Webs and Situated Abstractions

Abstract

In Chapter 2 we laid the foundation stones of our work and saw the problematisation of dichotomies such as formal/informal, concrete/abstract, contextualised/decontextualised as our primary challenge. We questioned the extent to which there was any justification in thinking of mathematical ideas in terms of a global hierarchy, and in particular, the arrangement of such a hierarchy along a concrete/abstract dimension. Our contention was that this misperception arose from restricting attention to what individuals learn, to the interaction between the individual and knowledge domain. By emphasising questions of meanings and their construction we hoped we would avoid conflict between mathematics as an object of study and mathematics in use. In particular, we sketched the framework for a theory of mathematical meaning which would transcend the purely situated view of cognition, while simultaneously respecting the epistemology of mathematics. There were secondary concerns: we were interested in how the resources of the setting structure and are structured by participants' mathematical activities and the meanings they construct. Following Vygotsky's ideas of tool use, we focused on the dialectical relationship of action and thought which synthesised the idea of thinking tools, communicative tools and (real) tools, and this led us to focus, in Chapters 3 and 4, on the ways in which computational tools could be exploited by learners in the construction of mathematical meanings.