Pre-service teachers’ development of bidirectional reasoning
Paoletti, Teo Joseph
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Researchers have indicated students and teachers do not maintain productive function and inverse function meanings. Other researchers have indicated that students are capable of developing foundational meanings for mathematical ideas through quantitative and covariational reasoning. This dissertation reports results of an investigation into the ways in which students can leverage quantitative and covariational reasoning to develop their function and inverse function meanings. Specifically, I examine two pre-service teachers’ activities in a semester long teaching experiment in which they addressed tasks intended to support them in developing bidirectional reasoning via quantitative and covariational reasoning. In my analysis, I identify two manifestations of bidirectional reasoning: reasoning bidirectionally with respect to axes and reasoning bidirectionally with respect to quantities. I distinguish instances of students’ activities exemplifying each type of reasoning and I characterize the interplay of the manifestations. In addition to exploring the students’ bidirectional reasoning, I examine implications this reasoning had for their other mathematical meanings. Engaging in bidirectional reasoning had immediate implications for the students’ understanding of conventions common to school mathematics as conventions rather than steadfast rules that needed to be followed. For example, whereas in a pre-interview the students unquestionably presumed the input of a graphed function was represented on the horizontal axis, by the end of the teaching experiment each student considered the quantity on either axis (or a implicitly defined third quantity) as representing the input to a graphed function. I also describe the students’ development of more sophisticated function and inverse function meanings that was supported by their reasoning bidirectionally. For example, whereas each student maintained inverse function meanings that relied on switching techniques in the pre-interview, by the end of the teaching experiment each student understood function and inverse function to be about defining input and output quantities. These findings have important implications for the teaching and learning of both pre-service teachers as well as for middle and high school students. I conclude with these implications as well as limitations of the study and directions for future research.