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| 125 9 The Trouble with Math Ralph Abraham Since 1994, the Ross School on Long Island has evolved a curriculum based on a core of world cultural history using an outline I wrote and a long essay written by the historian William Irwin Thompson for the Ross School in 1995. Thompson has devoted considerable time to working with teams of teachers to develop specific course materials for this program, now known as the Ross spiral curriculum. Since each grade at the Ross School is devoted to an epoch of cultural history, and the epochs and grades follow in chronological sequence, it is possible to integrate all of the traditional subjects in the matrix of relationships from which they originally evolved. For example, when the sixth grade is devoted to the axial age, then ancient Greek math, science, art, philosophy, and so on may be taught together, evoking the cultural ambiance of the ancient world. This is the program that has evolved, although the historical sequence and integration of math is not yet complete. This chapter is devoted to describing a proposed program, based on my ideas as well as those of Rupert Sheldrake, William Irwin Thompson, and Courtney Sale Ross, that would integrate math more completely with the core of the spiral curriculum. Since mathematics is the universal language of spacetime patterns and the common modeling strategy of all academic disciplines, math skills are highly advantageous for conceptually integrating historical data and understanding the gigantic complex system in which we live. We have therefore proposed that math skills and the concepts of chaos theory be given special emphasis throughout the spiral curriculum. My ideas for math education have evolved over some fifty years of university teaching at the University of Michigan, University of California Berkeley, Columbia, Princeton, and University of California Santa Cruz. My research in dynamical systems and chaos theory was radically changed by the arrival in Santa Cruz of computer graphics technology in 1974. Early adoption of computergraphicsinmyresearchinchaostheorydiffusedintouniversityteaching , with state-supported grants to introduce more visual representations in the introductory math courses, and the creation of the Visual Math Project at UC 126 | Ralph Abraham Santa Cruz. A sizable National Science Foundation grant in 1980 was devoted to the validation of the visual approach for math education. We showed that more visuals resulted in less math anxiety. In the 1980s, the Visual Math Project grew into a master’s degree program in applied and computational math, and then in the 1990s into a research program, the Visual Math Institute. In 1987, a brief wave of popularity of chaos theory prompted me to begin writing Chaos, Gaia, Eros (1994). In that book I divided world cultural history into three epochs—static, periodic, and chaotic—each of which was tied to a particular development in mathematics. I had become friends with Thompson a few years earlier, and out of respect for his superior knowledge of history I sent him draft chapters for comment. He replied that he had himself devised a similar scheme, most recently published in 1986 in his book Pacific Shift. His own scheme, in which world history is outlined as having four major epochs, or cultural ecologies, each characterized by a different mathematical style or mentality—arithmetic, geometric, dynamic, and chaotic—is credited in Chaos, Gaia, Eros. Later I learned that he had been working on these ideas since 1961 in his undergraduate honors thesis influenced by Gregory Bateson, Vico, and Heinz Werner. When Courtney Sale Ross contacted me in 1995, after reading Chaos, Gaia, Eros, to request that I help her turn my book into a curriculum for the Ross School, we brought in Thompson as well and together began the ongoing project to create the Ross spiral curriculum, a new school program integrated by world cultural history. This chapter discusses the problems and strategies of the math component of this history-based program for middle and high schools, as it has evolved in the context of my work with the Ross School over the past fourteen years, and then proposes a new program in outline form. This program has influenced the Ross spiral curriculum but has not yet been fully integrated, partly because of the pernicious influence of standardized tests, as discussed below. Bifurcations of Cultural History The word bifurcation is taken from the jargon of chaos theory, as explained in great detail in my book Chaos, Gaia, Eros (1994). But in our context, it means a major shift in history. My emphasis on historical...

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