Preamble: From Models to Modeling

The objective of this chapter is to situate this book by giving a global overview of the history of the change in perspectives on symbolizing and modeling in the mathematics education community. This history describes a shift from the use of symbols and models as embodiments of mathematical concepts and objects in instructional practice, design and theory, to explorations in semiotics as a central field of interest. Underlying this shift is a shift from correspondence theories of truth to contextualist theories of truth. The latter category encompasses constructivism and socio-cultural theory, which constitute the main background theories that are currently adopted in the mathematics education community. The chapter starts with a discussion of two instruction theories that have incorporated the classical use of manipulative materials and visual models. These concem the so-called ‘mapping theory’, which has emerged within the context of information-processing theory, and Gal’perin’s theory of the stepwise formation of mental actions. Next follows a sketch of the constructivist critique. This is followed by a discussion of the role of (cultural) tools from a socio-cultural perspective. Finally the change in ways of describing and conceptualizing symbolizations that has emerged recently is addressed. In relation to this, the semiotic notion of a sign as an integrated signifier/signified pair is discussed. This is complemented with a discussion of the notion of an inscription as the material correlate of a sign, and of the instrumentation of ict tools.