# 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.

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## Notes

1. We note that our argument about curricula is rooted in the United States context and may not be the case for countries that have worked to more substantively integrate discrete mathematics into their curricula (e.g., Spain, Hungary, Germany, Israel, etc.). However, we consider this issue an important point for discussion among international researchers—understanding in what ways and how different countries have taken up calls to integrate discrete mathematics in curricular materials.

2. Establishing the 3-D array as a representation for all ordered triples involves spatial operations, rotation and translation. We do not analyze those operations, here, because this paper is focused on the way Olive coordinated her MPS and BCS.

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## Acknowledgements

This research was supported by the National Science Foundation Grant DRL-1920538 and an Indiana University Proffitt Grant. The views expressed do not necessarily reflect official positions of the foundation.

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Correspondence to Erik S. Tillema.

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