Chaos, a word whose origins go back millennia to creation myths in both the Hebraic-Christian and Greco-Roman traditions, has emerged in the latter part of the twentieth century as one of the “new sciences”: chaos, fractal geometry, complexity. As part of the new sciences, mathematical chaos theory— still rich in fecundity and creation—is the study of equations which model the turbulence now found as natural to our universe. The nature of chaos and its implications for education are discussed in this entry.

Whereas Isaac Newton, in the seventeenth century, saw a simple order reigning in the universe––“Nature is pleased with simplicity”—scientists now find the starry skies filled with “dark” energy, pulsars, galaxies forming and exploding, and “black holes,” devouring all they ensnare. On Earth, an increasing number of earthquakes, tsunamis, hurricanes, tornadoes, and global warmings are devastating—and all unpredictable. This is the sort of turbulence in nature that chaos theorists model and try to predict. Hence the term, deterministic chaos.

Deterministic chaos is not an oxymoron but, rather, is recognition that the universe and world are basically dynamic in nature––not stable in a simple sense but full of “orderly disorder.” There is in nature a complexity that makes long-term prediction impossible and short-term prediction (as in weather forecasting) short range and only probabilistic. Further, this complexity, studied mathematically, shows an interweaving of order with chaos. Chaos concepts not only challenge traditional views of cause-effect (deterministic, predictable) and an either/or frame of order versus chaos, but carry educational implications for ways we teach and design curricula.

The concept that long-term prediction is impossible—prediction devient impossible—comes from Henri Poincaré’s realization in the late 1890s that the mathematics of calculus, useful for Newton’s theory of gravitation between two objects, does not work when a third object (sun, moon, earth) is introduced into the relationship. Nonlinear mathematics is needed to solve such interactions. Nonlinear relationships in a simple X−Y equation are different from the commonly used first order [X1] and second order [X2] equations, where the X’s are predetermined, with the Y’s (as functions of X [F(x)]) being solved for set Xs.

Nonlinear mathematics, by contrast, is recursive in nature; operating by feeding the Y (or function of X [F(x)]) answer back into the equation for an iteration. Thus the “answer” (Y or F(x) ) to the first statement of the equation becomes the next X. And so the process continues, continually recursing: the ninth statement of X depending on the eighth function of X (its Y). In this manner, the results of the iteration are deterministic––by tracing back the X/Y statements to the original X––but nonpredictable, except for a probable guess as to the next iteration. The educational implications of teaching and designing curricula from a nonlinear viewpoint challenge entrenched notions of preset syllabi, lesson plans, and methods of direct instruction.

In the 1960s, Edward Lorenz, a meteorologist with a feeling for the nonlinear, was doing mathematical simulations of weather patterns on his computer––old and slow by today’s standards. Wanting to relook at his data, he submitted the old data back into the computer, but, to save time, rounded off his 6-digit equations to 3 digits. Such a minor difference (.001) should have, according to a 1:1 linear, cause-effect correlation, yielded a minor, almost unnoticeable, difference in the new printout. Using the new printout, Lorenz could check his original work. To his surprise, the new 3-digit equations produced a different set of patterns. Small differences (causes) yielded major differences (effects). Thus, was born chaos theory’s famous dictum: “A butterfly flapping its wings in Rio can cause a typhoon in Tokyo.” Such an accelerated and accumulated sense of development throws into doubt an educational fixity of IQ and even brings into question the “averaging” of grades.

Supercomputers have allowed nonlinear mathematics to venture where mathematical theory has not gone before—into environmental biology, population dynamics, and information theory. All these deal with interactive relationships, where factors are proactive not merely reactive. Examples include population growth (birth-death relations), predator/prey, and message-noise relations. Such system dynamism, far from states with an equilibrium or central focus, is often framed in a logistic equation. This equation, following the nonlinear X/F(x) frame, already described, is F(x) = rx(1−x), where r is a constant––amount of food, space, information––and 1−x is an inversion of the original x, limiting (but not centering) the interactive relationship. Thus the relationship is bounded but still dynamically interactive. Again, the notion of a system being bounded but not centered, and dynamically changing, offers challenges and opportunities in the social sciences. Here the name of Niklas Luhmann stands out.

An interesting aspect of the logistic equation is that as r increases from 1 to 2, doubling occurs (the output in the equation vacillates between two numbers––a boom/bust, seven good years/seven bad years bifurcation). Another doubling occurs as r moves to 3; while at 3.57 (where the doublings are fast and furious) chaos sets in––hence mathematical chaos came from James

Yorke’s “Period Three Implies Chaos.” Within this chaotic area (3.57 and above), there are spaces of regular order. Thus, nonlinear dynamics, in a metaphorical sense, asks social theorists to see that chaos and order are not opposed but actually are entwined within one another. Such a worldview encourages moves beyond modernism’s dichotomous framings––Black/White, good/bad, either/or, right/wrong––to look at a “third space” where new possibilities emerge.

Chaos theory, the mathematical study of the nonperiodic order found in nature (as in the slight fibrillation of a healthy heart), provides new insights into nature’s way of being, insights which show creativity as part of that being. Its implication for education is potentially profound.

**Bibliography: **

- Bird, R. (2003). Chaos and life. New York: Columbia University Press.
- Doll, W., Fleener, J., St.Julien, J., & Trueit, D. (2005). Chaos, complexity, curriculum, and culture. New York: Peter Lang.
- Gleick, J. (1987). Chaos: Making a new science. New York: Penguin.
- Hall, N. (1991). Exploring chaos. New York: W. W. Norton. Kiel. L., & Eliot. E. (1997). Chaos theory in the social sciences. Ann Arbor: University of Michigan Press.
- Lorenz, E. (1993). The essence of chaos. Seattle: University of Washington Press.
- Luhmann, N. (1995). Social systems (J. Bedartz, Jr. & D. Baecker, Trans.). Stanford, CA: Stanford University Press.
- Tien-Yien, L., & Yorke, J. A. (1975). Period three implies chaos. The American Mathematical Monthly, 8(10), 985–992.

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