# Rational Numbers: An Integration of Research

By Thomas A. Romberg; Elizabeth Fennema et al. | Go to book overview

5
PROTOQUANTITATIVE ORIGINS OF RATIO REASONING

Lauren B. Resnick Janice A. Singer University of Pittsburgh

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?

The hypotheses developed in this chapter represent an extension of work by Resnick and Greeno ( Resnick, 1989; Resnick & Greeno, 1990) on the intuitive origins of mathematical concepts of number. By intuitive we mean knowledge that does not depend on formal instruction, knowledge that children construct on the basis of their everyday experience in the world. The broad psychological theory within which our nations about mathematical development have been formulated holds that learning is both situation

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