ABSTRACT
In this chapter, the author first lay out nature of informal, everyday mathematics knowledge and then develop in more detail the two hypotheses just mentioned concerning the sources of persistent difficulty in learning school mathematics. Much of elementary arithmetic has as its conceptual base the fact that all numbers are additive compositions of other numbers. The cases presented are analyzed in terms of the permissions and constraints on number operations that the additive composition principle embodies. A particularly rich set of examples of principle-based informal mathematical reasoning comes from a longitudinal case study the author conducted of a single child's invented arithmetic. The mathematics of protoquantities is likely to develop in situations of direct engagement with physical material and in situations in which protoquantitive properties of material are named and discussed. The mathematics of numbers and operators is not likely to develop informally-at least for most children-because situations required for their development are not part of most children's everyday lives.
