planned curriculum or an enacted one, with its different references to history, syllabi, classroom discourse, intended learning outcomes, and actual classroom events. Questions that arise include: which of these aspects is to be described and represented as a complex system and in terms of associated theories? How do the complex systems models for the different aspects of curriculum relate to one another? In which cases or for which phenomena do metaphors or analogies with complex systems hold true and under which conditions do they break down? In this article, two relatively simple examples – a parent–child exchange and the decision making concerning the (non) inclusion of a concept in a curriculum outline – are used to illustrate both modeling and complex-systems-theoretical thinking about curriculum-related issues. The article concludes with a critical interrogation of the limits and limitations of any analogy or metaphor derived from the complex systems theories. As a way of introducing the readers to the modeling of complex systems, consider the following exchange that also serves as a model of a simple teacher–student transaction and, therefore, for curriculum as process (praxis): Example 1 On the way from home to the vacation spot, a child (C) initiates an exchange with her parent (P). Episode 1 1 C: When are we on vacation? 2a P: We are on vacation. In this situation, we recognize the exchange as a question– answer pair. The parent (patiently) responds that they are on vacation, where the emphatic are indicates that the vacation has already started. Naive analysts might think that the child asked a question. Such an analysis does not allow constructing a model for all possible next instances
that evolve from the child’s utterance. That there are other ways of performing a next utterance leading to a different turn pair is evident from the parent’s reply later during the 3-day voyage following the same utterance by the child. Episode 2 1 C: When are we on vacation? 2b P: Stop bugging me. This time, the parent interprets the child’s utterance not as a question but as a form of annoyance. Of course, the parent could have responded in the same way as in episode 1, which is what happened throughout the trip between episodes 1 and 2. These instances illustrate that it is impossible to know in advance what the function of an utterance is until it actually occurs. In the first episode, the conversation unfolds as a question–answer pair; in the second instance, it is an annoyance–warning pair. Of course, many other forms of pair completion can be imagined. Any reader not yet convinced of this is encouraged to think of all those situations where an utterance was met with a response like ‘‘You hurt me!’’ and to which the initial speaker replies, ‘‘It was a joke’’ or ‘‘I didn’t mean to hurt your feelings but . . .’’ How can this multiplicity of turn pairs emerging from the same utterance be modeled, and how might this diversity be realized in a model that takes into account the complexity of social relations?
A Model for Representing Social Processes As a first explanation, one may draw on the catastrophe theory (Thom, 1979) as a framework that allows modeling the complexity articulated in the above episodes. The catastrophe theory is both a precursor and a parallel theory to the more popular chaos theory. It combines qualitative and quantitative mathematics to provide an easily intelligible way of modeling the two event categories. Figure 1 presents a two-state system for completing the child’s utterance as raising a genuine question versus aggravating the parent. To the right of point A, the system has but one state where the two turns together constitute
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Figure 1 A simple catastrophe theoretic model for the complexity of social life.
Curriculum and Complex Systems Theory
one question–answer pair. To the left of point B, annoyance occurs as expressed in the singly possible annoyance– warning pair. Between A and B, two stable states are possible and simultaneous (the dotted part of the curve denotes unstable states of the system): genuine question and bugging. The interesting aspect of this situation is that we cannot know what response will follow the utterance of ‘‘when are we on vacation?’’; that is, whether we are on the upper or lower branch until the event has taken place. The child’s utterance constitutes a bifurcation point where, for a moment prior to the parent’s reply, the system exits simultaneously on both branches and either response can occur; therefore, it is only when the parent replies that these possibilities collapse into one real event. Moreover, the system may jump from one branch to another in unpredictable ways, brought about by an infinitesimal change in conditions. We can see this as being the case when, at some point along the journey, the system flips and a different outcome results in episode 2 with the parent’s reply; or, equivalently, we can say that the reply flipped the system from one state into another. What becomes clear when relating this simple example to everyday events is that social life is so complex that we never know whether we are to the left of point B or to the right of point A, though habits may make us think we are in either place. The unpredictability of soci(et)al situations, therefore, suggests that we are always between A and B; however, in relatively predictable (stable) cases, we are closer to A (on the upper branch) or B (on the lower branch). Here, only two actors and two possible states are considered, whereas in real classrooms, there are 20 or more individuals and the possibilities for next turns are large and, likely, infinite. Moreover, because interactions are contingent in the sense that one person never knows what another is up to, real (complex) systems such as classrooms are not only contingent, but also exist in states of instability, which makes them unpredictable even with respect to their immediate future. The best we can do as educators then, is to develop curricula that anticipate what could be as opposed to ones that prescribe what (we believe) is (e.g., Pirie and Thom, 2001).
where the value of some variable xt at some point (in time) t is fed back into the equation to calculate its value xt + 1 at the next instant (t þ 1). The end result of this iteration depends on the constant k value. When k < 3, x will always asymptotically approach one value (Figure 2). However, at k ¼ 3, the system goes through a point of instability: to the left (k < 3.0), the animal population always stabilizes at one value, whereas to the right of the point (k > 3.0), the population oscillates between two values (i.e., attractors (Briggs and Peat, 1989)). Point k ¼ 3 is then considered to be a bifurcation point and infinitesimal perturbations – which in computer simulations arise from rounding errors – are sufficient to change the observable behavior of the system: it is nondeterministic (Prigogine and Stengers, 1979). As k increases, there will be further bifurcations where there are 4, 8, 16, and so forth stable values of x until an infinite number of asymptotic xt ¼ 1 are possible (Figure 2); here, periodicity ends and the population jumps unpredictably from one value to another. However, subsequently, with further increase of k, a small number of equilibrium states of x emerge from chaos. As a system moves through bifurcation points, it develops its own particular and contingent history, arising from the intertwining of deterministic laws of change (e.g., motion) and indeterminate (chance) fluctuations. This model (i.e., eqn [1] and Figure 2) directly relates to the conversation between the child and the parent. It is, in fact, equivalent to the catastrophe-theoretical model presented earlier and depicted in Figure 1 for 3 < k < 3.45, where x jumps between two values. The model allows us to understand the situation right after the child utters his/her sentence and prior to the parent’s response. At this point, the function of the child’s utterance in the collectively constituted conversation is not yet clear, and two possible responses are possible. For any k, therefore, the diagram provides the possibility space; as soon as the next person (parent) acts, the possibilities are reduced
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A Second Model for Representing Social Processes To model and better understand such complexity, one may use an analogy with another system (e.g., animal population) where complexity arises from recursion – a process of repetitive application to something, for example, a number. The simplest recursive system that leads to rather complex phenomena and can be used to model real events is given by the iterative equation xt þ1 ¼ kxt ð1 xt Þ
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Figure 2 A bifurcation diagram for eqn [1]: depending on the value of k, there are 1, 2, 4, . . . values that xt = 1 takes.
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to certainty and the value of x is certain. Readers will immediately realize that any moment in a real classroom situation corresponds to the situations with k > 3.6, where there are innumerable possibilities of the next answers setting up new situations and each will have innumerable possibilities of continuation. As k increases, windows (white panels (Figure 2)) open up, within which three stable numbers appear, followed by period doubling to 6, 12, 24, and so forth. These period doublings resemble the original image, but now occur at a new scale (insert in Figure 2). This is what is denoted by the terms self-similarity and scale invariance. In the classroom, these qualities could be observed where a whole-class discussion about a chaotic event mirrors the discussions that previously occurred in smaller groups where each group came up with its own explication of a phenomenon of interest (e.g., Roth and Duit, 2003). Self-similarity of content is evident across the different discussions and scale invariance exists as a result of the discussions emerging on both whole-class and small-group levels. It is also important to note that real human life situations do not correspond to stable k values that ecologists, for example, use to model repeating size patterns of animal populations; rather, human life is so complex that it corresponds to (unpredictable) changing values of k, and because of this, it is impossible to be absolutely certain what the effect(s) of our (discursive and practical) actions will be. In more poetic words, Frost (1966) makes the same point in ‘The road not taken’; there are always more than two roads and we can never know which one we are placing our first step on. Each time, we immediately find ourselves at another bifurcation point with similar complexity. In addition, regardless of the scale (e.g., at a microlevel) at which we examine a human event, for instance, what a student focuses on during a science demonstration, the complexity is so great that we cannot be certain about how many different ways even the simplest of science demonstrations might be perceived by the students (e.g., Roth, 2006).
A Third Model for Representing Social Processes Such perception phenomena can be modeled using morphodynamical (catastrophe-theoretical) models, which combine qualitative (typological) and quantitative (topological) dimensions. The discontinuities (e.g., in perception, seeing, hearing, and categorization) are the result of complex, continuous, dynamic, and, therefore, inherently unstable processes. Qualitative discontinuities are recovered mathematically as dynamic events of bifurcation (Petitot, 2003), where complex systems radically change their expression, here in a qualitative difference, for example, in the perception of a color, an edge, and so forth. These models therefore
synthesize objectivism and subjectivism into a dialectical identity of nonidentical entities. Language and speech are other important nonlinear (complex) dynamic systems that are of importance in curriculum theory. What a second-grade student hears during a reading lesson, for example, when her teacher utters what linguists transcribe as ‘‘bea’’ – bare, to bear, or bear (animal) – is a function of context. Each of the different possible hearing constitutes an attractor (Gaudin, 2005) similar to those that are observable in Figures 1 and 2.
A Fourth Model for Representing Social Processes Educators (teachers) who come into contact with the chaos theory frequently liken curriculum and the process of developing it to strange attractors where the trajectories of learning cannot be predicted but, in the end, captured in the potential of one or the other attractor, only a small number of possible learning outcomes – one in the extreme – are achieved. To illustrate how this model is used as a form of scientific representation, let us examine a high school experiment involving a magnetic bob that moves back and forth across two magnets after it begins at a certain position in space and with some initial velocity (Roth and Duit, 2003). This constitutes a two-state system in motion in which its endpoint is unknown because even infinitesimally minute influences along its trajectory change not only the trajectory, but also the endpoint. Thus, the only certain prediction we can make is that the system will end up near one of the two attractors. What we cannot predict is where the bob ends up – because over many trials and everything else being equal, the probability of it ending up at one or the other magnets is the same, that is, p ¼ 0.5. To study this system’s behavior as it moves from left to right and right to left, changing its speed (i.e., magnitude of velocity) as it approaches and recedes from the magnets, requires scientists and mathematicians to plot its trajectory in phase space – a set of variables that completely describe the system. In this instance, there are two one-dimensional variables, position and velocity, and, together, they create a two-dimensional phase space (Figure 3). The figure shows how, beginning with the same position and velocity, the pendulum is caught at one point of time in the left attractor (A1), whereas it is caught in the other attractor (A2) at the end of another run. Although such a model provides an entry point for high school students’ understanding of chaotic systems (e.g., Roth and Duit, 2003), it is a very limited analogy for understanding anything about curriculum planning or the enactment of it.
Curriculum and Complex Systems Theory
A Fifth Model for Representing Social Processes A more realistic system consists of a simple model for how people make decisions, such as choosing one of two options or classifying an object into one of two categories. For example, let us imagine a group of curriculum developers attempting to decide whether to include a particular concept in a curriculum (i.e., phase space). A model that is based on parallel distributed processing, and used for artificial neural networks, allows us to represent such a decision as constraint satisfaction (McClelland and Rumelhart, 1988). Each statement made in the decisionmaking process describes an aspect of the curriculum as a node (Figure 4); three statements support the inclusion of a particular concept under discussion (Figure 4, see 1–3) and three statements are made not to include the concept. Some statements support each other explicitly and are modeled as reinforcing links between the nodes (þ). Some pairs of statements are contradictory, each supporting a different decision, which is modeled as links between
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Figure 3 The trajectories of a simple chaotic system in phase space. Although the system begins at the same place and with the same velocity, its trajectory is unpredictable and, yet, the system will end up near one or the other attractor.
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Figure 4 A simple model for decision making (categorization) that relates each statement supporting one of two decisions (gray, white statements). When a statement is supported by another, it is positively linked to it, whereas when another statement contradicts it, it is negatively linked. The strength of support or contradiction corresponds to how emphatic the statement is presented in the meeting.
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nodes that weaken each other (–). Some statements do not relate to others and these are characterized in the model by the absence of a link. The final model consists of a network of statements, about a proposal, which mutually reinforce, weaken, or are neutral to one another. The outcome of the process is a solution with the least constraint in the network. Mathematically, the system finds itself somewhere up on a mountainside in a six-dimensional space and is moving (could we say flowing?) down the steepest gradients of the slope until it ends at some lowest place (lowest state). (Mathematically, the problem is the same as in eqn [1], the only difference being that there are now six mutually influencing variables rather than one.) The dynamics of the system in six dimensions (its phase space) is impossible to envision; however, the (Euclidean) distance that the system has from the two end states (yes and no) can be calculated, yielding a representation of the systems dynamics in two dimensions, which constitute the interpretation or decision-making space (Hutchins, 1995). The left panel of Figure 5 shows the trajectory of the decision-making process for different scenarios. In scenarios 1–3, the curriculum committee starts mildly favoring a decision to include the concept. However, because of different ways in which the same propositions are affecting others, the curriculum committee quickly comes to a supportive decision (trajectory 1), after initial consideration to the contrary (trajectory 2), or, during the considerations, shifts to support not including the concept (trajectory 3). The two end states – inclusion or noninclusion of the concept – constitute attractors where the system does end up. There is a third attractor, which does not constitute an absolute minimum (relaxed state of the system) but a local minimum in which the system can be trapped. (Imagine a dip somewhere on our six-dimensional (fractal) surface that can trap the system in its course.) Scenarios 4–6 (Figure 5, right panel) model the different situations in which the curriculum committee is hung because both decisions (inclusion and noninclusion) are concurrently being supported and, therefore, a consensus cannot be reached. The different trajectories model the different ways in which the committee reaches the state of indecision. In scenario 6, despite the almost certain starting position of not including the concept, the curriculum committee moves toward a state where both funding and not funding are supported equally well. Despite the complexity of the possible outcomes that this model can handle, it is still simplistic when we compare it to real-life social–psychological processes. Although it represents the changing strengths of the propositions 1 through 6, it cannot express the relation between them; it models all statements as if they arise simultaneously rather than unfolding over time, continuously modifying the context in which future statements are made; and it only models six statements as opposed to what is more realistic,
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Figure 5 Modeling group decision making when some propositions favor including a concept in the curriculum (1, 3, and 5 (Figure 3)) and others favor not including the concept (2, 4, 6 (Figure 3)). Slight variations in emphasis with which arguments are made lead to the inclusion (1, 2) or noninclusion of the concept under consideration (3). Sometimes, even with extended discussion, the decision-making group ends in a stalemate (4–6).
a potentially infinite number of statements committee members might make in the process. Nevertheless, the strength of this particular model is that it underscores the notion that minor variations in initial conditions can lead to substantially different outcomes – one decision over another. This model also sufficiently captures the complexity of classroom events in that it demonstrates how, once all statements made in some situation are known, the possible ways in which concrete classroom conversations come to a conclusion (Roth, 2001).
Limitations of Analogies and Metaphors in Curriculum Theorizing In this article, different models and situations are presented to exhibit the suitability of complex systems theories to curriculum theorists. It has been suggested that the mathematical models and graphical representations may be used in metaphorical and analogical ways. However, the effective use of metaphors and analogies requires knowledge not only of source and target domains, but also, and importantly so, the instances in and conditions under which the relationship breaks down. This is also the case for ideas, concepts, and theories from the domain of complex systems. All too often, concepts are taken from the complexity theory, applying them in loose ways that lack rigorous analysis but abound with generic advice (Fenwick, 2004). From this articulation of different models, it is reasonable to conclude that the complex systems theory for actual physical, sociocultural, and cultural–historical contexts may be very different from the one that appears in the simple models presented here and elsewhere, and which figures prevalently in the educational literature. We do not know whether the phenomena of interest in the curriculum
show scale invariance, such as the one depicted in Figure 2. Do terms such as self-similarity and recursion allow us to expand possibilities for curriculum theorizing or restrict such actions? Current thinking across many disciplines, including landscape and physiological, paleo-, freshwater, and estuarine ecology, population interactions, meteorology and climatology, as well as global change, suggests that this is not precisely the case: to forecast the evolution of the complex system, one needs to take into account the particular scale chosen or involve the required scaling rules that, however, have not yet been developed (Costanza, 2003). The influence of scale, resolution, and hierarchy between different levels has not been investigated to any extent and remains a key question in developing coherent models (theories) of complex systems. Thus, although some curriculum theorists use the notion of scale invariance, even the inventor of the concept of fractal geometry takes a more differentiated view: ‘‘no one believes that the world is strictly homogeneous or scaling. . . . One should not be surprised that scaling fractals should be limited to providing first approximations of the natural shapes tackled’’ (Mandelbrot, 1983: 18 and 19). What, if so, are these limits? To what extent do models for simple constructed phenomena such as question–answer pairs scale up and capture truly complex and real events that arise moment to moment in classrooms, schools, and the field of education at large? In what manners do these theoretical frames allow us to understand events such as test taking, differential achievement, or the similarity in transnational curriculum comparisons? In mathematics, the complexity theory is concerned with solving classes of problems (algorithms), some of which are vastly more difficult to solve (compute) than others. It is concerned not with the difficulty of solving the problem (algorithm), but with ‘‘infinite families of
Curriculum and Complex Systems Theory
problems where there would be a general algorithm for finding answers to all the problems of a single family’’ (Penrose, 1989: 141). In curriculum theorizing, we need a similar field concerned with tracking the solution(s) of infinite classes of problems. Fundamentally, the ideas that appear in much of the literature on complex systems are grounded in an epistemology of the same – recursion produced by some mathematical algorithm, the algorithm itself, self, similarity, self-similarity, and scale invariance are all terms that begin with the ideology of sameness and construct difference as the opposite, not the same, lying outside the boundaries of the same. The idea of the curriculum as a strange attractor, though implying the unpredictability of different trajectories, also implies one or more common endpoints that some enacted curriculum or curriculum committee might arrive at. Precisely here lies a problem that difference, which any real physical and cultural–historical system displays, is not modeled as a positive phenomenon, that is, as a phenomenon in and for itself. It is therefore necessary for the complex systems theories that have any value for describing and explaining real systems of interest to begin differently and recognize that any similarity is the outcome of a (decision making) process rather than a priori; sameness and similarity are human productions and not inherent characteristics of nature or culture (Deleuze, 1994). For example, although cooking pots and children’s cups that have a pair of handles appear to us as two different entities that are also substantially distinct on a practical level, from a mathematical perspective, they are considered to be topologically the same; that is, it is possible to morph one into the other without severing or reattaching any of its parts. Consequently, this raises an additional need for educators’ questioning and careful consideration of these forms of mathematics and the models they belong to as suitable structures to constitute the source of analogical thinking about curriculum. The examples and models presented here – social interaction and decision-making processes – are complex and unpredictable. They serve to highlight just how nonsimplistic the real world really is. At the very least, the complex systems theories provide us with ideas that are suggestive of how intractable the world is given that the simplest of phenomena already involve enormous and, perhaps, incalculable complexity.
Bibliography Briggs, J. and Peat, F. D. (1989). Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness. New York: Harper and Row.
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Costanza, R. (2003). A vision of the future of science: Reintegrating the study of humans and the rest of nature. Futures 35, 651–671. Davis, B. (2004). Inventions of Teaching: A Genealogy. Mahwah, NJ: Erlbaum. Davis, B. and Sumara, D. J. (2000). Curriculum forms: On the assumed shapes of knowing and knowledge. Journal of Curriculum Studies 32, 821–845. Deleuze, G. (1994). Difference and Repetition, Patton, P. (trans.). New York: Columbia University Press. Fenwick, T. (2004). A review of chaos theory and higher education: Leadership, planning and policy. Complicity: An International Journal of Complexity and Education 1, 113–116. Frost, R. (1966). The road not taken [1915; 1916]. In Herrig, L., Meller, H, and Su¨hnel, R. (eds.) British and American Classical Poems, p. 112. Braunschweig: Georg Westerman Verlag. Gaudin, F. (2005). Y a-t-il de la non-line´arite´ en se´mantique. In Se`ve, L. (ed.) E´mergence, complexite´ et dialectique, pp 280–288. Paris: Odile Jacob. Guespin-Michel, J. (2005). Re´seau de regulation, boucles de retroaction et dialectique. In Se`ve, L. (ed.) E´mergence, complexite´ et dialectique, pp 243–253. Paris: Odile Jacob. Hutchins, E. (1995). Cognition in the Wild. Cambridge, MA: MIT Press. Lakoff, G. and Johnson, M. (1980). Metaphors We Live by. Chicago, IL: University of Chicago. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. (updated and augmented edition). New York: W.H. Freeman. McClelland, J. L. and Rumelhart, D. E. (1988). Explorations in Parallel Distributed Processing: A Handbook of Models, Programs, and Exercises. Cambridge, MA: MIT. Penrose, R. (1989). The Emperor’s New Mind. Oxford: Oxford University Press. Petitot, J. (2003). Morphodynamical enaction: The case of color. Biological Research 36, 107–112. Pirie, S. E. B. and Thom, J. S. (2001). Thinking through ecological metaphors in mathematics education. In Gunn, S. and Begg, A. (eds.) Mind, Body, and Society: Emerging Understandings of Knowing and Learning, pp 45–52. Melbourne, VIC: University of Melbourne. Prigogine, I. and Stengers, I. (1979). La nouvelle alliance: Me´tamorphose de la science. Paris: E´ditions Gallimard. Roth, W. -M. (2001). Designing as distributed process. Learning and Instruction 11, 211–239. Roth, W. -M. (2006). Learning Science: A Singular Plural Perspective. Rotterdam: Sense Publishers. Roth, W. -M. and Duit, R. (2003). Emergence, flexibility, and stabilization of language in a physics classroom. Journal for Research in Science Teaching 40, 869–897. Thom, R. (1979). Woru¨ber soll man sich wundern. In Maurin, K., Michalski, K., and Rudolph, E. (eds.) Offene Systeme II: Logic und Zeit, pp 41–107. Stuttgart: Klett-Cotta.
Further Reading Elman, J. L., Bates, E. A., Johnson, M. H., et al. (1996). Rethinking Innateness: A Connectionist Perspective on Development. Cambridge, MA: MIT Press. Emmeche, C. (1994). The Garden in the Machine: The Emerging Science of Artificial Life. Princeton, NJ: Princeton University Press. Gleick, J. (1988). Chaos: Making a New Science. New York: Penguin. Hayles, N. K. (ed.) (1991). Chaos and Order: Complex Dynamics in Literature and Science. Chicago, IL: University of Chicago Press. Raijmakers, M. E. J., van Koten, S., and Molenaar, P. C. M. (1996). On the validity of simulating stage wise development by means of PDP networks: Application of catastrophe analysis and an experimental test of rule-like network performance. Cognitive Science 20, 101–136.
I am inclined to think . . . that unlike the basic question of computability itself, the issues of complexity theory are not quite the central ones in relation to mental phenomena. (Penrose, 1989: 145).
Educators are increasingly engaging in a discourse that includes terms such as chaos, complexity, emergence, and fractals to understand, describe, and provoke teaching and learning situations (e.g., Davis, 2004). These terms are founded in scientific and mathematical fields where modeling phenomena such as weather, the shape of ice crystals, galaxies, or mountain ranges has contributed to the development of, among other things, film animation and video games (Mandelbrot, 1983). However, actual mathematical modeling of curriculum development, curriculum as a process, or curriculum as praxis rarely, if ever, occurs in terms of the complex systems theory, although the mathematics of nonlinear dynamical systems may be the most appropriate tool. (At the time of this writing, the ISI Web of Science database did not list any educational publication under this search term.) There may be two main reasons for this. First, sociocultural phenomena such as real classrooms are so complex that they exceed the capabilities of the mathematical models and computers that generate them. In addition, if classrooms are more complex than any other type of natural system or ecology (Costanza, 2003), then the developmental speeds of sociocultural phenomena exceed those of natural systems by orders of magnitude. Second, to make sense of and to subsequently model even the simplest of complex systems necessitates not only the conceptual mathematical competency of the educator, but also his or her procedural mathematical competency. Given the enormous complexity of social systems, some may argue that concepts from the complex systems (complexity and chaos) theory be used primarily for analogical purposes, though the apparent simplicity underlying the models of complex systems appear to make it suitable for social systems as well (Guespin-Michel, 2005). However, as research on analogical reasoning shows, the user still has to understand the structures of both the source system and the target systems for any analogy to exist and be productive. For instance, if in the context of a fourth-grade lesson on electricity, the analogy linking a continuous train on the track to the electrons in the wire only proves conceptually effective when all the corresponding structural elements are precisely matched (track/train/people in station pushing/resistance and wire/current/battery/light bulb). Alternatively, one might
assume that the terms from the complex systems theory are to be used for metaphorical purposes, thereby drawing on the power of metaphors in understanding (Lakoff and Johnson, 1980). Even if this is so – and often, this is the case – using such terms for descriptive purposes does not eschew the need for assessing how these metaphors mediate one’s conceptions and enactions of curricula. Furthermore, if the meaning of a metaphor is not communicated and comprehended appropriately, it does nothing but hide or obscure, rather than explicate the structural relations among sets of concepts. For instance, second-grade students learning about geometrical objects will find it difficult to understand and even memorize the word cylinder if they do not know that it comes from the ancient Greek term for roller. Having such knowledge would allow them to relate the term to their experience of rolling cylinders down an inclined plane or rolling it between their two hands. The upshot of these considerations is that the use of concepts from the complex systems theory makes sense only if the speakers and their audiences know the source phenomena (the models related to them) and the target phenomenon (curriculum development and enacted curriculum). If this is not the case, one can use any imagery and does not have to refer to the complex systems theory at all. For example, it is possible to use the imagery of a branching tree and not have to refer to fractal geometry (e.g., Davis and Sumara, 2000) that allows producing a family resemblance between computer-generated images of trees and trees that can touched and climbed. If we do not understand the underlying mathematical processes at least conceptually, it does not make sense to refer to the complex systems theory, fractals, or chaos, or to choose its associated images as metaphors. Thus, the addition of fractal does not make the metaphor anymore effective.
Modeling the Complexity of Social (Classroom) Interaction Curriculum as a process (praxis) and curriculum development continue to be thought largely in terms of linear theories, whereby teachers translate a planned (written) curriculum into their classroom actions with the result of achieving specific learning outcomes in students. This is the case although every educator has had experiences where the planned curriculum and the curriculum that is actually lived out or enacted are not the same thing;
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where plans or instructions – for example, recipe, assembly instructions, or modes of operation – led to very different sets and trajectories of action. Failing to implement one’s own plans for making a dinner, or for operating a new gadget, points to the complexity of life. That said, one can only imagine how much more complex social actions are, such as a teacher bringing forth and carrying off a lesson in the company of a classroom of students. The linearity that underpins many accepted planning models does little to reveal the complexity of nature generally and, specifically, the many orders of greater complexity that is human life. This linearity, as should be clear from this article, underlies the fundamental inappropriateness that teachers experience in university classrooms or the irrelevance of the lesson plans they make to actual classroom events. Complexity is or at least it should be assumed inherent in the term curriculum itself: one might be talking about a planned curriculum or an enacted one, with its different references to history, syllabi, classroom discourse, intended learning outcomes, and actual classroom events. Questions that arise include: which of these aspects is to be described and represented as a complex system and in terms of associated theories? How do the complex systems models for the different aspects of curriculum relate to one another? In which cases or for which phenomena do metaphors or analogies with complex systems hold true and under which conditions do they break down? In this article, two relatively simple examples – a parent–child exchange and the decision making concerning the (non) inclusion of a concept in a curriculum outline – are used to illustrate both modeling and complex-systems-theoretical thinking about curriculum-related issues. The article concludes with a critical interrogation of the limits and limitations of any analogy or metaphor derived from the complex systems theories. As a way of introducing the readers to the modeling of complex systems, consider the following exchange that also serves as a model of a simple teacher–student transaction and, therefore, for curriculum as process (praxis): Example 1 On the way from home to the vacation spot, a child (C) initiates an exchange with her parent (P). Episode 1 1 C: When are we on vacation? 2a P: We are on vacation. In this situation, we recognize the exchange as a question– answer pair. The parent (patiently) responds that they are on vacation, where the emphatic are indicates that the vacation has already started. Naive analysts might think that the child asked a question. Such an analysis does not allow constructing a model for all possible next instances
that evolve from the child’s utterance. That there are other ways of performing a next utterance leading to a different turn pair is evident from the parent’s reply later during the 3-day voyage following the same utterance by the child. Episode 2 1 C: When are we on vacation? 2b P: Stop bugging me. This time, the parent interprets the child’s utterance not as a question but as a form of annoyance. Of course, the parent could have responded in the same way as in episode 1, which is what happened throughout the trip between episodes 1 and 2. These instances illustrate that it is impossible to know in advance what the function of an utterance is until it actually occurs. In the first episode, the conversation unfolds as a question–answer pair; in the second instance, it is an annoyance–warning pair. Of course, many other forms of pair completion can be imagined. Any reader not yet convinced of this is encouraged to think of all those situations where an utterance was met with a response like ‘‘You hurt me!’’ and to which the initial speaker replies, ‘‘It was a joke’’ or ‘‘I didn’t mean to hurt your feelings but . . .’’ How can this multiplicity of turn pairs emerging from the same utterance be modeled, and how might this diversity be realized in a model that takes into account the complexity of social relations?
A Model for Representing Social Processes As a first explanation, one may draw on the catastrophe theory (Thom, 1979) as a framework that allows modeling the complexity articulated in the above episodes. The catastrophe theory is both a precursor and a parallel theory to the more popular chaos theory. It combines qualitative and quantitative mathematics to provide an easily intelligible way of modeling the two event categories. Figure 1 presents a two-state system for completing the child’s utterance as raising a genuine question versus aggravating the parent. To the right of point A, the system has but one state where the two turns together constitute
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Figure 1 A simple catastrophe theoretic model for the complexity of social life.
Curriculum and Complex Systems Theory
one question–answer pair. To the left of point B, annoyance occurs as expressed in the singly possible annoyance– warning pair. Between A and B, two stable states are possible and simultaneous (the dotted part of the curve denotes unstable states of the system): genuine question and bugging. The interesting aspect of this situation is that we cannot know what response will follow the utterance of ‘‘when are we on vacation?’’; that is, whether we are on the upper or lower branch until the event has taken place. The child’s utterance constitutes a bifurcation point where, for a moment prior to the parent’s reply, the system exits simultaneously on both branches and either response can occur; therefore, it is only when the parent replies that these possibilities collapse into one real event. Moreover, the system may jump from one branch to another in unpredictable ways, brought about by an infinitesimal change in conditions. We can see this as being the case when, at some point along the journey, the system flips and a different outcome results in episode 2 with the parent’s reply; or, equivalently, we can say that the reply flipped the system from one state into another. What becomes clear when relating this simple example to everyday events is that social life is so complex that we never know whether we are to the left of point B or to the right of point A, though habits may make us think we are in either place. The unpredictability of soci(et)al situations, therefore, suggests that we are always between A and B; however, in relatively predictable (stable) cases, we are closer to A (on the upper branch) or B (on the lower branch). Here, only two actors and two possible states are considered, whereas in real classrooms, there are 20 or more individuals and the possibilities for next turns are large and, likely, infinite. Moreover, because interactions are contingent in the sense that one person never knows what another is up to, real (complex) systems such as classrooms are not only contingent, but also exist in states of instability, which makes them unpredictable even with respect to their immediate future. The best we can do as educators then, is to develop curricula that anticipate what could be as opposed to ones that prescribe what (we believe) is (e.g., Pirie and Thom, 2001).
where the value of some variable xt at some point (in time) t is fed back into the equation to calculate its value xt + 1 at the next instant (t þ 1). The end result of this iteration depends on the constant k value. When k < 3, x will always asymptotically approach one value (Figure 2). However, at k ¼ 3, the system goes through a point of instability: to the left (k < 3.0), the animal population always stabilizes at one value, whereas to the right of the point (k > 3.0), the population oscillates between two values (i.e., attractors (Briggs and Peat, 1989)). Point k ¼ 3 is then considered to be a bifurcation point and infinitesimal perturbations – which in computer simulations arise from rounding errors – are sufficient to change the observable behavior of the system: it is nondeterministic (Prigogine and Stengers, 1979). As k increases, there will be further bifurcations where there are 4, 8, 16, and so forth stable values of x until an infinite number of asymptotic xt ¼ 1 are possible (Figure 2); here, periodicity ends and the population jumps unpredictably from one value to another. However, subsequently, with further increase of k, a small number of equilibrium states of x emerge from chaos. As a system moves through bifurcation points, it develops its own particular and contingent history, arising from the intertwining of deterministic laws of change (e.g., motion) and indeterminate (chance) fluctuations. This model (i.e., eqn [1] and Figure 2) directly relates to the conversation between the child and the parent. It is, in fact, equivalent to the catastrophe-theoretical model presented earlier and depicted in Figure 1 for 3 < k < 3.45, where x jumps between two values. The model allows us to understand the situation right after the child utters his/her sentence and prior to the parent’s response. At this point, the function of the child’s utterance in the collectively constituted conversation is not yet clear, and two possible responses are possible. For any k, therefore, the diagram provides the possibility space; as soon as the next person (parent) acts, the possibilities are reduced
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A Second Model for Representing Social Processes To model and better understand such complexity, one may use an analogy with another system (e.g., animal population) where complexity arises from recursion – a process of repetitive application to something, for example, a number. The simplest recursive system that leads to rather complex phenomena and can be used to model real events is given by the iterative equation xt þ1 ¼ kxt ð1 xt Þ
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Figure 2 A bifurcation diagram for eqn [1]: depending on the value of k, there are 1, 2, 4, . . . values that xt = 1 takes.
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to certainty and the value of x is certain. Readers will immediately realize that any moment in a real classroom situation corresponds to the situations with k > 3.6, where there are innumerable possibilities of the next answers setting up new situations and each will have innumerable possibilities of continuation. As k increases, windows (white panels (Figure 2)) open up, within which three stable numbers appear, followed by period doubling to 6, 12, 24, and so forth. These period doublings resemble the original image, but now occur at a new scale (insert in Figure 2). This is what is denoted by the terms self-similarity and scale invariance. In the classroom, these qualities could be observed where a whole-class discussion about a chaotic event mirrors the discussions that previously occurred in smaller groups where each group came up with its own explication of a phenomenon of interest (e.g., Roth and Duit, 2003). Self-similarity of content is evident across the different discussions and scale invariance exists as a result of the discussions emerging on both whole-class and small-group levels. It is also important to note that real human life situations do not correspond to stable k values that ecologists, for example, use to model repeating size patterns of animal populations; rather, human life is so complex that it corresponds to (unpredictable) changing values of k, and because of this, it is impossible to be absolutely certain what the effect(s) of our (discursive and practical) actions will be. In more poetic words, Frost (1966) makes the same point in ‘The road not taken’; there are always more than two roads and we can never know which one we are placing our first step on. Each time, we immediately find ourselves at another bifurcation point with similar complexity. In addition, regardless of the scale (e.g., at a microlevel) at which we examine a human event, for instance, what a student focuses on during a science demonstration, the complexity is so great that we cannot be certain about how many different ways even the simplest of science demonstrations might be perceived by the students (e.g., Roth, 2006).
A Third Model for Representing Social Processes Such perception phenomena can be modeled using morphodynamical (catastrophe-theoretical) models, which combine qualitative (typological) and quantitative (topological) dimensions. The discontinuities (e.g., in perception, seeing, hearing, and categorization) are the result of complex, continuous, dynamic, and, therefore, inherently unstable processes. Qualitative discontinuities are recovered mathematically as dynamic events of bifurcation (Petitot, 2003), where complex systems radically change their expression, here in a qualitative difference, for example, in the perception of a color, an edge, and so forth. These models therefore
synthesize objectivism and subjectivism into a dialectical identity of nonidentical entities. Language and speech are other important nonlinear (complex) dynamic systems that are of importance in curriculum theory. What a second-grade student hears during a reading lesson, for example, when her teacher utters what linguists transcribe as ‘‘bea’’ – bare, to bear, or bear (animal) – is a function of context. Each of the different possible hearing constitutes an attractor (Gaudin, 2005) similar to those that are observable in Figures 1 and 2.
A Fourth Model for Representing Social Processes Educators (teachers) who come into contact with the chaos theory frequently liken curriculum and the process of developing it to strange attractors where the trajectories of learning cannot be predicted but, in the end, captured in the potential of one or the other attractor, only a small number of possible learning outcomes – one in the extreme – are achieved. To illustrate how this model is used as a form of scientific representation, let us examine a high school experiment involving a magnetic bob that moves back and forth across two magnets after it begins at a certain position in space and with some initial velocity (Roth and Duit, 2003). This constitutes a two-state system in motion in which its endpoint is unknown because even infinitesimally minute influences along its trajectory change not only the trajectory, but also the endpoint. Thus, the only certain prediction we can make is that the system will end up near one of the two attractors. What we cannot predict is where the bob ends up – because over many trials and everything else being equal, the probability of it ending up at one or the other magnets is the same, that is, p ¼ 0.5. To study this system’s behavior as it moves from left to right and right to left, changing its speed (i.e., magnitude of velocity) as it approaches and recedes from the magnets, requires scientists and mathematicians to plot its trajectory in phase space – a set of variables that completely describe the system. In this instance, there are two one-dimensional variables, position and velocity, and, together, they create a two-dimensional phase space (Figure 3). The figure shows how, beginning with the same position and velocity, the pendulum is caught at one point of time in the left attractor (A1), whereas it is caught in the other attractor (A2) at the end of another run. Although such a model provides an entry point for high school students’ understanding of chaotic systems (e.g., Roth and Duit, 2003), it is a very limited analogy for understanding anything about curriculum planning or the enactment of it.
Curriculum and Complex Systems Theory
A Fifth Model for Representing Social Processes A more realistic system consists of a simple model for how people make decisions, such as choosing one of two options or classifying an object into one of two categories. For example, let us imagine a group of curriculum developers attempting to decide whether to include a particular concept in a curriculum (i.e., phase space). A model that is based on parallel distributed processing, and used for artificial neural networks, allows us to represent such a decision as constraint satisfaction (McClelland and Rumelhart, 1988). Each statement made in the decisionmaking process describes an aspect of the curriculum as a node (Figure 4); three statements support the inclusion of a particular concept under discussion (Figure 4, see 1–3) and three statements are made not to include the concept. Some statements support each other explicitly and are modeled as reinforcing links between the nodes (þ). Some pairs of statements are contradictory, each supporting a different decision, which is modeled as links between
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Figure 3 The trajectories of a simple chaotic system in phase space. Although the system begins at the same place and with the same velocity, its trajectory is unpredictable and, yet, the system will end up near one or the other attractor.
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Figure 4 A simple model for decision making (categorization) that relates each statement supporting one of two decisions (gray, white statements). When a statement is supported by another, it is positively linked to it, whereas when another statement contradicts it, it is negatively linked. The strength of support or contradiction corresponds to how emphatic the statement is presented in the meeting.
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nodes that weaken each other (–). Some statements do not relate to others and these are characterized in the model by the absence of a link. The final model consists of a network of statements, about a proposal, which mutually reinforce, weaken, or are neutral to one another. The outcome of the process is a solution with the least constraint in the network. Mathematically, the system finds itself somewhere up on a mountainside in a six-dimensional space and is moving (could we say flowing?) down the steepest gradients of the slope until it ends at some lowest place (lowest state). (Mathematically, the problem is the same as in eqn [1], the only difference being that there are now six mutually influencing variables rather than one.) The dynamics of the system in six dimensions (its phase space) is impossible to envision; however, the (Euclidean) distance that the system has from the two end states (yes and no) can be calculated, yielding a representation of the systems dynamics in two dimensions, which constitute the interpretation or decision-making space (Hutchins, 1995). The left panel of Figure 5 shows the trajectory of the decision-making process for different scenarios. In scenarios 1–3, the curriculum committee starts mildly favoring a decision to include the concept. However, because of different ways in which the same propositions are affecting others, the curriculum committee quickly comes to a supportive decision (trajectory 1), after initial consideration to the contrary (trajectory 2), or, during the considerations, shifts to support not including the concept (trajectory 3). The two end states – inclusion or noninclusion of the concept – constitute attractors where the system does end up. There is a third attractor, which does not constitute an absolute minimum (relaxed state of the system) but a local minimum in which the system can be trapped. (Imagine a dip somewhere on our six-dimensional (fractal) surface that can trap the system in its course.) Scenarios 4–6 (Figure 5, right panel) model the different situations in which the curriculum committee is hung because both decisions (inclusion and noninclusion) are concurrently being supported and, therefore, a consensus cannot be reached. The different trajectories model the different ways in which the committee reaches the state of indecision. In scenario 6, despite the almost certain starting position of not including the concept, the curriculum committee moves toward a state where both funding and not funding are supported equally well. Despite the complexity of the possible outcomes that this model can handle, it is still simplistic when we compare it to real-life social–psychological processes. Although it represents the changing strengths of the propositions 1 through 6, it cannot express the relation between them; it models all statements as if they arise simultaneously rather than unfolding over time, continuously modifying the context in which future statements are made; and it only models six statements as opposed to what is more realistic,
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Figure 5 Modeling group decision making when some propositions favor including a concept in the curriculum (1, 3, and 5 (Figure 3)) and others favor not including the concept (2, 4, 6 (Figure 3)). Slight variations in emphasis with which arguments are made lead to the inclusion (1, 2) or noninclusion of the concept under consideration (3). Sometimes, even with extended discussion, the decision-making group ends in a stalemate (4–6).
a potentially infinite number of statements committee members might make in the process. Nevertheless, the strength of this particular model is that it underscores the notion that minor variations in initial conditions can lead to substantially different outcomes – one decision over another. This model also sufficiently captures the complexity of classroom events in that it demonstrates how, once all statements made in some situation are known, the possible ways in which concrete classroom conversations come to a conclusion (Roth, 2001).
Limitations of Analogies and Metaphors in Curriculum Theorizing In this article, different models and situations are presented to exhibit the suitability of complex systems theories to curriculum theorists. It has been suggested that the mathematical models and graphical representations may be used in metaphorical and analogical ways. However, the effective use of metaphors and analogies requires knowledge not only of source and target domains, but also, and importantly so, the instances in and conditions under which the relationship breaks down. This is also the case for ideas, concepts, and theories from the domain of complex systems. All too often, concepts are taken from the complexity theory, applying them in loose ways that lack rigorous analysis but abound with generic advice (Fenwick, 2004). From this articulation of different models, it is reasonable to conclude that the complex systems theory for actual physical, sociocultural, and cultural–historical contexts may be very different from the one that appears in the simple models presented here and elsewhere, and which figures prevalently in the educational literature. We do not know whether the phenomena of interest in the curriculum
show scale invariance, such as the one depicted in Figure 2. Do terms such as self-similarity and recursion allow us to expand possibilities for curriculum theorizing or restrict such actions? Current thinking across many disciplines, including landscape and physiological, paleo-, freshwater, and estuarine ecology, population interactions, meteorology and climatology, as well as global change, suggests that this is not precisely the case: to forecast the evolution of the complex system, one needs to take into account the particular scale chosen or involve the required scaling rules that, however, have not yet been developed (Costanza, 2003). The influence of scale, resolution, and hierarchy between different levels has not been investigated to any extent and remains a key question in developing coherent models (theories) of complex systems. Thus, although some curriculum theorists use the notion of scale invariance, even the inventor of the concept of fractal geometry takes a more differentiated view: ‘‘no one believes that the world is strictly homogeneous or scaling. . . . One should not be surprised that scaling fractals should be limited to providing first approximations of the natural shapes tackled’’ (Mandelbrot, 1983: 18 and 19). What, if so, are these limits? To what extent do models for simple constructed phenomena such as question–answer pairs scale up and capture truly complex and real events that arise moment to moment in classrooms, schools, and the field of education at large? In what manners do these theoretical frames allow us to understand events such as test taking, differential achievement, or the similarity in transnational curriculum comparisons? In mathematics, the complexity theory is concerned with solving classes of problems (algorithms), some of which are vastly more difficult to solve (compute) than others. It is concerned not with the difficulty of solving the problem (algorithm), but with ‘‘infinite families of
Curriculum and Complex Systems Theory
problems where there would be a general algorithm for finding answers to all the problems of a single family’’ (Penrose, 1989: 141). In curriculum theorizing, we need a similar field concerned with tracking the solution(s) of infinite classes of problems. Fundamentally, the ideas that appear in much of the literature on complex systems are grounded in an epistemology of the same – recursion produced by some mathematical algorithm, the algorithm itself, self, similarity, self-similarity, and scale invariance are all terms that begin with the ideology of sameness and construct difference as the opposite, not the same, lying outside the boundaries of the same. The idea of the curriculum as a strange attractor, though implying the unpredictability of different trajectories, also implies one or more common endpoints that some enacted curriculum or curriculum committee might arrive at. Precisely here lies a problem that difference, which any real physical and cultural–historical system displays, is not modeled as a positive phenomenon, that is, as a phenomenon in and for itself. It is therefore necessary for the complex systems theories that have any value for describing and explaining real systems of interest to begin differently and recognize that any similarity is the outcome of a (decision making) process rather than a priori; sameness and similarity are human productions and not inherent characteristics of nature or culture (Deleuze, 1994). For example, although cooking pots and children’s cups that have a pair of handles appear to us as two different entities that are also substantially distinct on a practical level, from a mathematical perspective, they are considered to be topologically the same; that is, it is possible to morph one into the other without severing or reattaching any of its parts. Consequently, this raises an additional need for educators’ questioning and careful consideration of these forms of mathematics and the models they belong to as suitable structures to constitute the source of analogical thinking about curriculum. The examples and models presented here – social interaction and decision-making processes – are complex and unpredictable. They serve to highlight just how nonsimplistic the real world really is. At the very least, the complex systems theories provide us with ideas that are suggestive of how intractable the world is given that the simplest of phenomena already involve enormous and, perhaps, incalculable complexity.
Bibliography Briggs, J. and Peat, F. D. (1989). Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness. New York: Harper and Row.
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Costanza, R. (2003). A vision of the future of science: Reintegrating the study of humans and the rest of nature. Futures 35, 651–671. Davis, B. (2004). Inventions of Teaching: A Genealogy. Mahwah, NJ: Erlbaum. Davis, B. and Sumara, D. J. (2000). Curriculum forms: On the assumed shapes of knowing and knowledge. Journal of Curriculum Studies 32, 821–845. Deleuze, G. (1994). Difference and Repetition, Patton, P. (trans.). New York: Columbia University Press. Fenwick, T. (2004). A review of chaos theory and higher education: Leadership, planning and policy. Complicity: An International Journal of Complexity and Education 1, 113–116. Frost, R. (1966). The road not taken [1915; 1916]. In Herrig, L., Meller, H, and Su¨hnel, R. (eds.) British and American Classical Poems, p. 112. Braunschweig: Georg Westerman Verlag. Gaudin, F. (2005). Y a-t-il de la non-line´arite´ en se´mantique. In Se`ve, L. (ed.) E´mergence, complexite´ et dialectique, pp 280–288. Paris: Odile Jacob. Guespin-Michel, J. (2005). Re´seau de regulation, boucles de retroaction et dialectique. In Se`ve, L. (ed.) E´mergence, complexite´ et dialectique, pp 243–253. Paris: Odile Jacob. Hutchins, E. (1995). Cognition in the Wild. Cambridge, MA: MIT Press. Lakoff, G. and Johnson, M. (1980). Metaphors We Live by. Chicago, IL: University of Chicago. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. (updated and augmented edition). New York: W.H. Freeman. McClelland, J. L. and Rumelhart, D. E. (1988). Explorations in Parallel Distributed Processing: A Handbook of Models, Programs, and Exercises. Cambridge, MA: MIT. Penrose, R. (1989). The Emperor’s New Mind. Oxford: Oxford University Press. Petitot, J. (2003). Morphodynamical enaction: The case of color. Biological Research 36, 107–112. Pirie, S. E. B. and Thom, J. S. (2001). Thinking through ecological metaphors in mathematics education. In Gunn, S. and Begg, A. (eds.) Mind, Body, and Society: Emerging Understandings of Knowing and Learning, pp 45–52. Melbourne, VIC: University of Melbourne. Prigogine, I. and Stengers, I. (1979). La nouvelle alliance: Me´tamorphose de la science. Paris: E´ditions Gallimard. Roth, W. -M. (2001). Designing as distributed process. Learning and Instruction 11, 211–239. Roth, W. -M. (2006). Learning Science: A Singular Plural Perspective. Rotterdam: Sense Publishers. Roth, W. -M. and Duit, R. (2003). Emergence, flexibility, and stabilization of language in a physics classroom. Journal for Research in Science Teaching 40, 869–897. Thom, R. (1979). Woru¨ber soll man sich wundern. In Maurin, K., Michalski, K., and Rudolph, E. (eds.) Offene Systeme II: Logic und Zeit, pp 41–107. Stuttgart: Klett-Cotta.
Further Reading Elman, J. L., Bates, E. A., Johnson, M. H., et al. (1996). Rethinking Innateness: A Connectionist Perspective on Development. Cambridge, MA: MIT Press. Emmeche, C. (1994). The Garden in the Machine: The Emerging Science of Artificial Life. Princeton, NJ: Princeton University Press. Gleick, J. (1988). Chaos: Making a New Science. New York: Penguin. Hayles, N. K. (ed.) (1991). Chaos and Order: Complex Dynamics in Literature and Science. Chicago, IL: University of Chicago Press. Raijmakers, M. E. J., van Koten, S., and Molenaar, P. C. M. (1996). On the validity of simulating stage wise development by means of PDP networks: Application of catastrophe analysis and an experimental test of rule-like network performance. Cognitive Science 20, 101–136.