Supporting Students’ Understanding of Variables in an Inverse Context through Targeted Questions
While international and U.S. mathematics standards expect secondary students to develop connected meanings of inverse functions (Bergeron & Alcantara, 2015), previous research has found post-secondary students, pre-service teachers, and practicing teachers often struggle to construct these productive meanings (Brown & Reynolds, 2007; Engelke, Oehrtman, & Carlson, 2005; Even, 1992; Lucus, 2005; Paoletti, et. al, 2017; Vidakovic, 1996). We began our study by using Anotemos software to qualitatively code (Corbin and Strauss, 2008) two implementations of a problem-based lesson on inverse functions that we designed in collaboration with a teacher from a large midwestern public school. We coded for student conceptions of inverse (Stevens et. al, 2020) as well as problem statements and questions posed by the teacher (Milewski & Strickland, 2020). We observed the teacher asking students to consider “any value” and connected this idea to the concept of conceptualizing quantities as variables (e.g., Thompson & Carlson, 2017). We then created a table matching the questions that ask students to consider “any value” with the inverse conceptions expressed by the student the teacher is talking to. We argue the teacher selectively asked these “any value” questions to students he viewed as having inadequately generalizable inverse conceptions. Finally, to learn how teachers view this way of launching an exploration of inverse function, we designed three different ways of launching the problem (solving for an unknown value, generalized number, variable) and asked eight teachers to depict the remainder of the classroom discussions. Teachers employed different conceptions of variables to make sense of the different framings.
