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Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff &amp; Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks.

Through the looking glass, and what algebra found there: historically informed conceptual metaphors of algebraic substitution and Gaussian elimination