UC Irvine

Different kinds of knowledge are acquired in different ways. In this chapter, I consider four kinds of knowledge: (1) Core knowledge, which is implicit, nonlinguistic and learned easily and automatically; (2) Explicit, linguistic knowledge that fits into an existing conceptual structure (i.e., a new word for an old concept); (3) Explicit, linguistic knowledge that does not fit into any prior conceptual structure (e.g., how to knit); and (4) Explicit, linguistic knowledge that is incompatible with some prior conceptual structure, making it very difficult to learn (e.g., knowing that the earth is a round ball floating in space.) It is this fourth type of learning that Carey (2009) calls ‘conceptual change.’ In this chapter I argue that mathematics includes all four types of knowledge, but that natural numbers are an example of the fourth and most interesting type—knowledge that is the result of conceptual change.