The basic hypothesis of the teaching experiment, The Child’s Construction of the Rational Numbers of Arithmetic (Steffe & Olive 1990), was that children’s fractional schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because when a new scheme is established by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. In that case where children’s fractional schemes do emerge as accommodations in their numerical counting schemes, I regard the fractional schemes as superseding their earlier numerical counting schemes. If one scheme supersedes another, that does not mean the earlier scheme is replaced by the superseding scheme. Rather, it means that the superseding scheme solves the problems the earlier scheme solved but solves them better, and it solves new problems the earlier scheme did not solve. It is in this sense that we hypothesized children’s fractional schemes can supersede their numerical counting schemes and it is the sense in which we regarded numerical schemes as constructive mechanisms in the production of fractional schemes (Kieren, 1980). Relevance: This paper relates to Ernst von Glasersfeld’s reformulation of Piaget’s concept of scheme.