ABSTRACT
This chapter lays the groundwork for a theory of the intuitive origins of proportion and ratio reasoning. We argue that children have a set of protoquantitative schemas that allow them to reason about ratio- and proportionlike relations without using numbers. Among others, (a) a fittingness schema —or the idea that two things go together based on an external dimension —and (b) a covariation schema —or the idea that two size-ordered series covary, either directly or inversely—form the basis of the protoquantitative knowledge. In the course of elementary schooling, children also learn, separately, about properties of numbers, including their factorial structure. At the heart of our theory is the proposal that these two types of knowledge — protoquantitative schemas about physical material in the world, and factorial number sense—eventually must merge to give children a means to model quantitatively situations that require the use of ratios and proportions.
We know that ratio and proportion are difficult concepts for children to learn. They constitute one of the stumbling blocks of the middle school curriculum, and there is a good possibility that many people never come to understand them. What makes ratios so hard to learn? What resources exist for teaching them more effectively and learning them more easily?