New approaches for two-variable inequality graphs utilizing the Cartesian Connection and the APOS theory

Abstract

This study utilizes the Cartesian Connection and the action, process, object, and schema (APOS) theory to investigate how preservice mathematics teachers constructed and justified graphs of two-variable inequalities. The analysis shows that most teachers were not equipped with the skills and knowledge that were prerequisites for action, process, and object conceptions of two-variable inequality graphs. Many teachers used informal rules, rather than mathematical ideas such as the Cartesian Connection, to construct and/or justify inequality graphs. They also perceived inequality graphs as shapes rather than as representations based on the meaning of inequalities. I present a genetic decomposition for inequality graphs, through the concepts of variable and of parameter, as well as tasks associated with the genetic decomposition. I call for more studies on inequalities, including those relating to the implementation of the suggested genetic decomposition and tasks.