ABSTRACT
Written numbers and equations, drawings of graphs and geometrical diagrams, and many other symbols are pervasive signs of mathematical activity, both in mathematical teaching and learning as well as in professional work. These symbolic expressions are often seen as external representations of mental ones (Dufour-Janvier, Bednarz, & Belanger, 1987; Goldin & Shteingold, 2001; Kaput, 1998). Whatever the differences between mental and physical objects, the assumption is that of a causal resemblance between the symbols written on different surfaces and the mental images of those who understand them, so that the former ones are an “external” version of the “internal” ones and vice versa. By means of internalizing and externalizing, what is outside goes inside and what is inside goes outside. Whereas external representations-those marks on paper or computer screens that can be moved and looked on-can be observed directly, the internal or mental ones have to be inferred from what the symbolizer says or does. In this chapter we lend support to an alternative point of view: Using mathematical representations is not a matter of holding correspondences between an outside and an inside, but of inhabiting symbolic places that embrace both the symbol-user and the world in which he or she lives.
