On the role of mathematics in explaining the material world: Mental models for proportional reasoning

Abstract

Contemporary psychological research that studies how people apply mathe-matics has largely viewed mathematics as a computational tool for deriving an answer. The tacit assumption has been that people first understand a situa-tion, and then choose which computations to apply. We examine an altema-tive assumption that mathematics can also serve OS a tool that helps one to construct an understanding of a situation in the first place. Three studies were conducted with 6th-grade children in the context of proportional situations because early proportional reasoning is a premier example of where mathe-matics may provide new understanding of the world. The children predicted whether two differently-sized glasses of orange juice would taste the same when they were filled from a single carton of juice made from concentrate and water. To examine the relative contributions and interactions of situational and mathematical knowledge, we manipulated the formal features of the problem display (e.g., diagram vs. photograph) and the numerical complexity (e.g., divisibility) of the containers and the ingredient ratios. When the problem wa