CHAPTER FOUR - INTEGRATION

CHAPTER FOUR - INTEGRATION

• The Industrial Age Model
• The Chaotic Model
  • The Archetype
    • The tidal bore.
    • Soliton theory.
  • Dissipative Structures Research
    • Physics - Benard cells.
    • Chemistry.
    • Biology.
    • Brains.
    • Organizations.
    • Summary.
  • Fractals
  • The Chaotic Model Revisited
• Education
  • The Developmentalists
  • Functionalists and Structuralists
  • Intuition
  • Instructional Design and Technology
• The Taxonomy Revisited
• Inquiry

CHAPTER FOUR. INTEGRATION

The Industrial Age Model

As we have defined reductionism, the breaking of a system into its constituent parts, nonlinear chaotic systems fall short of this definition. The chaotic model shows only that some few parts can be taken out and controlled in the short term or observed in the long term, that a complete break-out of the parts and their interactions is impossible. This concept that a complete break-out or reduction is impossible is also not a new idea to the proponents of the industrial model. But their recognition of this problem required that theory be developed that said that lack of consideration of some factors of various degrees of importance was inconsequential. This work was discussed in chapter two, including the Simon-Ando theorem and the Ando-Fisher theorem, which state that "...even when influences which have been neglected have had time to make themselves fully felt, the relative behavior of the variables within any set will be approximately the same as would have been the case had those influences never existed..." (Fisher & Ando, 1971, p.192). Such a defense of reductionism indicates that even these reductionists realized the inherent problem with the axiom. With the consideration of chaotic dynamics now behind us, it is easier to understand how the extreme sensitivity of chaotic dynamics provides the reason and the circumstances to falsify these theorems.

Figure 4-1. Industrial Age Paradigm

Researchers working with these reductionist and convergent axioms of the industrial paradigm also developed certain perspectives on distribution. As was previously discussed in chapter two, "...many, if not most of the statistical tests that are used by researchers can be derived from a single theory... a distribution called the normal distribution" (Marascuilo & Serlin, 1988, p.248). Again chaos theory provides a countering view. One of the characteristics of chaotic dynamics is their fractal dimension. In Benoit Mandelbrot's study of fractals, The Fractal Geometry of Nature, Mandelbrot takes a contrary position on distribution:

Many scholars resort to the Gaussian probability distribution in their disquisitions, without feeling that this choice has to be justified. Either it is the only distribution they know intimately and trust, or they believe it accounts for the distribution of every random quantity in Nature, from conscripts' heights to astronomers' errors of measurement.

"Actually, this last belief is quite without foundation. This Essay includes many examples that show the world to be full of grossly non-Gaussian phenomena" (Mandelbrot, 1983, plate C10).

The third axiom to be considered is mechanism. Mechanism is the mechanical assemblage of the engineer reduced system for the purpose of controlling the actions of a machine. This axiom implies that there is no butterfly effect in machines. But as with reductionism, the machinists have always known about the butterfly in their machines. No matter how much oil they put on it, friction and entropy grind all machines to a halt with time. However, machinists have so confined the butterfly that users of the machines can leap to a fresh one (to a system whose initial points are still well defined) as the old model becomes too erratic for practical use. For machinists to succeed in building chains of cause and effect, the depth of interaction between the components must be nil. To the degree this independence is successful, reductionism works. Independence must prevail or dominate.

Chaos theory derives from the observation of systems at the other end of the continuum of the depth of interaction, of systems whose components are greatly dependent on each other. Through nonlinearity, that is feedback, they change each other. Chaos theory does not deny that machines exist, only that other important aspects of behavior also exist. In such nonlinear systems, the model of the hierarchical or top-down designed machine fails and is bound to fail. This problem of linearity is at the root of the failure of the artificial intelligence community to deliver on its promises (Dreyfus, 1972; Dreyfus & Dreyfus, 1986). Chaos theory raises the question of whether this problem of linearity is also at the root of the failure of educational generalizations (Cronbach, 1975) to endure. Reductionist researchers must face the difficult task of arguing that education and learning are not just primarily linear events, but totally linear, for as has been discussed and will be discussed further, even the smallest nonlinear term in a linear network is sufficient for chaotic dynamics to build.

This concept of a continuum along the characteristic of depth of interaction, Hegel's internal and external interaction, deserves a further look. Defining the world we study based on just the stark features of each end of this continuum can imply a false discontinuity. All manner of variations in between highly divergent and highly convergent systems are possible. Further, as the logistic equation shows, the variations could easily be descriptions of various behaviors of one system. But for systems dominated by the potential for great depth of interaction among the constituent parts, such systems do not seem a best fit for any level of the industrial age paradigm. Order in professional practice for such nonlinear systems is not best achieved, cannot be achieved by the reductionist and mechanistic designs of the industrial age paradigm.

Further detail, however, must be added to the industrial age paradigm to show the vertical nature of the model. (See Figure 4-2. Industrial Age Paradigm.) Drawing back from the detail at the various levels of the industrial age paradigm, the overall relationship between the levels is more apparent. The arrows show that the relationships between the levels are hierarchical. The upward arrow indicates that the beliefs and assumptions at this level reappear to guide problem formation and solution at levels three and four. It is important to remember that problems in levels three and four occur on a variety of scales, from big to small, yet the foundational levels guide thought and solution in all cases. The directives of this foundational level serve as the spectacles that filter and inform perceptions.

Figure 4-2. Industrial Age Paradigm.

In this industrial age model, the arrow from the higher levels is deflected from reaching and serving as an interactive reality check confirming or disconfirming the lower foundational principles. The deflection occurs as a result of the previously discussed logical possibility argument. This argument takes irregularities and turns them into problems for basic research to rethink with new and different experiments. This deflection occurs in part because of the assumption of level one, that systems converge given sufficient time. Once those convergent patterns are identified, the researcher is to build on them to find further pattern. The patterns of reductionism, mechanism and determinism had already been found and had been repeatedly found useful. If systems converge, there is no reason to look deeper at older convergent patterns when irregularities occur, rather these patterns are the stand from which new attacks are made on the higher level problem until it cracks. One works from the known to the unknown. The continued absence of a repeating controllable pattern merely indicates that one must creatively find a different way or different means to search for it at the basic research level.

In summary, the puzzle piece of chaos theory, does not fit well with the industrial paradigm. Note that this is not a criticism of the internal consistency of the industrial model, but rather a recognition of the range within which the model is effective and the boundaries beyond which it is not. Any criticism should be reserved for the degree this model has been applied beyond its boundaries in education.

A number of educators have criticized this model for precisely this reason, application beyond its boundaries. Further examples of this criticism, more focused on issues of educational research and instructional practice will be presented. These examples are not direct support for the chaotic model, but rather relate to Kuhn's observation of paradigm shifts. It takes major criticism of the prior paradigm for new paradigms to come to center stage.

Romberg & Stewart (1987) primarily focused on assessment, but also focused on industrial age aspects of the student, the teacher and the curriculum [the quotes are from selected chapters from Romberg & Stewart, 1987]:

Assessment:

This chapter argues that the nature, forms, purposes and design of major models of assessment are dominated by the prevailing, old world view, helping to perpetuate it, and that there is an iterative relationship which inhibits change. [Author's notation: for example, in this chapter the authors refute content by behavior matrices or profile achievement tests.] (Romberg & Zarinnia, 1987, p.153)

...the accompanying objective testing invariably results in poor, minority and handicapped students placing at the low end of the curve. It stamps with failure the groups most dependent on the educational system for improvement and acts as a dangerous social filter. (Romberg & Zarinnia, 1987, p.166)

Students:

...(A) focus on isolated parts essentially trains students in a series of routines without educating them to grasp the overall picture.... ...This may well be a major barrier to the development of expert thinking, which usually focuses on function rather than form; naive perceptions persist (Resnick & Gelman, 1985). (Zarinnia & Romberg, 1987, p.41)

Teachers:

Attempts to prevent emasculation of curricula include a focus on textbook selection and conscious efforts to bypass teachers by relegating them to the role of manager and the consequence of this was to invalidate the teacher's skills. (Zarinnia & Romberg, 1987, p.45)

Math is segmented into a hierarchy of behavioral objectives; next, the steps through that hierarchy were mechanized via textbooks, worksheets, and tests. Teaching was dehumanized to the point that the teacher need do little but manage the production line. (Romberg, 1987, p.137)

Curriculum:

Behaviorism reflects the application of the engineering approach of scientific management to the problems of education. Scientific management rested on three basic principles: specialization of work through the simplification of individual tasks, predetermined rules for coordinating the tasks, and detailed monitoring of performance (Reich, 1983). These microprinciples pervaded American education with the same thoroughness with which they were applied in the economy. They dominated the breakdown of administrative processes, the building-block approach of Carnegie units, the content and structure of textbooks, belief in the textbook as an effective tool for transmitting content, the structure of university education, and monitoring and evaluation. Hence emerged the notion of progress through the mastery of simple steps, the development of learning hierarchies, explicit directions, daily lesson plans, frequent quizzes, objective testing of the smallest steps, scope and sequence curricula.

Unfortunately, these are only the more obvious aspects. One consequence of such meticulous planning is that it renders the unplanned unlikely. A second is that a system designed to eliminate human error and the element of risk also eliminates innovation. A third is that, like factory work, it is crashingly dull, uninspiring, and unmemorable except for its boredom, for personal involvement and the mnemonics of the unexpected are nonexistent. (Romberg & Zarinnia, 1987, p.161)

There are a number of criticisms of the industrial model resonant with the Romberg and Zarinnia critique and also related to the axioms of the industrial model. These include: curriculum as a linear course to be run; testing as both objective and predictive; curriculum that should be uniform for all; an IQ that cannot be changed appreciably; factor analysis; the school system as a machine; the normal curve as a grading ideal; the top-down hierarchical school clocked by tight time schedules and rules (e.g., Doll, 1986; Sawada & Caley, 1985). Such criticisms do not propose or confirm an alternative, but accelerate the search for one.

The Chaotic Model

Figure 4-3. Chaotic Dynamics Paradigm

A conceptual emphasis of the study of chaotic dynamics in the last chapter was the unpredictability brought about by the sensitivity of the models to the slightest variation in initial conditions. However, looked at from another perspective, the dynamics also contain an implicit and intricate order, an emphasis of this chapter. The water drip study (see figure 4-4: water drip data) reacquaints us with chaotic dynamics research design. In measuring the time intervals between drips from a leaking faucet, it would be reasonable to expect a random process that graphed to a shapeless blob. Instead the nonlinear interaction between springiness of the drip and its weight reveal a well defined pattern that matches graphing of Henon's chaotic attractor. This patterning and organizing feature of chaotic dynamics provides a lead to the axiom of self-organization.

Figure 4-4. Water drip data

Self-Organization Theory

As with building an understanding of chaos, some simple metaphors aid in the creation of intuition for a new perception of order. This perception assumes or begins with an interactive background of chaotic dynamics. Weather served as a prototypical example of chaotic dynamics. The tidal bore soliton can serve the same role for the paradigm's axiom of self-organization.

The Archetype

The tidal bore.

While riding his horse along a canal, he noticed that the sudden stop of a boat formed a 30 foot long wave that maintained its well defined shape till he lost it after a couple of miles in the windings of the canal. The study and understanding of such waves possessed him for the rest of his life. The value of such work failed to appear relevant to his peers and not until the 1970's did such work play an important role in science (Briggs & Peat, 1989). All over the earth, on a similar though larger scale, when strong tides push waves into shallower rivers, nonlinear coupling creates solitary waves that can travel upstream for long distances. These tidal bores have been recorded up to 30 feet in height and and traveled for hundreds of miles upstream, that is against the natural current of the river. Today physicists refer to this unnatural wave as a soliton or solitary wave.

Soliton theory.

Computer modeling shows that the addition of even minute nonlinear terms is sufficient to dominate or transform well-behaved linear models into nonlinear forms with the potential to create solitons, instead of creating a gradual even disbursement of energy throughout the system. The Fermi-Pasta-Ulam (1965) model of this phenomena done on the computer Manaic I in the 1950's:

...shows that the nonlinear world is holistic, it's a world where everything is interconnected, so there must always be a subtle order present. Even what appears on the surface as a disorder contains a high degree of implicit correlation. Sometimes this below the surface correlation can be triggered and emerges to shape the system. Soliton behavior is, therefore, a mirror of chaos. On one side of the mirror, the orderly system falls victim to an attracting chaos; on the other, the chaotic system discovers the potentiality in its interactions for an attracting order. On one side, a simple regular system reveals its implicit complexity. On the other, complexity reveals its implicit coherence. (Peat & Briggs, 1989, p.127)

The triggering of self-organization does not require an immediately prior sharp triggering event. The nonlinear self-focusing can occur in mid-ocean without seismic initiation and such giant waves have been photographed by satellite, that is as a by-product of extended turbulence. The triggering primarily occurs as a by-product, a derivation of far-from-equilibrium turbulent chaos. Communication and information are intrinsic components of these nonlinear, that is, feedback driven systems. The outcome of this information transfer, a self-organized system, may "...appear simple and regular. But this is deceptive, for the feedback in simple-appearing orders like the soliton wave is also unanalyzably complex" (Peat & Briggs, 1989, p.143). But, eventually even solitons dissipate away.

The theory then of the chaotic paradigm is saying not that its three axioms are synonymous, but that they theoretically are strongly linked and overlapping distinctions derived from observable, describable and to an extent, quantifiable, phenomena. This analysis raises the further question as to why the chaos concept would be singled out to label the paradigm when all three play such strong roles. Though I see chaotic dynamics as the more common background phenomena, a primordial soup, the question has merit. The reader should be on the lookout for concepts shared by all axioms that might serve better as an overarching label. This concern will be discussed in greater depth later.

The soliton is but an intense form of a variety of self-organizing structures that Prigogine (1984) refers to as dissipative structures, open systems which take advantage of energy flow and produce entropy (more chaos as waste by-product). "As the moon takes advantage of gravity to stay in orbit, so dissipative structures take advantage of entropy" (Briggs & Peat, 1989, p.139). The study of self-organization or synergetics, as it is also called (Haken, 1984), provides a number of examples from physics, biology and chemistry (see also Schieve & Allen, 1982).

Dissipative Structures Research

Physics - Benard cells.

Benard cells are just one more example or model of an inherent tendency of the simplest forms of matter to organize themselves when conditions fall within critical ranges, ranges however interspersed with chaotic regimes. "True, the type of complexity achieved is rather modest, but nevertheless it presents characteristics which were usually ascribed exclusively to biological systems. More importantly, far from challenging the laws of physics, complexity appears to be an inevitable consequence of them when suitable conditions are fulfilled" (Nicolis, 1989, p.319).

Chemistry.

The common chemicals of the Belousov-Zhabotinskii (BZ) reaction provide classic examples of such development. "A typical preparation consists of cerium sulphate Ce2 (SO4)3, or another cerium salt, malonic acid, CH2 (COOH)2 and potassium bromate, KBrO3, dissolved in sulphuric acid. The evolution of the system can be followed by the naked eye by the use of a colouring substance such as, for instance, ferroin, which gives a red colour when there is an excess of ions of Fe2+ and a blue one when there is an excess of Fe3+" (Nicolis, 1989, p.321). A well stirred BZ reaction with a slow pumping reaches chemical equilibrium. Faster rates of pumping yield such a correlation of all the molecules that a pattern forms of red, blue, red, blue and so on, a chemical clock. If the stirring ends, a variety of beautiful forms of circles, and rotating spirals and multi-armed spirals can be formed. This physico-chemical self-organization begins to look rather life-like.

Biology.

Abstracting from this process and other biological processes, it is "...natural to conjecture that these gradients provide the tissue with a 'coordinate system' conveying positional information to the individual cells, thanks to which they can recognize their position with respect to their partners" (Nicolis, 1989, p.328). A gradient is by definition, not in equilibrium. That is, nonequilibrium is essential to the concepts of time and space.

At another step up the biological ladder, the intense interaction between early cellular species is now generally viewed as creating the opportunity for innovative new species that combined separate species into new ones. A primary point of Margulis and Sagan (1986) is that the survivors were those that cooperated, not competed to determine which was the stronger. Many biologists now see the mitochondria which continue to possess their own separate DNA as at one time a separate species that produced oxygen that was poisoning themselves and other species but when merged gave great power to new forms of life. Augros and Stanciu (1987) and others (Gould, 1988) argue that this same conclusion of evolution by cooperation can be reached by the study of the interaction of higher species. Competition is rare across species for the species interact with each other and their environment to provide a unique niche for themselves.

Human beings can also be seen, from their brain to their toes, as a highly interactive network of special teams of microbes. Our existence depends on their cooperation. "In fact, all life is a form of cooperation, an expression of feedback arising out of the flux of chaos" (Briggs & Peat, 1989, p.156)).

Brains.

Guilleman and others surveyed a number of studies of brain cells. "Whereas neuroendocrine physiology has been dominated by theories based on deterministic feedback loops in the context of control theory, these studies suggest nonlinear, cooperative metabolic interaction among the cells, multidetermined function, and emergent global properties --in short, a synergetic system" (Guilleman et al., 1983, p. 160). As a consequence the structure of nervous systems can also be seen as a highly open-ended hierarchy or seen as a heterarchy. "The positive feed-back between effectiveness of connections and synchrony of signals is the basis of an organization process. It is proposed that the cells can dynamically express grouping into blocks by synchronizing their activity in time (with a resolution of a few milliseconds) (Malburg, 1983, p. 248-9). In A Comparative Analysis of Structure and Chaos in Models of Single Nerve Cells and Circadian Rhythm, it was concluded that:

...in all the examples studied here, the parameters can be chosen in such a way that asymptotic states are nearly attained after 20 or 30 iteration steps. This is consistent with the time constants typical of the buildup of short-term memory on 1 hand (the order of a second or a little less), and, on the other hand, with what is required for "evaluating" a statistical moment in a neuronal network, namely a few tens of milliseconds. (Bienenstock, 1983, p. 262)

Skarda and Freeman's study (1985) of the olfactory bulb of rabbits (its chaotic aspects discussed in chapter three) showed that the olfactory bulb left its low level chaotic state and organized momentarily when an odor was detected. New odors eventually lead to unique patterns for a particular smell.

Prigogine summarized this work on the brain by noting the symbiotic aspect of the discussed axioms: "chaos is what makes life and intelligence possible. The brain has been selected to become so unstable that the smallest effect can lead to the formation of order" (Briggs & Peat, 1989, p.166).

Organizations.

Prigogine notes two paths out of chaos into spontaneous self-organization, through "oscillations or to spatial structures" (1984, p. 135). He also notes that such dynamic systems do not allow for the possibility of assuming the dynamics of closed systems (fixed or convergent systems). Their stability comes from elsewhere (Prigogine, 1984) as with the nonlinear feedback formation of solitons. A spinning top could serve as a reasonable metaphor. Its spin, along with its ample supply of energy, provides dynamic resistance to tilting and perturbation. The social organism is kept stable through a common desire to communicate and common means of communication. The more successful organizational structures provide a more fluid dynamic means of communication.

Experiments by Alex Bavelos (1952) reveal much about management structures. Bavelos investigated the strategies and feelings of people with different expertise engaged in tasks of various difficulty from simple to complex. For example, in one experiment, a group of five members must find the only common symbol in a deck of cards. Each member is isolated in soundproof rooms with slots for messages. The configuration of the channels for the slots can be changed. In this experiment, two systems of communication were compared. The circle and the star configuration are respectively, a and b. The degree of task difficulty is varied for each configuration. (See figure 4-5.) As soon as members of the group believe they know the common symbol, a key is pressed. After all press the same key, the session ends.

Figure 4-5 Circle-Star Communication Comparison.

When symbol description was easy, the members of the circle configuration believed they were fast and efficient, felt fine and spread identification of leadership evenly over all members. The members of the star configuration produced very different results. Though these groups were twice as fast, they felt themselves slow and inefficient, for which some idiot in the team was to blame. For this group, 94% identified the center of the configuration to be the leader.

When symbol description is difficult, the democratic group works just as well, however, somewhat slower than before. They still feel fine and think they are doing well. The dramatic change is with the star groups: depending on the "strangeness" of the symbols, the groups disintegrate sooner or later. Participants walk out in anger, the "idiots" multiply, and blame is passed from one to others. Indeed, when later the communication records are studied, the star performers soon stop talking about symbols. They start calling each other names.

...Expanding language is what keeps the people in the circle configuration going. As the records show, names for the funny looking things are soon invented, some referential, "lion", "cow", etc. or new ones "splots", "bimbim", etc. names that are either kicked around, modified, or kept; and when adopted by the group, the find-the-common-symbol problem is back to finding common symbols, by ignoring fuzzy objects. (Von Foerster, 1984, p. 22)

Malik and Probst (1984) note that basic assumptions are made about the cause of social order. It is currently accepted wisdom that non-human living systems (ecologies and organisms) change through a gradual process of differentiation, random mutation and competitive selection that is called evolution. However, human systems, our social institutions, are seen in the context of intentional planning and design. Theirs emerges, ours is constructed, theirs is non-rational, ours is rational and goal oriented. Further, the assumption is made that constructed, goal oriented systems are better. But if we allow evolution to also apply to social organizations, then a different perspective can develop:

At the center of this is the view that a firm together with its context is a self-organizing system which can be organized and guided only to a limited extent through conscious planned intervention. This view is in contradiction with the vast majority of dominant views in the worlds of science and practice...but not a sufficient reason to reject it lock, stock and barrel. (Malik & Probst, 1984, p. 108)

Malik and Probst buttress their case with three arguments. First, there is ample evidence to support the view that there is another system besides command hierarchy systems. Not only have polycentric systems as self-organizing systems long been promoted by small group research but, command hierarchies have a narrower range of effectiveness:

The really decisive difference between a system based on the pattern of a command hierarchy and one based on self-organization and polycentric systems consists in the fact that a self-organizing system displays considerably greater adaptability than the command hierarchy. A polycentric system is able to process much more information and to perform mutual adjustment of a larger number of relations than the other type of system. This can also be easily proved quantitatively. (p. 110)

Second, it is not necessary for anyone to intentionally produce rules. "...they arise in the process of evolution as a result of the interaction of individuals with one another and with their environment, through a process analogous to mutation and selection. They are often not known to the persons concerned, but their functioning can be observed" (p. 110). This idea should not be so surprising. The field of anthropology devotes much of its work to revealing hidden and invisible rules and their function. "If these rules did not predate the attempt by any form of management to intervene, then the task of managing a system would probably be impossible" (p. 111).

Third, evolution-oriented social theory has discovered that "there are exceedingly appropriate social organizations which are neither completely natural nor completely artificial, but belong to a third, largely ignored category, namely those systems which are indeed the result of human action but not of human intention, goals or planning" (p. 109). Further, such results are common. "Activity in the social context often produces unintended and unforeseen secondary effects" (p. 109). They note that most managers have discovered that the outcomes of negotiations often do not correspond to the intentions of negotiations. They further note: "The same applies to the raising of children. Here too we have much less control over events than we like to imagine. We cannot command a child to love us or respect us; we can only create conditions which favor the genesis of desirable reciprocal relationships" (p. 113).

Whereas Weidlich and Haag (1983) previously noted that many social systems were at times "planned", Malik and Probst (1984) state that even those that are planned are much less planned then commonly believed. Von Foerster notes that democratic self-organizing structures are much more effective than autocratic structures as the level of complexity and stress in the environment increases. What a curious irony that increasingly chaotic environments require increasing structural chaos to maintain order, that is to retain a common goal and group unity. Chaos overcomes chaos. Order requires chaos.

Summary.

Self-organizing systems begin "... as the search for new attractors arising when a system is driven away from its state of equilibrium" (Nicolis, 1989, p.331). These attractors are necessary conditions to meet environmental change and to make changes to the environment. "Clearly, the coexistence of multiple attractors constitutes the natural model of systems capable of showing adaptive behavior and of performing regulatory tasks" (Nicolis, 1989, p.332). But such capacity requires considerable chaotic activity or chaotic charge:

One of the main conclusions emerging from the preceding sections is that self-organization rests on the ability of nonlinear far-from-equilibrium dynamical systems to create and sustain states of matter displaying regulatory and other remarkable properties, which would be exceedingly improbable under equilibrium conditions. (Nicolis, 1989, p.336)

Nicolis makes a further statement that has considerable relevance for teachers seeking to understand an individual's capacity: "We can therefore say that nonequilibrium reveals the potentialities hidden in the nonlinearities, which remain 'dormant' at or near equilibrium" (Nicolis, 1989, p.332). To reveal potentiality, educators must reveal to students the need to explore, to risk and to interact at levels of increasing depth, for these acts un-center the learner and build interest and motivation.

The robustness and omnipresence of life on the planet perhaps obscure its frailty, a frailty also present in the engagement to learn and to risk. Self-organization is bounded:

In somewhat anthropomorphic terms, order appears to be a compromise between two antagonists: the nonlinear chemical-like process, which through fluctuations sends continuously but incoherently 'innovating signals' to the system, and the transport-like process which captures, relays and stabilizes them. Disturbing the delicate balance between these two competing actors leads to such qualitative changes as an erratic state in which each element of the system acts on its own, or on the contrary, a homeostatic fossil-like state in which fluctuations are crushed and a full uniformity is imposed. Complexity and self-organisation appear to be limited on both sides by two different kinds of states of disorder. (Nicolis, 1989, p.341).

Lack of interaction and communication is also a major contributor to isolated behavior. Equally, one way communication in human systems, hierarchies, dampen and crush self-organization.

As in any paradigm, research in the paradigm must work within the fundamental guidelines laid out in the explanatory levels. To inquire about self-organizing systems, research methodology changes:

Nonlinear models differ from linear ones in a number of ways. Rather than trying to figure out all the chains of causality, the modeler looks for nodes where feedback loops join and tries to capture as many of the important loops as possible in the system's "picture." Rather than shaping the model to make a forecast about future events or to exercise some central control, the nonlinear modeler is content to perturb the model trying out different variables in order to learn about the systems critical points and its homeostasis (resistance to change). The modeler is not seeking to control the complex system by quantifying it and masterings its causality; (s)he wants to increase her "intuitions" about how the system works so (s)he can interact with it more harmoniously. Thus, the development of the system model exemplifies the shift that the science of chaos and change is making from quantitative reductionism to a qualitative holistic appreciation of dynamics" (Briggs & Peat, 1989, p.177).

These observations of self-organizing phenomena yield one more axiom for the explanatory level, the axiom of self-organization: systems in the chaotic regime far from equilibrium will develop self-organizing systems,.

Fractals

A fractal was defined earlier as an irregular fragment, the most elemental unit of an irregular discontinuous or grainy pattern that can be iterated to create the larger pattern. But fractal or fractals also refer to the larger pattern, the arrangement that is the result of the interaction over time between chaos and the nonlinear pulsings of self-organization:

Mandelbrot sees reality as discontinuous, where the sequence of change is independent of scale or magnification. In mathematics, this is analogous to the cantor set. Mandelbrot's contribution was to find a way of measuring its variation, or what he calls its fractal dimension. There is a rich variety of example for which there is adequate record: the change of daily price and monthly price change in cotton; the change of noise in telecommunication signals over 1 minute versus one hour or one day; the change in the levels of the Nile river; the change in the severity of earthquakes over a geographic area; the change in the distribution of personal income in a free market economy; the change in the distribution of animal circulatory systems (Gleick, 1986).

But fractals are also the means by which self-organization creates structure. In this sense, all living structures and forms are fractal, and their fractal nature has something to say about the nature of the turbulence and the patterns of nonlinear organization and their respective competition and cooperation. Fractal structures are nature's way of saying that given the innovative form of organization and this type of turbulence, this is how it is for now. The mathematician Barnsley (1986) provides a simple way to conceptualize fractals: "Fractals...are often the tell-tales of the interfaces between order and chaos. Coastlines separate the churning sea and the stationary land" (Barnsley, 1986, p.53).

For researchers to generate a fractal, one must also define the process of iteration. Of course, when we see the fractal image, we do not see the iteration process behind it that had to exist to make the image or organism. All living forms have achieved some means of transferring energy waves into structures, of iterating basic designs, however alterable in the future. Those that are most adaptable would have the best means for altering their fractal structure, a process that could also be called re-invention, a term discussed earlier that is drawn from innovation theory.

Educators require more than the motivation (chaos) and the means for an innovative idea (self-organization). They require the means for retention and storage and further action. Fractals assess the outlines of such storage and the process, discontinuous, irregular and bumpy. Clearly visible at each level of self-organization there is a concrete physical element, a structure or structural change or in terms of human evolution, a tool or model. If examined in ever finer detail this fractal organization continues to reveal a world of discontinuities and infinite scale. Such fractal structure suggests that educational assessment could reveal a similar pattern in human understanding.

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The Chaotic Model Revisited

Further features can be added to show the relationships between the levels. (See figure 4-6.) An important distinction between the chaotic paradigm, figure 4-6 and the industrial paradigm (figure 4-2) is that the logical possibility deflection is gone and that the arrows show that all higher levels interact with all lower levels. The model itself is heterarchical. The logical possibility deflector is removed for several reasons. The logical possibility defense was already weakened by the previously discussed subjectivity of the quantum argument and the indeterminancy of Godel's axioms, but there is another claim made by chaotic dynamics itself. It is a point driven by the level one assumptions of figure 4-6, the industrial age paradigm. There is no logical possibility of ever getting ahead of diverging systems, of multiplying margins of error. Compared with the precision of weather data, educational margins of error are huge, further reducing prediction and chance of control. The infinitely precise measurement of an infinite time for the measurement is not a logical possibility either. The industrial age paradigm and its scientific method are simply unlikely containers for chaotic systems.

The introduction of chaotic dynamics at level one and the concepts of holism, self-organization and chaos (indeterminism) of level two comprise the explanatory strategy, the most fundamental level. The upward arrow indicates that the beliefs and assumptions at this level reappear to guide problem formation and solution at levels three and four. It is important to remember that problems in levels three and four occur on a variety of scales, from big to small problems, yet the foundational levels guide thought and solution in all cases and that the directives of this foundational level serve as the spectacles that filter and inform our perceptions. Yet, the arrow into the explanatory strategy sections indicates that levels three and four can serve also as an interactive reality check confirming or disconfirming foundational principles.

Figure 4-6. Relationships in the Chaotic Paradigm.

Level three (figure 4-6. the chaotic paradigm) holds the previously presented basic research on chaotic dynamics of a number of disciplines that education has historically referred to (physics, biology, sociology and others), but also indicates the formation of new fields within and across these disciplines that need further mining by educators: ecology and evolutionary biology and nonlinear science in general. The study of art will be seen later to have an important place here too. In summary, the continued critique of the industrial age paradigm model has shown the contents of figure 4-2 (the industrial paradigm) to contain a number of serious problems that suggest the need for an alternative model. Figure 4-6, the chaotic paradigm, shows that such an alternative is possible to conceive across all levels of the model.

Education

How dynamic to make models for instructional design has been a central issue for at least the field of educational technology. The educational system`s capacity to forecast and then deliver what students need to learn is at the root of the debate (Streibel, 1986, 1988, 1989). Streibel's critics continue to treat the debate as a technical one, technical issues for which they have found and can find answers. It would be of aid to that dialog to recall that this issue is a long standing one with deep philosophical roots as well. It would equally be of benefit to this offering to recall that others without benefit of chaos theory have claimed an indeterministic and unpredictable aspect of education is both realistic and of value. Archambault summarizes Dewey's position on unpredictability in the curriculum:

The exact aims of instruction cannot be legislated, for they depend on a cluster of variables that are unique to a particular place and time. For a given situation, short-range aims must be relatively specific. Yet any school situation, since it is experimental in nature, has a quality of unpredictability. To try to specify, in exact detail, the precise knowledge that a student is to achieve, is to consider ends as remote, distinct, and separate from practical contingencies and the dynamic purposes of pupils. (1964, p.xxiii)

This thread of appreciation for the chaotic and indeterministic was also carried forward by Piaget. Piaget's work is in part precedent for Prigogine's work, and though the contribution is not direct, at the end of Piaget's career he was in communication with and empathetic with Prigogine's work in the years before such work would earn Prigogine the Nobel prize in chemistry.

The Developmentalists

Like the development of a seed, the process of growth and learning is an internal process played out according to inner values or eigenvalues, an unfolding according to the nature of the particular seed. Hall's 1883 article "The Contents of Children's Minds" is concerned with the "...natural order of the development of the child" (Kliebard, 1986, p.28). With a firm belief in the power to determine cause and effect, Hall proposed that the:

...curriculum riddle could be solved with ever more scientific data, not only with respect to the different stages of child and adolescent development, but on the nature of learning. From such knowledge, a curriculum in harmony with the child's real interests, needs and learning patterns could be derived. The curriculum could then become the means by which the natural power within the child could be unharnessed. (Kliebard, 1986, p.28).

Hall also recognized that strong forces and turbulence characterized children, noting that "the pupil is in the age of spontaneous variation which at no period of life is so great" (Kliebard, 1986, p.14). Later, Kilpatrick's generalization of Stimson's agriculture home-project method in his 1918 article The Project Method provided a primary means for pupils to act on their personal interests and values. This in turn developed into activity or experience curriculum (Kliebard, 1986). Maria Montessori also contributed significantly to the movement toward greater independence and individual responsibility. The great sensitivity of chaotic environments reminds one of her concern for the teacher to provide just a hint, touch or key and let the rest develop on its own. "The idea is to open the door only a little, rather than give a guided tour into the interior of all the world's knowledge" (Gebhardt-Steele, 1985, p.7). She gave specific attention to this sensitive period.

As a child developed, certain periods of special sensitivity appear from time to time, and disappear. When they are present the child shows a particular interest in certain objects and exercises, and is able most readily to cope with and learn the matters to which his special sensitivity applies.... (Connell, 1980, p.135)

This does not deny that there are also dissonances in her own work with such openness. There is a tension between the open system side of human growth and development and the closed system side encouraged by industrial age perception. Montessori remained a zealot about the use of specific curriculum materials, their sequence, method and manner. She adhered to the formal disciplinist tradition. She wrote "It is exactly in the repetition of the exercises that the education of the senses consists...[this is] true intellectual gymnastics" (Connell, 1980, p.135).

Whereas developmentalists sought to capture and control these sensitive periods, chaos theory implies otherwise. But whether they can be controlled or not, the chaotic paradigm would see that the project method is an effective contribution toward meeting individual need for self-organization. But more than that, the nonequilibrium necessary for innovation and self-organization requires an intensity and in-depth involvement, not more project exercise.

Functionalists and Structuralists

There are also connections here to chaotic dynamics. In Piaget's last work (Piaget, 1975/1985), when explaining dynamic processes, he cites Prigogine's 1971 work on dynamics (p.3). Prigogine would capture the Nobel prize later in 1977 for his work on dissipative systems, which in turn led to his previously cited work, Order Out of Chaos (1984). But chaos theory has ties to significant features of both of these developmental groups.

There are aspects of structuralism that chaos theory complements and there are others that it critiques. It complements aspects of equilibration. Intelligence forms through a balancing of assimilation against accommodation in a series of stages of equilibrium. Disequilibrium forces accommodation which leads to the next equilibrium (Piaget, 1975/1985). It complements it by meeting its most serious objection. "The principal objection to our hypothesis has, however, been to say that we confine ourselves to description and provide no explanation" (p. 147). Chaos theory demonstrates a functional model that can lead to both oscillations of the Piagetian model. For in the causal chain of chaos models, increased energy input increases the variation of system behavior while low energy input can continue the convergent type operations of assimilation. Yet, as self-organization research has shown there are powerful forms of assimilation motivated by strong energy input. In summary, the information age paradigm as previously delineated, meets all of Piaget's conditions for being a structural explanation. Beilin quotes Piaget, "(i)n short, the notion of structure is composed of three key ideas: the idea of wholeness, the idea of transformation and the idea of self-regulation" (1983, p. 17). These three ideas map onto the previously articulated principles of the information age paradigm and further argues for their completeness: holism, chaos and self-organization.

A conflict occurs, though, with Piaget's insistence on central tendency or equilibrium. He utilizes convergent assumptions about system behavior to describe the process. "There is no need to insist on functional reasons. It is clear that if knowledge is due to assimilation and accommodation, those activities must be equilibrated" (Beilin, 1983, p. 148). The chaos paradigm shows that this is not necessarily so, though its axiom of self-organization does not undo equilibration. Rather, chaotic dynamics adds a stage of disequilibrium, one at least as important as equilibrium. From the chaos paradigm's perspective it is more important, for chaos is the background dynamic that is a prerequisite for powerful nonlinear orders such as solitons to emerge. Piaget's model also provides no description, let alone explanation of the sudden nature of understanding, which the sensitivity of chaotic systems and self-organizing systems describes and explains.

Aspects of functionalism also benefit from and conflict with chaos theory. The chaotic paradigm implies both bottom up development and self-organization that complements functionalism.

Structures such as schemata, to organizational entities that Piaget and Barlett made popular, are conceived of by functionalists as dynamically organized representations. ...This conception of schemata differs considerably from... Piaget's theory, in which the constructs are given a much more logical and theoretical interpretation" (Beilin, 1983, p.23).

Chaos theory implies a unity across diverse and overlapping functions, negating Beilin's criticism of the functionalists that, "Information theory is a collection of models lacking a coherent theory... Without such a coherent theory, an endless number of models is possible for any given task" (Beilin, 1983, p.19). This is precisely the point made by chaotic dynamics. Information theory also lacks explanation of transformation and self-regulation, because the feedback loops and cybernetic principles are used within specific procedures, not interactively across them (Beilin, 1983), yet chaotic dynamics serves as an easy extension of this view.

On the other hand, there is conflict between chaotic dynamics and the functionalists with regards to their "bias towards environmental control of behavior" (Beilin, 1983, p.9). The unpredictable nature of chaotic systems suggest that such dominance is unlikely and unhealthy in the long term. The previous discussion of brain research concurs with such criticism of external control:

the contrast of this concept of organization to the cybernetical (psuedo-) concept of organization is extreme. In no sense are the organized states arbitrarily programmable, nor are they preformed in any palpable or coded form. The organized state is produced by the local interactions within the system, without there being the need for any global or centralized control. (Malburg, 1983, p. 240).

Chaos theory, then, appears to be able to play a role in criticizing and revitalizing aspects of the structuralist dialog and aspects of their work also provide positive feedback for the concepts of the chaotic paradigms. Other aspects of educational theory also reasonably integrate with chaotic dynamics to provide a new model for educational inquiry, notably Nodding's work on intuition.

Intuition

Nodding and Shores (1984) identify many aspects of what intuition is. It is a form of seeing, a "capacity that reveals" (p.53), a feeling of direction, of something at hand. It is separate from cognition (reason) and Will. It is multi-functional, it: operates momentarily and intensely; turns perception inward upon the objects of conception where it gives objects to the analytical side and to Will for further contemplation; gives direction to experience; contacts objects directly in phenomena. Will controls our access to our receptivity to intuition, a receptivity which requires that we "put aside the urge to control and impose" (p.54) but it is not part of our unconscious mind, that is "we cannot catch ourselves in the act of intuiting for intuition is consciousness supreme and therefore unanalyzable" (p. 55)

Intuition is also not accelerated unconscious analytic thinking which the authors refer to as the nonserious view of intuition, a view which would cause problems for a reasonable definition of inference.

To draw an inference, is precisely, to make conscious connections between prior statements (premises) and conclusions (inferences). The view is particularly unsatisfying when we consider the affect that accompanies intuitive accomplishment. Why should the intuitive conclusion come to us with the surprise, clarity and beauty of perception if it is merely a rapid version of reasoning? Why should we so often find ourselves inarticulate when the intuition comes to us?" (p. 50).

We can use chaotic dynamics to theorize some answers to these questions. Gardner (1984) postulated that our brains housed many forms of intelligence of various degrees of development, and though his conception emphasized more their autonomy, interaction between his divisions is not denied. In a more recent summary of the brain, Miskin, Mortimer and Appenzeller (1987) review a highly interactive nonlinear brain in which all divisions are in rapid and simultaneous contact with other divisions. From this and the previously discussed research on brain activity, I choose to view the brain as a vast sea of turbulent activity whatever the divisions, but as Prigogine noted, so sensitive that the slightest impulse is sufficient to generate order and thought. Our will is sufficient to direct many important aspects of thought processing, but the dynamic parallel processing brain can, like soliton formation in mid-ocean produce or generate perceptions of great power and duration, which Will could choose to ignore, but contrary to the account of Nodding and Shores could crash in unbidden even against our Will and unattached to chains of reason. This account provides some explanation for the surprise and inarticulateness that can accompany the arrival of intuitive thought. It is certainly not a mystical account, but it does not deny mystic and religious use of the term intuition to represent revelation and enlightenment.

Nodding and Shores then are quite right to separate truth and knowledge. Intuition can provide us with knowledge, and increase the liklihood of it being truth because it is not filtered by the biases of will or the faulty logic of reason. But it cannot guarantee truth, for the brain can only build nonlinear interactions based on the ideas, constructs and so forth that the brain has gathered. As they say in computer science, GIGO, or garbage in, garbage out. But the potential here is significant, for our chains of logic seem to be bound by strict limits to the number of competing concepts we can entertain in our chains at any one time. But the nonlinear interactions of that turbulent neuronal sea are not so constrained and can mix a far greater number of wave forms in creating intuitive solitons and perhaps other dissipative structures.

The chaos literature also raises the concept of chaos as health and the concept of self-organization as a superior mode for processing information. In this light, heavy use of rational inference and deduction driven thought could be seen too a degree as mentally unhealthy and limiting to the best of our abilities. This is so because the view of intuition is one of being open or receptive, not directive so that it is therefore encouraging nonlinear thought by not letting linear operations dominate.

So far, chaos theory has been integrated with educational concerns in a variety of ways: with the major tenets (holism and self-organization) of the alternative paradigm of the reconceptualists; with a major strand of curriculum theory (the developmentalists), with a recent approach to cognition (intuition). Finally, it will be integrated with a debate in the field that studies instructional design and the means for delivery, in some places called educational technology. As before, such integration considers central aspects of the area and the critique and adjustment that the chaos paradigm involves.

Instructional Design and Technology

The status quo in instructional design and technology (IDT) represents a belief system that claims that the ends in curriculum design can be determined, that objectives can be developed and measured, that delivery systems for a hierarchy of prerequisite skills are available to teach what needs to be taught in accordance with the design and objectives and that computer technology is the delivery system of choice both inside and outside of schools. The status quo is represented by the recent publications of Reigeluth (1989), Heinich (1988) and Merrill (Shore, 1989) and to an extent Damarin (1988).

The reconceptualist case could be represented by Apple (1987) and Streibel (1986, 1988, 1989). A central claim is that curriculum means and ends cannot and should not be so determined. However, the technology (especially computer technology) is deterministically bringing about negative social consequences including the destruction of jobs, deskilling of labor (and teachers and learners) and computers exacerbate the lack of social opportunities for the poor (Apple). They also have other negative consequences for education, reducing intelligence by restricting student agency. That is, it is the deterministic nature of computers to bypass, narrow, impose and constrain the learner (Streibel).

The chaotic paradigm supports the reconceptualist position that the goal of predictability in IDT has serious difficulties, both for the larger question of curriculum design and the more personal question of the learning event and the appropriate educational individualized response. But just as chaos theory rejects the belief in curriculum determinism of IDT, both as a possibility and as personal and social benefit (that is even if it could be so determined, it would be unhealthy) so it consistently rejects technological determinism and concurs with IDT that computers should play a significant and active role with learning. Further, allied with the larger concerns of the paradigm, the chaotic paradigm would support any and all aspects of computers that increase interactions so that new forms of self-organization and turbulence will appear.

In this vein, chaos theory would integrate the conceptual work of artists using computers. Their experience makes an important contribution to this debate and argues strongly for the further integration of the inquiry of both art and science. Pope (1988) not only argues for the expressive potential of computers but notes that the computer is revitalizing persistent ideas in the history of art that have lacked an appropriate medium until now: kineticism, participation and cybernetics. But the questions raised here go deeper than the potential for new art forms. They touch on the capacity of humans to reject that which is not beneficial and the capacity to reinvent if given the opportunity. Part of the problem is that the reconceptualists taken by technological determinism underestimate human capacity. The problem is not with the technology, it is with the inadequacies of the paradigm employed. A deeper look at artistic perception better reveals the location of the problem.

The criticism that technology, especially computers, deterministically forces design and use and therefore is suspect would have a different reception among artists who consider the computer their expressive medium and perhaps among artists in general. For to make such a criticism is to locate your level of development in the stages of apprentice, journeyman and master. From the art perspective it is understood that all mediums have limitations and restrictions whether clay, paint, ballet or the trumpet. There are skills to be learned, concepts to be explored and mastered, limitations to be faced and perhaps overcome. But the status of art and of the artist comes not from pieces dictated by the medium, but by artists of sufficient ability to produce work that rises above and yet through the medium, artists that make innovative and unique contributions linked intimately to the needs of the spirit and time in which they are made and yet, make visible enduring aspects of what it means to be human.

This is of course not so harsh a criticism for a field to say of itself that it is not yet capable, that it is not beyond the apprentice level. But for a field to be constrained from creative acts by its technology and to so indicate to others that such is the case is to announce that the field has reached the end of its time and its issues should be passed out to other fields for further pursuit. Chaos theory allows a different emphasis, one that would revitalize computer utilization. It allows for the computer as an instrument capable of increasing interaction in numerous ways, ways that extend human agency and expressiveness. It refutes equally the curriculum determinism of the status quo and the technological or computer determinism of the reconceptualists.

The Taxonomy Revisited

Disorderly nonlinear chaotic dynamics, in the more complicated view in chapter four, can now also produce patterns of convergency, e.g., the periodicity and torus forms of a soliton and eventually dissipate in the manner of fixed point systems. All the prior convergent categories are not replaced however, as they arise from a different source of explanation. Even more curious, the branches of the taxonomy interact on the nonlinear level, with the increasing complexity of life building on layers which transpose from convergent to divergent categories and back again, layers which thrive in feedback to each other. We have come a long way from Krathwohl's models of cause and effect. But we have borne out Krathwohl's hunch that attention to cause and effect is important, for pursuit of their ramifications suggests a new paradigm for educational inquiry.

Inquiry

The question of an overarching label for this paradigm was raised previously. Is "chaotic" the best description for this paradigm? It was previously argued that chaotic phenomena were the dominant event in nature and that the potential in the turbulence and interaction accounted for the initiation of the structures that exist, and thereby being more fundamental, deserved to be the labelling flag. But if the criteria was the development of identifying terminology that represented values for education suggested by the chaotic paradigm, then the goal of chaos, is inappropriate and one-sided. Education would find no value in chaos for chaos's sake. Education requires both chaos and self-organization to explore the parameters of human potential. But aspects of both what chaotic dynamics manipulates and how organization "becomes" are significant to educational values.

A selection from these values seems a better source for the term to communicate the meaning of this paradigm for education. Chaotic dynamics passes information from one variable to another and in doing so changes the variables and thereby creates more information. Information manipulation is the grist for the educational mill. On the other hand, the concept of becoming is about process, but in chaotic dynamics a process of building that comes not from advance planning, but from the contingencies of the present, a process that I would call emergence. To call the paradigm the information paradigm would highlight concern for communication and the manipulation and creation of information. Yet the phrase does not convey the dynamicism of the paradigm. The better fit is the phrase the emergent paradigm, (see figure 4-7. the emergent paradigm). Yet, chaotic dynamics emerges as a phrase that resonates with the social role that educators must fill, generators of a dynamic that allows individuals to develop to their fullest in an increasingly turbulent and divergent world.

Figure 4-7. The Emergent Paradigm.

A second issue arises in this summation of theory and research surrounding this now titled emergent paradigm. In the process of applying the radical model of chaotic dynamics to a reasonable integration with educational theories, the reasons for a major division in human inquiry and curriculum organization lose their relevance. The chaotic paradigm implies that as science's ability to falsify has decreased another giant step, science should give greater emphasis to its creative side, the stage, that as Krathwohl said, is prerequisite to any effort to confirm or falsify. This suggests the vanishing of the chasm between science and art. The physicists Bohm and Peat have proposed (1987) that science move closer to art from the perception that "scientific truth, like artistic truth, is a matter of endless nuance... [and that] theory is an abstraction of the whole and therefore is in a sense, an illusion" (Briggs & Peat, 1989, p.200). This illusion is an important inoculation of irony for both scientific and artistic inquiry. In the chaotic paradigm, then, art and science lose one of their common distinctions. But there is a second aspect to this. The falsification of the rationale for specialization and reduction requires that any area of inquiry cannot ignore any other. As a consequence, essential to formation of the larger holistic view, art and science must integrate their programs. This development does not foreclose experimental work, destroy statistics or end academia as we know it, but rather will put them to different ends, predominantly creative ends.

As interdisciplinary study cuts against the grain of the general organization of curriculum and instruction in American education, reform along interdisciplinary lines poses special challenges for educational reform. The study of chaotic dynamics serves as case in point. The problems and needs of interdisciplinary study of nonlinear and chaotic phenomena have been reported in conjunction with the National Academy of Sciences and other national policy groups (Feigenbaum et al., 1987). Among their several recommendations, there should be greater international contact with stronger nonlinear programs of study abroad, especially with the French and the Soviets. Typical United States universities are often rigidly defined along traditional disciplines and lack the proper flexibility to handle such study. Also, institutions should create more focused centers for such study. "In any effort to guide this research, however, it is imperative that nonlinear science be recognized for what it is: an inherently interdisciplinary effort not suited to confinement within any single conventional discipline or department" (Feigenbaum, 1987, p. 54). Hence administrative arrangements will need more attention. The reward for participating faculty is a "remarkable breadth of application and the potential to influence both our basic understanding of the world and our daily life" (p. 54). The chaotic paradigm of this thesis provides an opportunity for educators to offer leadership for the study of phenomena of great interest to the field of education and to all disciplines from the field of education which by its very nature has many interdisciplinary needs and interests.

Finally, certain prescriptions for research and practice appear to follow from the emergent paradigm theory, prescriptions that suggest further areas for research and test. From the concept of the butterfly effect, it follows that highly sensitive human environments would benefit from explicit instruction in ethics, supporting Lincoln and Guba's contention "that it is possible to shape affairs in a desired direction, albeit with a good deal of uncertainty" (1985, p.151). From the fractal structure of self-organizing systems (Barnsley, 1986), it follows that educational assessment that seeks to map the edge of an individual's knowledge should seek a fractal edge with its expected discontinuities. But perhaps the principle conclusion derived from chaos theory and the larger emergent paradigm for education is simple and basic: interact. To interact is a fundamental directive for education as a system and the individual as a learner. But to simply interact is insufficient in itself. There are certain qualitative levels that must be reached. Taking to heart the assumption of chaotic dynamics as health in a system, a healthy educational program stimulates interaction into the chaotic regime. A principle contribution of the industrial paradigm for professional practice has been the development of hierarchy, which is a principle tool for conservation. Hierarchy, the organizational pyramid, is inimical to meeting the needs of a rapidly changing world. It is a fundamental roadblock to reaching the chaotic regime in education and learning and must be destroyed. Further, in those systems that have emerged from the chaotic regime, those with the longest survival rates are those that have retained an intimate link with the level of chaotic dynamics beneath them. Educators must find and use the many tools and networks now available that will enable them to stay in touch with and contribute to chaos and self- organization.

Emergent education and the adoption of the chaotic or emergent paradigm await the march of time, but their metaphors in particular provide motivation and hope for their elaboration. To believe in emergent education and the infra-structure of chaos theory is to believe in the butterfly effect metaphor, that a single individual's simplest gesture has the potential to cause significant change even beyond the horizon. No perturbation is too small to be of influence (Ekeland, 1988). To accept self-organization metaphors like the tidal bore or the giant red spot of Jupiter, is to believe that the collective force of individuals acting in light of their own eigen-value, or personal values can be enduring and powerful. The former contrasts with the vanity of individual effort in the face of the overwhelming averaging effect of central tendency. The latter connotes that significant collective effort does not require central authority. Both raise the value and status of teachers and learning.

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