Revisiting the notion of equations in a capstone course: forging internal and external connections

Undergraduate students of mathematics continue to solve equations in virtually any course they attend, just as they did in secondary school–yet what do they learn about equations and their solutions at university? Are they capable of combining elements of abstract algebra and real analysis to assess what it means to solve an equation, in particular, what it means for an equation to be solvable–or not? In this paper, we present a mathematical and didactical analysis of these questions, illustrated by examples from a capstone course for future Danish high school teachers. We also provide further perspectives for similar foundational themes to be revisited from the “higher standpoint” of university mathematics, as proposed by Felix Klein.