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This study examines the phenomenon of backward transfer in the context of high school students learning and reasoning about linear and quadratic functions. Using quantitative methods, this study provides statistical evidence that it is possible to produce intended productive backward transfer effects on students’ prior ways of reasoning about linear functions with quadratic functions instruction that emphasizes quantitative and covariational reasoning. Using qualitative methods, this study characterizes the quality of the backward transfer effects on students’ quantitative and covariational reasoning. The significance of these results is that if intended productive backward transfer is possible, then it represents a new way for mathematics education to be improved. 

Student focusing and noticing, which drive reasoning, are important but under researched aspects of student learning. Quadratic functions representations are perceptually and conceptually complex and thus, offer much for students to focus on and notice. Our study compared a teacher’s goals for student focusing and noticing during quadratic functions instruction with what students actually focused on and noticed. Qualitative analysis revealed some alignment but also informative ways that the teacher’s goals and student outcomes for focusing and noticing were misaligned. These results will further the field’s understanding of how students learn about quadratic functions and may have implications for student focusing and noticing of other mathematics topics as well. 

The topic of study in this report is student focusing and noticing. Specifically, we examined a teacher’s goals for student focusing and noticing and the student outcomes for focusing and noticing. The mathematics context for this research was quadratic functions and covariational reasoning. Two whole-class discussion episodes were analyzed. Results showed ways that the teacher’s goals and student outcomes were aligned and three ways that they were misaligned. These results could inform how quadratic functions are taught and how teachers can improve the alignment between their goals for student focusing and noticing and student outcomes for focusing and noticing. 

This study was conducted to gain understanding about potential influences that learning about quadratic functions has on high school algebra students’ action versus process views of linear functions. Pre/post linear functions tests were given to two classrooms of Algebra II students (N=57) immediately before and immediately after they participated in a multi-day unit on quadratic functions. The purpose was to identify ways that their views of linear functions had changed. Results showed that on some measures, students across both classes shifted their views of linear functions similarly. However, on other measures, the results were different across the classes. These findings suggest that learning about quadratic functions can influence students’ action or process views of linear. Furthermore, the instructional differences between classes provide insights into how to promote those influences that are productive for students’ views.