Investigating Two Teachers? Development of Combinatorial Meaning for Algebraic Structure

This paper reports on the results of a four-day teaching experiment that supported two algebra teachers to develop a combinatorial meaning for algebraic structure. The purpose of the teaching episodes was to support the teachers (a) to establish a combinatorial understanding for algebraic structure (Tillema & Burch, 2022) by generalizing the cubic identity, (a + b)3 = a3 + 3a2b + 3ab2 + b3, as a symbolization of quantitative and combinatorial relationships out of a contex-tualized problem (Tillema & Gatza, 2016) and (b) to develop a combinatorial meaning as a mobilization of their understanding through a series of algebraic tasks (cf. Thompson et al., 2014). The findings from this study contribute to research literature on teachers' mathematical meanings within secondary algebra by investigating how teachers' combinatorial meanings developed and how differences in their combinatorial meanings impacted their algebraic reasoning. The findings demonstrate a combinatorial pathway for supporting the development of expanding and factoring as reversible polynomial operations (cf. Sangwin & Jones, 2017).