Paramathematical notions and Klein's Plan B: the case of equations

TL;DR

This paper explores the mathematical and didactical understanding of solving equations at the university level, emphasizing the integration of abstract algebra and real analysis through examples from teacher education.

Contribution

It provides a novel analysis of how university students comprehend equation solvability, combining mathematical theory with didactical insights.

Findings

Students often solve equations without deep understanding of solvability.

Integration of algebra and analysis enhances comprehension of solutions.

Didactical approaches improve students' conceptual grasp.

Abstract

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 to combine 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.