Using combinatorics problems to support secondary teachers understanding of algebraic structure

Abstract

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers’ combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of polynomials. The data in this paper examines the learning that occurred as one teacher transitioned from making a generalization from a sequence of contextualized combinatorics problems to applying her combinatorial reasoning to symbolic problems common in algebra curricula. The findings from the study include the identification of three planes of learning that can be used to differentiate among ways that combinatorial reasoning can be used to engage in binomial expansion. The highest plane involved constructing a combinatorial scheme for binomial expansion, a scheme that supported the teacher to produce the equivalence, $\left(x+a\right)\left(x+b\right)\left(x+c\right)={x}^{3}+\left(a+b+c\right){x}^{2}+\left(ab+ac+bc\right)x+abc$), and to see important algebraic structure in it. The contributions of the study include: (a) expanding earlier arguments about the ways that combinatorics can be integrated into goals of extant curricula (e.g., Maher et al. in Combinatorics and reasoning: Representing, justifying and building isomorphisms. Springer, 2011); and (b) proposing how reflecting abstraction can be used to study the transition between generalizations learners make from contextualized problem situations to operating with and on generalizations expressed with  conventional algebraic symbols. This second contribution is an under-researched area in the algebra literature (Dörfler in ZDM - Int J Math Educ 40(1):143–160, 2008), and points to an important role that combinatorial reasoning can play in algebra learning.