CHAPTER I - THE NATURE OF THE STUDY: THE PARADIGM PROBLEM

CHAPTER ONE - THE NATURE OF THE STUDY, THE PARADIGM PROBLEM

• Assessment of the Current Paradigms in Education
  • Mental Provisioning
  • An Ecological Problem
  • Problem and Hypotheses
• Hypotheses Rationale
  • Foundational Relationships
    • The Educational Paradigm Framework
    • The Industrial Age Framework
      • Figure 1.1. Educational Paradigm Model
        
    • New Directions. Reconceptualist Axioms
      • Figure 1-2. Industrial Age Paradigm
        
    • Chaos and Determinism
      • Figure 1-3. Chaotic Dynamics Paradigm
        
• Limitations and Guidelines
• Organization of the Study

Chapter I

THE NATURE OF THE STUDY: THE PARADIGM PROBLEM

Overall, this study seeks to appraise an alternative vision of education and research that will require painting in broad brush strokes in the manner of a generalist, cutting across vast reaches of time, content and research areas. Here the concept of a paradigm is understood to refer to a broad world view composed of many interacting levels. This analysis considers four levels in the ecology of education: practice, research methodology and the explanatory levels of axioms and systems assumptions. In part, the study seeks to clarify the issue of interaction in empirical research, which will raise the issue of the nature and the effect of empirical research itself on educational practice and the issue of the incommensurability of the multitude of educational paradigms. It is then, in part, a critical analysis of historical and conceptual concerns. It is also a normative study, for in its appraisal it notes implications that make general recommendations for policy and practice in the field. It is analysis of old theory and creation and appraisal of new in the spirit of transformational change for the purpose of stimulating educational renewal.

Assessment of the Current Paradigms in Education

Mental Provisioning

One assumption is that there is an identifiable dominant paradigm in educational research. As noted by Popkewitz, "one of the ironies of contemporary social science is that a particular and narrow conception of science has come to dominate social research" (1984, p.2). Second, key tenets of this paradigm are under persistent non-trivial attack from a variety of sources (e.g.: Allender, 1986; Eisner, 1985; Lincoln and Guba, 1985; Lincoln, 1989; Pinar, 1987; Popkewitz, 1989; Romberg, 1987; Sawada and Caley, 1985). Further, the number of alternatives to the educational research of this dominant industrial age paradigm are increasing in number and in vigor (Allender, 1986). "It has never been more true that standards for educational research are threatening to change. How much, or whether they should change is part of an energetic and exciting debate that has proponents from all sides" (Allender, 1986, p.173).

Some initial labeling of theoretical camps is needed for further discussion. The dominant paradigm will be referred to as the industrial age paradigm or the scientific method paradigm. This position is meant to include both the logical positivists as well as the post-positivists as exemplified by Phillips (1987). The paradigm tested in this study will be referred to as the chaotic dynamics paradigm or chaos paradigm. However, there is some difficulty in contriving a label for the current paradigms that have arisen to challenge the industrial paradigm. The current challengers need further ideas to resolve "their uncoordinated positions" (Allender, 1986, p.177). As Phillips (1987) notes, they appear primarily united by their dislike of the industrial age paradigm, but as the phrase "anti-industrial age paradigmists" is awkward, the term reconceptualists or the phrase reconceptualist paradigms shall be used. (Pinar (1988) notes there is not universal acceptance of this label which he brought to prominence in an earlier decade (1975). No slant is intended by its use here.) The reader should not at this point find in this labelling an absolute taking of sides for the reconceptualists by the proposed chaos paradigm, since the perspective of the chaos paradigm presented in later chapters will make contributions to both of these theoretical camps. As Mazurek (1988) has shown in the savaging of some reconstructionists (Lincoln & Guba, 1985), their case has its problems as well. Though a more dispassionate reviewer (Tierney, 1988) at least grants them respect in their pointing to ground-breaking positions, for new constructions require the denial and failure of important aspects of the dominant paradigm (Kuhn, 1970).

In the area of educational practice, another assumption is that the focus has moved beyond debating whether change should occur to focusing on what and how much change should occur. The debate about practice is equally as vigorous and composed of equally as diverse ideas as in the debate over research theory and methodology. The plethora of national reports serves as testament to this (e.g.: Adler, M. J., 1982; Boyer, E.L., 1983; Goodlad, J. I., 1983; National Commission on Excellence in Education, 1983; Twentieth Century Fund, 1983). The production and analysis of these reports continues (Howe, 1984; Lazerson, 1986). In spite of their divergent suggested solutions and lack of consensus, these national reports are united in the claim that dramatic and significant change is needed in our educational system.

So, the situation at the level of research and theory is characterized by an uneasy detente and at the level of practice is characterized by heading in many seemingly conflicting directions at the same time. Lack of agreement in theory makes it difficult to deny or choose between or guide competing programs of practice. It implies a relativity in which all things are justifiable and in which inter-paradigm communication is difficult if not impossible. Inquirers seeking development past this detente in theory and splintered direction in practice may find radical innovation essential.

An Ecological Problem

In the industrial paradigm, certain foundational principles undergird research and practice. They include a number of axioms (reductionism and mechanism), which will be considered more fully in a moment. I claim that the axioms of this paradigm in turn rest on a more basic perception of systems. Stated succinctly, with time, systems converge. In the language of the industrial paradigm:

Since experimental design does not enable scientists to make exact predictions of the outcome of individual experiments, statistical analysis remains dependent on an assumption [my emphasis] that random fluctuations must in the long-run average out - indeed, at a rate and with a probability that has been calculated in the tables. With that assumption [my emphasis], it has become possible since Fisher not only to identify causes, but to estimate their magnitude.

The considerable degree to which this assumption represents hidden curriculum became more manifest after searching through library shelves of introductory and higher level statistical textbooks without finding explicit reference to this basic operating assumption. This observation supports the contention of David Krathwohl, a former president of the American Educational Research Association, that "causal explanation... [is] a concept routinely used in research method books, but rarely examined in them" (1985, xiii).

From the perspective of the dominant paradigm, research productivity depends on steering away from random aspects of systems and towards convergent aspects. "Research should be conducted in such a way that those who question its outcome can repeat it and obtain the same results" (Williamson, et al., 1982, p.14). Repeatable patterns can be validated by the scientific method. They therefore naturally have garnered the lion's share of scientific attention, for research empowers through the detection of convergency which provides predictability and the potential for control. Where aspects of the social and individual system behavior are important to human interests, the absence of convergency can be considered a failure of method and the motivation to seek another means of finding such convergency.

In a contrasting paradigm, the reconceptualist paradigm, the axioms of holism and self-organization undergird the levels of research and practice. The belief that the formation and long term nature of systems important to education are generally random or indeterministic supports these axioms (e.g., Cherryholmes, 1987; Derrida, 1981; Foucault, 1980; Nodding, 1984). On this point, some reconceptualists are forward and explicit, "it is impossible to distinguish causes from effects" (Lincoln and Guba, 1985, p.38). (Important variations on indeterminism will be considered later.) As proof of the non-deterministic nature of the dominant paradigm, critics cite the dismal failure of educational researchers to produce the practical results claimed for the goal of convergent research in a number of areas, e.g. the study of teaching (e.g., Garrison, 1987; Giordano, 1983; Broudy, 1983) Within this perceptual ecology, two possible frames for this problem emerge. In one frame, the problems of educational research and practice come from the failure to correctly do research, the lack of research or the failure of practitioners to utilize what is known from correctly done research. This tact represents a point of view of long standing. In the other frame, the frame of this study, the problems come from the failures of basic perceptions on which much research and practice is based. These perceptual problems will be seen as based on differing views or models of system behavior. For example, the messy nature of professional, social and educational practice (Schon, 1983) and the unique nature of instructional design espoused for situated learning (Streibel, 1989) resonates with the indeterministic view. In contrast, the more rigid work of certain theorists (e.g., Phillips, 1987) and logical-empiricists (e.g., Krathwohl, 1985; Rogers, 1983, 1989) or the general practice of educational computing (e.g., Streibel, 1986) can be seen to favor a deterministic view.

There is a third view that may link these two ends of the problem of the ecology of educational inquiry. However, it seems to rescue the problem of theory and practice with but another paradox. There are deterministic systems that become indeterministic. There are two such types of deterministic systems (Ekeland, 1988): 1. the interacting values of the variables are assumed to an infinite number of decimal places; 2. the interacting variables have values of limited accuracy. In this second type, the interaction of the variables in these models so multiplies the margins of error, that scientific validation may only be possible for some very limited period of time. It is this second type of model, with variables of limited accuracy, which appears the best fit of these two choices for educational environments and will be further explored.

The discovery of problems with interaction was raised for educators by Cronbach (1975) with some emphasis. Interaction has since been seen as a steady source of confounding in a variety of areas, for example in innovation theory (Downs, 1976) leading to a "virtual paucity" (Mahajan, 1985, p.71) of prediction with diffusion models. More recent examples would include educational computing (Clark, 1985) and interactive video (Clark, 1989). As Romberg and Zarinnia note of the correlation matrix, "the basic problem is not that the matrix was wrong, but merely that it was inadequately simple and insufficiently flexible to accommodate new theoretical developments. It makes the assumption that items, cells, columns, and rows are independent" (1987, p.170). Interaction, then, appears to be a common problem in educational research.

The breakthroughs in dealing with interaction, unfortunately for social science, came from researchers in the hard sciences. What has emerged from this empirical scientific work is a body of interdisciplinary literature focusing on interaction and the chaotic dynamics of systems. The concept is sometimes referred to as deterministic chaos. The dynamics of these systems demonstrate the opposite of the assumption stated by Porter (1986) that random fluctuations must in the long-run average out. That is, chaotic dynamics shows fluctuations that do not converge and average out over time.

The issue of interaction is important from another point of view as well, for it raises another curious paradox in education. Interaction for the purpose of validating research has a negative effect (Cronbach, 1975) and therefore suppressing the interaction of variables is generally sought. At the classroom level, interaction has more than a positive effect, it is a pre-requisite for learning and the facilitation of higher-order thinking skills, and therefore stimulating greater depth of interaction is sought. These higher-order skills emphasize innovation and creativity which in turn promote indeterminism. The divergency inherent in innovation and creativity carries interesting parallels to the divergency inherent in the nonlinear system dynamics of deterministic chaos. Yet it appears that what must be minimized at the research level to promote predictability, must be maximized at the practice level to promote unpredictability. The paradox raises concerns about the negative impact of types of research on education. Has the broad acceptance of the industrial age paradigm produced a general conflict of interest between educational research and practice?

This concern over the general behavior of systems and their relationship to the large ecology of theory-practice linkage forms a general problem area from which many questions can be drawn: Are there variations to the understanding of convergency? To what degree is this belief in convergency dominant in educational thought? Is it adequate in describing educational events? If it is not adequate, what would be the implications of the interaction of this inadequacy with the other levels of the educational paradigm? What can replace it? Can the system dynamics research on the unpredictability of chaotic systems be applied? Do the implications of chaos provide greater coherence to the challengers of the industrial age paradigm of education? Could an integration of chaotic dynamics with the reconceptualist paradigms provide a synthesis that can replace the dominant paradigm?

Problem and Hypotheses

From this general question, this thesis derives two hypotheses:

1. the model of interaction developed in the study of chaotic dynamics is a more compelling model for education than the model of interaction generally assumed by the dominant and prior reconceptualist paradigms;

2. applying this radical model of interaction, other aspects of educational theory and practice can reasonably integrate with chaotic dynamics to provide a new model for educational inquiry.

Hypotheses Rationale

The second hypothesis derived from the larger educational problem is motivated by the stifling of innovative practice and with the lack of timely adjustment of the educational system to changing social needs. Variance and thereby innovation is reduced through both explicit and implicit phenomena. A characteristic of educational systems is their fixed hierarchies with top-down control structures. Educational systems have largely moved towards certain designs for the purpose of achieving greater control and greater prediction (and improved conditions) (Kliebard, 1986). These hierarchies are part of a system "that is almost ideally designed to thwart change" (Oettinger, 1969). Some observers find that the rigidity in practice noted by Oettinger is equally linked to and highly influenced by the dominant paradigm, hence their reference to the factory atmosphere and graded lock-step nature of classrooms (Romberg et al, 1987; Sawada & Caley, 1985). The implicit aspect of the system consists of researchers and policy makers settling down on a best recommendation and then bringing other parts of the educational system into the business of cloning or converging on their approach. Even the very study of innovation itself, within which the study of educational innovation consumes significant attention, has its research perceptions bounded by the industrial age belief in long term convergence (Rogers, 1983). As a consequence of this dominance, each new educational innovation is highly likely to be influenced by this paradigm. For example, educational computing appears to be particularly influenced by the dominant paradigm (Streibel, 1986, 1989). In summary, this suggests that the common systems of research and practice and the industrial age paradigm of which they are part, have become part of the problem, and that even radical changes that stay within the industrial age paradigm may not serve as part of the solution.

A beginning step in this analysis is examination of the industrial paradigm's assumption of and need for convergency. At its root is the belief that the isolation of convergency was essential for problem solving, that the rational way for humans to solve their problems was to develop top-down hierarchical structures. Central to critiquing root assumptions is the examination of the alternatives to convergency, especially the divergency created by chaotic dynamics, and to examine the spread of its application to other disciplines and to consider its study and relevance to education.

Foundational Relationships

The Educational Paradigm Framework

Figure 1-1. Educational Paradigm Model

The Industrial Age Framework

More precise definition of these key terms is in order. Reductionism (Ackoff, 1974):

...consists of the belief that everything in the world and every experience of it can be reduced, decomposed, or disassembled down to ultimately simple elements, indivisible parts. These parts were taken to be atoms in physics; elementary substances in chemistry, cells in biology; monads, directly observables, and basic instincts, drives, motives and needs in psychology; and individual or primary groups in sociology. Reality was taken to reside in these elements.

Reductionism gave rise to an analytical way of thinking about the world, a way of seeking explanations and, hence, of gaining understanding of it. Analysis consists, first, of taking apart something to be explained - disassembling it, if possible, down to the indivisible (and, hopefully, independent) parts of which it is composed; secondly, of explaining the behavior of these parts; and, finally, of aggregating these partial explanations into an explanation of the whole.

Mechanism represents a commitment to causal thinking. ...The world was a machine.

Determinism consists of the belief that effects are completely determined by causes. ...All relationships between parts and between parts and the whole were believed to be explainable by using only one ultimately simple relationship, cause-effect. One thing or event was taken to be the cause of another, its effect, if it was both necessary and sufficient for the other.

There are other aspects of mechanism that deserve further note. Mechanism is derived from the word "machine." Our understanding of machines includes the idea that machines do not make themselves, they are designed and are controlled in a top-down command hierarchy. In addition, they are made of parts and can be explained by the actions of the parts in cause and effect fashion.

Ackoff's articulation of these concepts is consistent with recent definitions in the Random House Dictionary of the English Language (1987):

reductionism: 1. the theory that every complex phenomena, especially in biology or psychology can be explained by analyzing the simplest most basic physical mechanisms that are in operation during the phenomena. 2. the practice of simplifying a complex idea, issue, condition or the like, especially to the point of minimizing, obscuring or distorting it.

mechanism: the theory that everything in the universe is produced by matter in motion; in philosophy, the view that all natural processes are explicable in terms of Newtonian mechanics.

determinism: the doctrine that all events, including human choices and decisions have sufficient causes.

The concept of determinism deserves further discussion. This thesis does not challenge the assumption that all events, including human ones, have causes. It chooses to examine the nature of the belief in another assumption linked to determinism, that humans can determine those causes simplistically enough to do so in a reliably repeatable fashion. Many of the problem areas in the industrial age modeling were assumed to be too complex for the present but hope was still held that further simplification was possible or greater computing power would be possible. Hence, one could define two levels of determinism in this model, the simple and the intractable. The next chapter expands this discussion of models of determinism.

These three aforementioned axioms undergird an approach to inquiry commonly known as the scientific method:

The scientific process requires the formation of a research question in order to fit an experimental design. This involves simplification of the desired question. For the scientific method to work, all questions must be reduced to a small number of material observables and to an answerable question.

This definition of the process is consistent with the historical definition and other more modern writers:

Descartes (1596-1650), one of the founders of modern science, referred to the foundations of "the admirable science" as being built upon those "ideas easiest to grasp, the simplest, and which can be most directly represented." The method of reason is to reduce involved and obscure propositions step by step to those that are simpler, and then, starting with the intuitive apprehension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.

Much later in this thesis, Dewey's case for the introduction of the scientific method into the humanities and education will show a similar perception of the scientific process or method.

It is worth noting that not all scientists are so carefully neutral in their definition of the scientific method:

According to the fairy-tale, the success of science is the result of a subtle, but carefully balanced combination of inventiveness and control. Scientists have ideas. And they have special methods for improving ideas. The theories of science have passed the test of method. They give a better account of the world than ideas which have not passed the test.

The above definitions are written with full awareness that there are further distinctions that can be made about the philosophy of science including logical positivism, operationalism, empiricism and more (Flannery 1988). However, it is felt that the ideas presented thus far are adequate introduction for the purposes of the arguments to be presented.

The next figure sketches the industrial age paradigm. (See Figure 1-2.) At the highest level, level four, stands educational practice which is hierarchical in nature, deemed generally inadequate by numerous national reports and due for significant change. In addition, all the levels of this model can be seen as part of a hierarchy controlled from the explanatory level, and especially by level one.

Figure 1-2. Industrial Age Paradigm

Level three of figure 1-2 indicates that the primary methodological tool of educational research is the scientific method, composed of a discovery phase ("...a not to be circumscribed creative act" (Krathwohl, 1985, p. xiii) and a confirmation stage (Krathwohl, 1985). The undergirding explanatory level consists of two levels. Level two holds the axioms of reductionism and mechanism. This level in turn depends on an assumption about the nature of deterministic system behavior in level one. In this case the essential level one assumption is convergency (Porter, 1986).

The characteristics of each individual level have a considerable amount of historical experience behind them in education. Yet, as has been noted, the educational system of which they are a part is under significant pressure to change and improve. It is reasonable to conclude that a need has developed to supersede this classic paradigm. However, a feature of a dominant paradigm is its resilience. "It is important to realize that living under the normal metaphor means that any initiative fighting to overthrow the metaphor will be met with awesome stabilizing forces" (Sawada and Caley, 1985). To take the risk of abandoning the tenets that have demonstrated such scientific power when there is little promise of a better paradigm poses serious difficulty. The alternative must supersede by a better account with potentially more advantageous results.

New Directions. Reconceptualist Axioms

Holism supplements yet supersedes reductionism. There is a sense that the whole is greater than the sum of its parts. "A system, viewed structurally, is a divisible whole; but viewed functionally it is an indivisible whole in the sense that some of its essential properties are lost in taking it apart" (Ackoff, 1974, p.3). A separated subsystem is different when independent than when integrated. The action of integrated systems can create behavior that no subsystem can do independently, e.g. writing. In other words, the interaction of parts creates new variables or parts. When the interaction ends through the removal of selected parts for isolated and controlled examination, these interaction-created variables can disappear, assuming they were visible in the first place.

This idea of holism is closely related to self-organization, which supplements yet supersedes mechanism through observing that internal fluctuations in a system can create their own unique order without any centralized command (Prigogine, 1977, 1984). The members organize themselves. Critical amounts of turbulence caused by energy and/or information flow allow members of the system to connect without top-down directions.

Jantsch (1980) notes a wide body of work in this area providing accounts of higher level organization:

Intuitive attempts to apply the same basic principles of self-organization, which are found at the levels of simple chemical and precellular systems, also to higher levels of evolution, have resulted in astonishingly realistic descriptions of the dynamics of ecological, sociobiological and sociocultural systems (Eigen and Winkler, 1975; Jantsch, 1975; Prigogine, 1976; Nicolis and Prigogine, 1977; Haken, 1977).

A parallel claim for a new paradigm based on holism and self-organization can also be seen emerging in medicine (Foss and Rothenberg, 1987). The paradigm shell represents a modification of their paradigm model.

Certain curriculum scholars have also identified and discussed the value of these concepts of holism and self-organization (Doll, 1986; Romberg, 1984; Romberg et al, 1987; Sawada and Caley, 1985) for building a new educational paradigm. These scholars in turn were following the tracks laid out by general system theorists (Ackoff, 1974; Bertalanffy, 1968) and later the nonlinear system theory of Prigogine (1977, 1980, 1984).

They have used the concepts of self-organizationist writers like Prigogine in a variety of ways. They make their points about holism and self-organization clearly and in detail. However, the concept of unpredictability receives much less explanation and emphasis. Just how this unpredictability comes about is not clear and curious linkages indicate a need for further thought. For example, integrating holism and self-organization with a multi-causal model that is to be judged by its "predictive power" (Romberg et al, 1987) takes a stand that chaos theory would open to question.

Various reconceptualist camps have put these axioms to their own purposes in various ways. As issues in determinism are central to my theme, the variations and conflicts with regard to determinism will receive further discussion. Keeping in mind that the idiosyncratic nature of the reconceptualists makes establishing agreement on any issue complex, Popkewitz's position will be used to represent the general position of the critical theorists and the position of Lincoln and Guba will represent the general position of the naturalists. The overall position taken here is that the explanations of determinism from both theoretical camps suffer equally from inadequacy and/or contradiction.

Popkewitz (1984) takes positions that assume both deterministic and indeterministic frames of reference. An indeterministic view seems required for his dialectical contention that systems of belief continually arrive and fade from power beyond anyones ability to control. This supports a political agenda which seeks to undermine faith in domineering political systems. Further, corrective possibilities are "not contained in the common belief that science provides specific guides for the organization of the present and future. Science, as in any form of knowledge, is not able to provide such positive guidance" (p.193). But a deterministic and top-down hierarchical view is necessary to believe that change can be directed and controlled: "The function of critical theory is to understand the relations among value, interest, and action and, to paraphrase Marx, to change the world, not to describe it" (p.45). Though this change is not open to creation, but "rather, the possibilities of the intellectual are in taking a negative stance towards our social conditions" (p.193) that is to articulate problematic issues such as race, gender and poverty and to act to refute them. The critical theorists appear to need a model that bridges or explains a fairly wide gap between their indeterminism and their determinism.

Lincoln and Guba (1985) appear to take a simpler position. Nature is indeterministic. One can only describe and appraise. This is not to say that action would not change things, but rather that any particular action provides no guarantee that any predicted outcomes will occur. Aspects of their writing categorically reject results suggestive of deterministic human events. But their reference or evidence is inadequate to the degree to which they emphatically reject determinism. Further, they also must face internal inconsistencies.

Lincoln and Guba's primary reference is Werner Heisenberg's indeterminacy principle:

This principle tells us that (1) at a subatomic level the future state of a particle is in principle not predictable, and (2) the act of experimentation to find its state will itself determine the observed state.... It means that ambiguity about the future is a condition of nature.

There are two problems with this idea. First, the logic has a fault. It is a long way from subatomic particles and the quantum effect to Western culture. By their own theory, particles self-organize and the number of self-organizing scales between the quantum level and U.S. classroom culture is enormous. There is no part of their theory that would explain why the classroom scale or other scales could not organize in a fashion facilitating deterministic analysis. This is not to say that such a theory is impossible, but rather they do not provide such. Second, one would surmise that the field of atomic particle physics would have died after Heisenberg's ideas became widely known, yet this is hardly the case, for the resulting quantum mechanics became "...one of the most successful theories in the history of science. It has made accurate prediction about a host of atomic, optical and solid state phenomena" (Briggs & Peat, 1989, p.28). The reason for its power is simple enough. Even though results of 1 or a few particles could not be determined exactly, the actions of large numbers of particles did respond deterministically and could be and were treated statistically through Boltzmann's 1870 innovative introduction of probability to physics. Unfortunately for the naturalist argument, at its base, quantum mechanics is a deterministic view based on linear mathematics (Shimony, 1989). But it is important to note that considerable indeterminism is part and parcel of this linear perspective. A major contention of this thesis is that in some classes of deterministic system behavior, probability theory can effectively counter this indeterminism and in others it cannot.

Finally, like the critical theorists, naturalists are reluctant to completely give up the role of the intellectual in controlling events, hence after considerable determinism and cause-effect bashing ("...indeterminism is now the basic belief that determinism once was" (p.113)), they conclude that "we believe it is possible to shape affairs in a desired direction, albeit with a good deal of uncertainty" (Lincoln & Guba, 1985, p.151). The account of the naturalists then also contains both contradictory and inadequate foundational elements with regards to the concept of determinism.

There is, then, in this writing of the reconceptualists a strong intuitive sense that human events are unpredictable and indeterministic, but there is no adequate underlying theory to explain it or justify their paradoxical interest in retaining bits of deterministic value structures.

Chaos and Determinism

Chaos has become a scientific term for systems that diverge exponentially with time. (Chapter three will be devoted to explaining this topic.) In brief, in repetitions of a system where as few as two variables mutually interact at a sufficient depth of interaction, the outcomes of the system diverge. Such systems are extremely sensitive to changes in variables so that differences several decimal places beyond human perception at critical points of interaction are sufficient to place the prediction of the system beyond human capacity to follow. This is the equivalent of saying that margins of error multiply so rapidly that prediction is lost. The general mathematical concept is that the variables are functions of each other:

S = f(P,B)

B = f(S,P)

P = f(B,S)

X = R X (1 - X)

The equations are deterministic. Exactly the same inputs yield exact outcomes at all iterations, but miniscule differences may yield dramatically different outcomes. One commonly used metaphor is the idea of a stone poised at the top of a hill. The tiniest of deviations in the beginning shove or in shoves along the hill may produce very different results in the end. Prediction may be possible for some period of time, but given the nature of chaos and that there are different routes into chaos, even the length of this time is not predictable.

The evidence that such aforementioned diverging behavior is integral to healthy systems gives support to the claim for a more advantageous account of systems in education. Specific examples of how biological and human physiological systems use chaotic dynamics as integral to their design not only raises the perspective of chaotic dynamics as a sign of health in living systems, (contrary to its effect in mechanical systems), but indicate that reliably repeatable phenomena may be a sign of disease (Pool, 1989; Rapp, 1985, 1987; Skarda & Freeman, 1987).

Given this new perspective on determinism, and as holism and self-organization are not found wanting, only their indeterministic base, the alternative paradigm proposed here, is a wedding of these two axioms to the more rigorous view of divergent cause and effect espoused in modeling chaotic dynamics. The model in figure 1-3 briefly represents some of the basic ideas of the proposed chaotic dynamics paradigm.

In general, in this paradigm, the emphasis in educational practice within level four shifts from hierarchy to heterarchy or to distributed bottom-up structures. For now, level three should be understood as a common meeting ground for new approaches to the educational research (theory and methods) that borrow from both creative (discovery) science (e.g.: Reason and Rowan, 1981; Romberg, 1987) and art (e.g.: Eisner, 1985), a level that generally denies the possibility of an objective version of Krathwohl's (1985) confirmation phase. The explanatory level two consists of the previously discussed axioms: holism, self-organization. The system level, level one, assumes divergent behavior over time, that is deterministic chaos.

Figure 1-3. Chaotic Dynamics Paradigm

For both paradigm models, further details with regards to level three and four will await chapters four and five. The level two concerns with holism and self-organization and the systems concepts of level one await more detailed discussion in chapters two and three.

The result of this general shift to alternative paradigm concepts is described by Toulmin: "This 'modern' science ...has begun to be superceded by 'post-modern' science; and in certain crucial respects, as a result, scientists have broken through bounds and restrictions that were placed on the scientific enterprise by its original founders" (1981, p.70). The means to break these bonds have come from surprising quarters. The last major product of the industrial age was a condensation of all that a machine could be conceived to be, the ultimate deterministic factory in a box, the computer. Yet, it was the advent of deterministic computer and calculator technology (Franks, 1989; Pagels, 1988) that was ironically necessary for the relatively recent mathematical discovery of chaos and of the universality of the circumstances for fundamental unpredictability in chaotic dynamics (Feigenbaum, 1978, 1979, 1981). Mathematicians knowledgeable about Poincare's work on the stability of the solar system may disagree with Franks's and Pagel's claim for the role of computer technology, but as Mandelbrot (1986) notes, Poincare and others of his era (including Cantor, Peano, Hausdorff and Sierpinski) failed to "see and develop a kinship among their constructions, and handled each of them as a 'monster' or 'exceptional set' which thoroughly missed their true significance" (p. 159). Poincare later abandoned this work, saying "...these things are so bizarre that I cannot bear to contemplate them" (Briggs & Peat, 1989, p.29). Lorenz's computer-based discovery of a chaotic system (1963a, 1964, 1966) and its later embellishment by many others adds to this century's set of beliefs about fundamental indeterminancy established by Poincare, Einstein, Heisenberg, Godel and others. One of the implications of chaotic dynamics for educators is that the last strands in the bond of education to the industrial age paradigm can be broken. This is similar in spirit to the thought of the physicist Joseph Ford (1989) that the discovery of the fundamental indeterminism in nonlinear system experiments breaks the last bond to Newton for science in general. This bond is the bond of long range predictability.

But while setting up the grounds to contrast these alternative paradigms, it must be noted that we are also establishing potential grounds for communication across these paradigms. The grounds for communication will be an exploration of the types and range of behaviors that systems are now known to inhabit. In other words, the creation of a taxonomy which could end the incommensurability of the alternative paradigms (Lincoln & Guba, 1985) is an important development on the route of testing whether chaotic dynamics provides a more compelling model for education and reasonably integrates with other aspects of educational theory and practice.

In summary of this introduction to foundational relationships, the reigning paradigm of intellectual inquiry in education has grown through, contributed to and benefited from the long rise of the industrial age. It developed practices based on the axioms of reductionism, mechanism and determinism. The reconceptualists began a challenge to the explanatory layer of this paradigm. The new concept of systems behavior provided by chaotic dynamics contributes to solving fundamental problems with the reconceptualists' position. Greater analysis of chaotic systems will be needed to reveal to what degree it supplements their position and to what degree it could change it. To the degree this new challenge withstands criticism, the claim can transform the paradigm for educational inquiry into a new frame of reference for the information age and implicate basic changes in curriculum and instruction.

Limitations and Guidelines

This thesis employs the work of specialized areas but seeks the horizontal perspective of a generalist. I have traded the exhaustiveness of the specialist for comprehensiveness of dimension, a comprehensiveness not only in the number of layers of the proposed paradigm, but in the multidisciplinary and multi-specialty approach to validation of the concept. One of the limitations of such a horizontal frame of reference is that it is open to the specialist's concern that significant developers of a piece of the puzzle have not been given their due. The interdiscipinary view is wide: philosophy, art, the history of science, mathematics, physics, chemistry, biology, medicine, neurology, sociology, and organizational management. The educational perspective was informed by several areas of the educational spectrum.: early childhood, innovation theory, educational computing, learning theory, history, and statistics. However to prevent the overall framework from being lost in detail, many important developers may not be recognized. I believe, though, that the footnotes and bibliographies of those cited are of significant enough character that they provide an avenue for those who would seek to pursue further any individual piece of the overall design.

Mumford comments on the important value of the generalist's view:

...(T)he generalist has a special office, that of bringing together widely separated fields, prudently fenced in by specialists, into a larger common area, visible only from the air. Only by forfeiting the detail can the over-all pattern be seen, though once that pattern is visible new details, unseen even by the most thorough and competent field workers digging through the buried strata, may become visible. The generalist's competence lies not in unearthing new evidence but in putting together authentic fragments that are accidentally, or sometimes arbitrarily, separated, because specialists tend to abide too rigorously by a gentlemen's agreement not to invade each other's territory. Although this makes for safety and social harmony, it ignores the fact that the phenomena studied do not hold to the same principles.

Mumford also notes certain rules the generalist must follow when seeking a more meaningful mosaic from scattered pieces: do not chip a piece to force it to fit; look in unlikely places for pieces; and grant specialist's authority on the shape of a piece.

The first and last points of Mumford's rules are simple enough when working within the range of "normal science" (Kuhn, 1970). But when confronting a paradigm shift, the shape (perception) of a piece changes when passing from one paradigm to another. This observation that hypothesis strongly influences what is observed is referred to as the theory-laden nature of observation as discussed by Hanson (1958). The specialist from the old or dominant paradigm may believe the piece "has been chipped" while to one who has seen the light of a different paradigm, the piece has been "transformed." This in turn creates a debate among specialists to which a vote on the facts contributes little, generally splitting on "party" or paradigm lines. As a consequence, the question of "right" cannot be settled by a debate on the facts. Instead the debate depends on both the depth of dissatisfaction with the old paradigm and the rhetoric and vision of the new. These complications do not completely invalidate Mumford's points but rather suggest they should be followed with some flexibility and awareness of their limitations.

There are further guidelines that can be suggested in reviewing certain areas of the literature. Much of the chaos literature is of such a new light that the possibility of mistranslating or the building of the reader's mistrust through the paraphrasing of novel ideas must slant the style in favor of longer and fuller quotations than perhaps necessary were the literature reviews covering ground that was more familiar territory to readers. This however does not relieve the writer from an obligation to clarity by including paraphrases of long quotations where the thread of the argument is in danger of breaking.

Another guideline requires establishing priority in tackling broad issues of theory and practice. The priority sought here is related to Hanson's point about the theory-laden nature of observation that continues to be emphasized in the literature (Garrison, 1988). Significant reform in the theory and practice of curriculum and instruction requires significant reform of theoretical foundations. As a consequence, primary space is devoted to formulating the groundwork for practical applications related to curriculum content, media, teachers and students, though the practical concerns will be considered in the last chapter as implications. Their presentation should be seen more as suggestive than as a complete analysis of all possibilities. But the theory laden nature of observation further emphasizes the importance of this underlying explanatory level for our practice of research and our practice of education through curriculum and instruction. Our expectations and actions as learners, teachers and curriculum designers interact heavily with these elemental perceptions.

The rationale for defense of the hypotheses of this thesis ultimately rests on the structure for a paradigm shift made by Kuhn (1970). For Kuhn, a scientific revolution represents a paradigm shift. A dominant or reigning paradigm frames our approach to the world and to our specialties. It is "prerequisite to perception itself" (Kuhn, 1970, p. 113). From Kuhn's point of view, to change the paradigm does change the world and changes our perception of our personal specialty. The choice of paradigm determines the general nature of the results. Inseparable levels of practice, theory, method, law and instrumentation compose a paradigm. One does not pick out a piece here and there to consider themselves members of a paradigm. Challengers to a paradigm must equally provide replacement for all. Past achievements are the foundation upon which continued research (normal science) stands. Normal science works by solving puzzles with the paradigm. Ad hoc modifications of theory begin the divergence from a paradigm. Some modifications are always necessary. But when the pressure to modify to fit the system becomes too great, when the fit is increasingly perceived as poor, alternative paradigms which are at the periphery become candidates for successor. This debate over successorship forms metascience. The dethroned become a special case of the new theory. Kuhn's ideas, then, form a critical perspective for the hypotheses in question. Does the proposed paradigm deny important aspects of the dominant model while presenting a creative coherent structure of its own?

This chapter noted the nature of the significant pieces of the new paradigm puzzle that have already been identified and seeks to complete the puzzle through concentration on particular issues. Does the model of interaction developed in the study of chaotic dynamics produce a more compelling model for education than the model of interaction generally assumed by the dominant and reconceptualist paradigms? Applying this radical model of interaction, do other aspects of educational theory reasonably and coherently integrate with chaotic dynamics to provide a new model for educational inquiry?

Organization of the Study

Does the model of interaction developed in the study of chaotic dynamics produce a more compelling model for education than the model of interaction generally assumed by the dominant and reconceptualist paradigms? Chapter two considers the old range of system behavior and its understanding of interaction. It focuses on determinism and convergence and their relationship to educational aspects of the industrial age paradigm. Chapter three describes the basic features of chaos theory. It then tracks the concepts of chaos and divergence across disciplines to those bordering education, noting that the educationally applicable ideas of this concept increase as they approach the field of education.

Can other aspects of educational theory and practice reasonably integrate with chaotic dynamics to provide a new model for educational inquiry? The latter part of chapter three discusses various qualities that emerge in the consideration of the chaos literature and their general application to education. Chapter four presents more detailed models to aid our organization and comparison of the industrial age and the chaotic dynamics paradigm. In appraising the proposed chaotic paradigm, chapter four notes the value and the problems of an accommodation that would support a meta-paradigm that would include both the industrial age and chaotic dynamics perspective. In the close of chapter four a new paradigm emerges.

Chapter five provides a divergent approach to considering the educational implications of nonlinear dynamics. This "chapter" delivers classroom activities and education that connect the new paradigm that emerges from chapter four to hands-on procedures for the improvement of curriculum and instruction in the present and future. It cannot best do this in the context of a formal chapter of writing. In the spirit of action research or in the model of researcher as participant observer, chapter five is a web site in which the experience transforms the engaged reader into both a writer, a learner and a reflective participant.

Top of Chapter One.	Chapter Two - Part 1.	A Chaotic Paradigm: Table of Contents
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