RUME 28
2026
Alexandria, Virginia
Conceptual Foundations for Connecting School and Abstract Algebra: A Historical Analysis
Page: 933
This paper uses a conceptual analysis of histories of algebra to identify a kind of reasoning, combinatorial reasoning, that was essential to the solution of polynomial equations. Historically, mathematicians increased reliance on combinatorial reasoning in the solution of polynomial equations co-occurred with the rise of symbolic algebra. Algebraic symbols were often the context in, and on, which they engaged in combinatorial reasoning. Combinatorial reasoning, however, is often in the background of proofs or arguments—present but not framed as an essential kind of reasoning that could be developed to understand the proofs or arguments. The purpose of foregrounding this reasoning in this paper is to argue that it can serve as a conceptual foundation for designing coherent algebra curricula across grades 9-16—curricula that allow students and teachers to see deeper connections between school and abstract algebra.