Secondary teachers’ guided reinvention of the definitions of reducible and irreducible elements

Informed by Realistic Mathematics Education, we designed a hypothetical learning trajectory on graduate students’ guided reinvention of reducible and irreducible elements in rings. We created experientially real context problems for use in a teaching experiment, in which secondary in-service and pre-service teachers used algebra tiles as an emergent model of factoring integers and quadratics in $Z\left(x\right)$. In their mathematical activity, this became the teachers’ model for abstracting the shared structure of (ir)reducible elements in $Z$ and $Z\left[x\right]$, which they used to formally define (ir)reducible elements. In this paper, we discuss the progression of the teachers’ reasoning and defining activities that were evident as they reinvented the definitions of reducible and irreducible elements of integral domains.