How concrete is concrete?

Abstract

If we want to make something concrete in mathematics education, we are inclined introduce, what we call, 'manipulatives', in the form of tactile objects or visual representations. If we want to make something concrete in a everyday-life conversation, we look for an example. In the former, we try to make a concrete model of our own, abstract, knowledge; in the latter, we try to find an example that the others will be familiar with. This article first looks at the tension between these two different ways of making things concrete. Next another role of manipulatives, will be discussed, namely that of means for scaffolding and communication. In this role, manipulatives may function as means of support in a process that aims at helping students to build on their own thinking while constructing more sophisticated mathematics.