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

Contemporary psychological research that studies how people apply mathemati cs has largely viewed mathematics as a computational tool for deriving an a nswer. The tacit assumption has been that people first understand a situati on, and then choose which computations to apply. We examine an alternative assumption that mathematics can also serve as a tool that helps one to cons truct an understanding of a situation in the first place. Three studies wer e conducted with 6th-grade children in the context of proportional situatio ns because early proportional reasoning is a premier example of where mathe matics may provide new understanding of the world. The children predicted w hether two differently-sized glasses of orange juice would taste the same w hen they were filled from a single carton of juice made from concentrate an d water. To examine the relative contributions and interactions of situatio nal and mathematical knowledge, we manipulated the formal features of the p roblem display (e.g., diagram vs. photograph) and the numerical complexity (e.g., divisibility) of the containers and the ingredient ratios. When the problem was presented as a diagram with complex numbers, or "realistically" with easy numbers, the children predicted the glasses would taste differen t because one glass had more juice than the other. But, when the problem wa s presented realistically with complex numbers, the children predicted the glasses would taste the same on the basis of empirical knowledge (e.g., "Ju ice can't change by itself"). And finally, when the problem was presented a s a diagram with easy numbers, the children predicted the glasses would tas te the same on the basis of proportional relations. These complex interacti ons illuminate how mathematical and empirical knowledge can jointly constra in the construction of a new understanding of the world. We propose that ma thematics helped in the case of successful proportional reasoning because i t made a complex empirical situation cognitively tractable, and thereby hel ped the children construct mental models of that situation. We sketch one a spect of the mental models that are constructed in the domain of quantity-a preference for specificity-that helps explain the current findings.