Examining the Concept Images of Function Held by Preservice Secondary Mathematics Teachers with Varying Levels of Prior Mathematical Experience

Abstract

This multiple case study examines the function concept images of preservice secondary mathematics teachers (PSMTs) enrolled in a mathematics content course designed specifically for PSMTs at a large urban university in the southwestern United States. The primary research question explores the changes in PSMTs’ function concept images when they engage with research-based explorations designed to elicit function-related cognitive conflicts. Furthermore, this study explores the extent to which there are differences in the function concept images of advanced undergraduate mathematics majors and those with the minimum prerequisite knowledge. Thematic analysis is applied to identify PSMTs function-related associations and characterize their function concept images before and after their interaction with the research-based explorations. Data analysis reveals five function concept image categories and 19 themes as well as a general shift in participants’ conceptions related to the vertical line test, algebraic and graphical representations of functions, and the types of sets on which functions can be defined.

In this qualitative study, seven participants completed pre- and post-interviews, during which they answered open-ended questions designed to reveal function-related associations. Two participants started the study with the minimum prerequisite mathematical knowledge; three students completed advanced studies in undergraduate mathematics; and the mathematical backgrounds of the two remaining students fell between these two categories. Thematic analysis methods (Braun & Clarke, 2006) were used to analyze recordings, transcriptions, and student work from the interviews revealing 19 function concept image themes. Analyzing the emergence of these themes across the pre- and post-interviews did not indicate a difference in the overall concept-definition-consistency of an individual’s conception based on their mathematical background; however, the majority of participants’ function concept images adjusted to include functions that cannot be represented on a coordinate plane or algebraically, functions that do not “pass the vertical line test,” and functions defined on non-numerical sets. These findings raise important questions regarding the types of course experiences that contributed to the shifts in participants’ function concept images and the transferability of these types of experiences to other undergraduate mathematics courses.