years, even scientists assumed that bird flocks must have leaders. It is only recently that scientists have revised their theories, asserting that bird flocks are leaderless and self-organized. A similar bias toward centralized theories can be seen throughout the history of science, with scientists remaining committed to centralized explanations, even in the face of discrediting evidence.
The history of research on slime mold cells, as told by Evelyn Fox Keller ( 1985), provides a striking example of centralized thinking. At certain stages of their life cycle, slime mold cells gather together into clusters. For many years, scientists believed that the aggregation process was coordinated by specialized slime mold cells, known as "founder" or "pacemaker" cells. According to this theory, each pacemaker cell sends out a chemical signal, telling other slime mold cells to gather around it, resulting in a cluster.
In 1970, Keller and a colleague proposed an alternative model ( Keller & Segel, 1970), showing how slime mold clusters can form if every individual cell follows the same set of simple rules, involving the emission and sensing of chemicals. Nevertheless, for the following decade, other researchers continued to assume that special pacemaker cells were required to initiate the aggregation process. As Keller and Segel ( 1970) wrote, with an air of disbelief: "The pacemaker view was embraced with a degree of enthusiasm that suggests that this question was in some sense foreclosed" (pp. 152-153). By the early 1980s, researchers had begun to accept the idea of aggregation among homogeneous cells, without any pacemaker, but the decade-long resistance serves as some indication of the strength of the centralized mindset.
For many years, there has been a self-reinforcing spiral. People saw the world in centralized ways, so they constructed centralized tools and models, which further encouraged a centralized view of the world. Until recently, there was little pressure against this centralization spiral. Even if someone wanted to experiment with decentralized approaches, there were few tools or opportunities to do so.
The centralization spiral is now starting to unwind. New computational tools based on the paradigm of massive parallelism are supporting and encouraging new ways of thinking. In some cases (as seen in the turtles-in-a-circle and rugby examples), these new tools can encourage new approaches to mathematical problems and new ways of conceptualizing mathematical ideas. In other cases (such as the traffic example), the tools can support explorations into the workings of real-world systems. Overall, these new tools provide an opportunity for students (and others) to move beyond the centralized mindset, suggesting an expanded set of models and metaphors for making sense of the world.
This chapter was originally prepared for the NATO Advanced Workshop on "The Design of Computational Media to Support Exploratory Learning" ( diSessa,