Stage 3 high school students’ generalization of a cubic identity

This paper reports on one study in a series of design research studies that have taken as a guiding design principle that combinatorial and quantitative reasoning can serve as a constructive resource for high school students to establish algebraic structure between a polynomial and its factors. Within this framing, we report on an interview study with eight 10th-12th grade students whose purpose was to investigate their progress towards generalization of the cubic identity ${(x+1)}^3={1\bullet (x}^3)+3\bullet \left(x^2\bullet 1\right)+3\bullet \left({x\bullet 1}^2\right)+1\bullet (1^3)$. The students worked on this generalization by solving cases of a 3-D combinatorics problem and representing their solutions using 3-D arrays. Findings include the identification of how differences in students’ combinatorial reasoning impacted their reasoning with 3-dimensional arrays and their progress towards a general statement of the cubic identity.