An investigation of an undergraduate student's reasoning with zero-divisors and the zero-product property

The zero-product property (ZPP), often stated as 'if ab = 0, then a = 0 or b = 0,' is an important concept in secondary algebra (as a tool for solving equations) and abstract algebra (as a property of integral domains). This study analyzes results from a teaching experiment to investigate how an undergraduate mathematics major might intuitively reason with zero-divisors and the ZPP. There are two primary findings. First, a procedurally embodied view of equation solving might preclude students' attention to the algebraic properties (including the ZPP) that justify the equivalence of two equations. Second, students might not carefully attend to zero-divisors because they are employing the converse of the ZPP instead of the ZPP itself. These findings advance a hypothesis about why students might view abstract algebra as a different subject than school algebra and also affirm the utility of the student-centered theoretical perspective that guided the instructional design and analysis of student activity.