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This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students' reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between associated amounts, students identified sections (e.g., an interval on a graph) in which they could make comparisons between amounts of change in quantities. Although such reasoning is useful for interpreting a Cartesian graph as a representation of covarying quantities, it does not foster attention to variation in the intensity of change in covarying quantities (e.g., a decreasing increase). Focusing on the kinds of relationships students make between amounts of change in covarying quantities might provide further insight into how students could develop a robust understanding of rate of change.

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (nStudents = 170, nTime-points = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed. [ABSTRACT FROM AUTHOR]

This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students' reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between associated amounts, students identified sections (e.g., an interval on a graph) in which they could make comparisons between amounts of change in quantities. Although such reasoning is useful for interpreting a Cartesian graph as a representation of covarying quantities, it does not foster attention to variation in the intensity of change in covarying quantities (e.g., a decreasing increase). Focusing on the kinds of relationships students make between amounts of change in covarying quantities might provide further insight into how students could develop a robust understanding of rate of change. [ABSTRACT FROM AUTHOR]