ABSTRACT

This chapter talks about quantities, their relationship to other quantities, and the way linked quantities change together. It explores two messages. The first is that children have a great deal of experience and intuitive knowledge. Whenever possible, it is best to build upon that knowledge. It is not the case that students cannot reason about quantities and change; it is simply that no one has asked them to think about these things before. Second, many of the powerful ways of thinking have roots in simple visual and verbal activities. The visualization and verbalization activities and discussions prepare students for another more abstract notion: the invariance of quantities. The most elementary forms of mathematical and scientific reasoning, whether it be logical, arithmetic, geometric, or physical reasoning, are based on the very simple principle of invariance of quantities. Mathematical similarity entails much more than merely looking at the two objects. It requires a search for quantities and relationships.