The emergence of algebraic structure: students come to understand units and zero-divisors

Abstract

Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concepts of zero-divisor and, more generally, elements with no multiplicative inverse are among those for which group theory has no analogue. In order to better understand how students come to understand the corresponding ideas of units and zero-divisors, this paper presents results from a study that investigated how students can capitalize on their intuitive notions of solving equations to reinvent the definitions of ring, integral domain, and field. In particular, the emergence and progressive formalization of the concept of zero-divisor at various stages of the reinvention process are detailed and discussed. Findings include a conceptual framework characterizing the emergence of the concept of zero-divisor and unit. This framework, in addition to documenting the emergence of these concepts, suggests that the concept of zero-divisor arises in direct contrast to the concept of unit.

Keywords:

• abstract algebra
• units
• zero-divisors
• guided reinvention
• emergent models
• Realistic Mathematics Education

Acknowledgements

The author would like to thank Tim Fukawa-Connelly, Sean Larsen, and Jason Martin for their guidance in executing this project.

Notes

To accommodate the students’ activity with the operation tables in preparation for solving equations, I introduced a preliminary phase occurring before the original situational setting.

Only the finished, refined definitions are shown here. For more detail on the defining process, see Cook [3].

Haden is referring to the zero-divisors they noted in Figure 9.