# Two students’ mathematical thinking and activity across representational registers in a combinatorial setting

## Abstract

In recent decades, there has been considerable research that explores the teaching and learning of combinatorics. Such work has highlighted the fact that understanding and justifying combinatorial formulas can be challenging for students, and there is a need to identify ways to support students’ combinatorial reasoning. In this paper, we contribute to research that explores effective ways to foster students’ combinatorial reasoning by highlighting the role that shifts in representational registers can play in supporting students’ combinatorial reasoning and activity. In particular, we present a case in which two students, aged 12 and 14, reasoned about, and developed a formula for, binomial coefficients. This occurred in a teaching experiment designed to examine their generalizing activity. In these interviews, the students first solved problems about determining the volume of a cube and then shifted to a combinatorial interpretation of this task, leveraging binomial coefficients as a way to consider growth in higher dimensions. The students demonstrated sophisticated reasoning about combinatorial tasks and were ultimately able to generate and understand a formula for binomial coefficients. In framing this case, we focus on the students’ use of different representations, highlighting the power of being able to transition between representational registers as a way of supporting students’ combinatorial reasoning. In this way, our case demonstrates the value of representational registers specifically as a mechanism by which to potentially improve the teaching and learning of combinatorics.

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## Notes

1. We note the similarity between their guess and the expression for the 3-dimensional case, which was 3N2 + 3 N + 1.

2. We note that because of how the students wrote 1s, they sometimes referred to the 1s as Is.

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## Acknowledgements

This research was supported by the National Science Foundation (DRL-1419973).

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Correspondence to Elise Lockwood.

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Lockwood, E., Ellis, A.B. Two students’ mathematical thinking and activity across representational registers in a combinatorial setting. ZDM Mathematics Education (2022). https://doi.org/10.1007/s11858-022-01352-8