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  1. 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. 
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    Free, publicly-accessible full text available October 25, 2024
  2. Lamberg, T ; Moss, D. (Ed.)
    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. 
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    Free, publicly-accessible full text available October 4, 2024
  3. Ayalon, M. ; Koichu, B. ; Leikin, R. ; Rubel, L. ; Tabach, M. (Ed.)
    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. 
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    Free, publicly-accessible full text available July 21, 2024
  4. Backward transfer is defined as the influence that new learning has on individuals’ prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students’ prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from lowsocioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions. 
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  5. 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. 
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