"Emergent Mathematical Ideas from Complex Conceptual Systems"
In many classrooms, mathematics is taught as a set of rules or theorems that need to be mastered through exercises of increasing difficulty, before they can be used in real world problems. In contrast, this paper suggests that mathematical ideas are emergent properties of holistic conceptual systems that students develop while mathematizing problematic situations. These ideas cannot be learned in isolation of the conceptual systems from which they derive their meanings. Instead, they develop as students notice relations and patterns in systems involving transformations on mathematical (structural) elements, such as rates, proportional quantities, sequences, and measures of chance. Furthermore, the evolution of these ideas in a classroom environment resembles the evolution of communities of complex adaptive systems, struggling for survival in the presence of diversity, selection, reproduction and communication. In this paper, we give some examples of such mathematical ideas and their associated conceptual systems, and we describe features of learning environments that foster their development.