Groups and Group-Like Structures

Piaget defined logico-mathematical operations as reversible and composable mental actions. Because of this, formal mathematical structures, such as algebraic groups, played important roles in Piaget’s genetic epistemology. Group-like structures describe ways that mental actions can be organized and composed with one another. For example, mental actions of partitioning can be composed with one another to partition a partitioned whole, and such mental actions can be reversed by iterating one of the parts to reproduce the whole. This chapter details the ways that Piaget relied on groups and group-like structures to build models of mathematical development. Whereas research in mathematics education often refers to schemes as structures for organizing mental actions, it rarely mentions group-like structures. The chapter draws on the example of the splitting loope/group to illustrate how formal algebraic structures—as researcher constructs—can be useful in modeling children’s mathematical development. It also includes related exposition on Piaget’s INRC group, Klein’s Erlangen program, Noether’s algebraic invariants, and the mother structures identified by the Bourbaki.