ABSTRACT
This chapter develops some specific hypotheses about the origins and development of ratio and proportional reasoning in children. It presents evidence of intuitive schemas for reasoning about densities and rates-both of which count as proto-quantitative forms of reasoning about ratios and proportions but also suggest that quantification of these schemas does not proceed smoothly or directly. The chapter develops evidence for proto-quantitative schemas for intensive relations: that is, schemas for density and rate that do not require exact quantification of two amounts being set in relation to one another. It looks at ways in which protoquantitative intensive schemas might be quantified, giving rise to true rates and ratios. The chapter describes evidence that here makes it clear that intuitive bases for ratio, proportion, and functional reasoning are available to young children. It suggests that even kindergarten and primary-grade children's capacity to reason about covariation is quite robust as long as they are not pressed to think in numerically quantified terms.
