The Microevolution of Mathematical Knowledge: The Case of Randomness

In this article, we explore the growth of mathematical knowledge and in particular, seek to clarify the relation between abstraction and context. Our method is to gain a deeper appreciation of the process by which mathematical abstraction is achieved and the nature of abstraction itself, by connecting our analysis at the level of observation with a corresponding theoretical analysis at an appropriate grain size. In this article, we build on previous work to take a further step toward constructing a viable model of the microevolution of mathematical knowledge in context. The theoretical model elaborated here is grounded in data drawn from a study of 10 to 11 year olds' construction of meanings for randomness in the context of a carefully designed computational microworld, whose central feature was the visibility of its mechanisms-how the random behavior of objects actually "worked." In this article, we illustrate the theory by reference to a single case study chosen to illuminate the relation between the situation (including, crucially, its tools and tasks) and the emergence of new knowledge. Our explanation will employ the notion of situated abstraction as an explanatory device that attempts to synthesize existing micro- and macrolevel descriptions of knowledge construction. One implication will be that the apparent dichotomy between mathematical knowledge as decontextualized or highly situated can be usefully resolved as affording different perspectives on a broadening of contextual neighborhood over which a network of knowledge elements applies.