Evolving Communities of Mind-In Which Development Involves Several Interacting and Simultaneously Developing Strands

Abstract

Abstract If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s). These research-based sequences are referred to as learning trajectories. Other researchers emphasize that conceptual development can involve interactions among ideas expressed using a variety of representational media and can occur along a variety of "dimensions" such as concrete-abstract, simple-complex, or situated-decontextualized. Therefore, different paths can be appropriate for different students, and trying to funnel development along any single developmental path can be inappropriate for some students. These researchers often envision trajectories to be specific paths within a branching tree diagram that portrays the space of possibilities. This article emphasizes a third type of situation called model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, 2000). They are problem-solving situations in which goals include developing more powerful constructs or conceptual systems. Therefore, significant conceptual developments occur because students are challenged to repeatedly express, test, and revise their own current ways thinking-not because they were guided along a narrow conceptual path toward (idealized versions of) their teacher's ways of thinking. That is, development looks less like progress along a path, and it looks more like an inverted genetic inheritance tree in which great grandchildren trace their evolution from multiple lineages that develop simultaneously and interactively.