Browsing by Author "Gartland, Sara"
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Item Backward transfer, the relationship between new learning and prior ways of reasoning, and action versus process views of linear functions(Mathematical Thinking and Learning, 2022-02-16) Hohensee, Charles; Willoughby, Laura; Gartland, SaraBackward 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 low-socioeconomic 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.Item "Every day I'm happy to go to class and experience math": opportunities for equity within instruction that supports mathematical learning and social-emotional learning(University of Delaware, 2021) Gartland, SaraThe problem driving this dissertation is a lack of knowledge of instruction that supports both mathematical learning (ML) and social-emotional learning (SEL). Without this knowledge, rich opportunities for providing equitable mathematics instruction are going unnoticed, which limits access, identity development, and power-sharing in mathematics classrooms. Guided by research questions targeting instruction, teacher characterizations of instruction, and student perceptions of instruction, this qualitative embedded case study directly addresses this problem by advancing a framework that connects supports for ML and SEL with opportunities for equitable instruction and culturally sustaining mathematics pedagogy (CSMP). ☐ Observations in three ninth-grade mathematics classes in a mid-Atlantic high school, interviews with two teachers, and interviews with four focal students from each class revealed: 1) instruction that supports ML and SEL created opportunities for equitable instruction by increasing access, supporting positive learner identity development, and redistributing power; 2) teachers framed their characterizations of such instruction via descriptions of relational interactions, prior-lived experiences, and/or tension between instruction and assessment; 3) students, individually and/or as a class, can be at different stages of noticing, taking up, and generating opportunities for equitable instruction. These findings contribute to the growing knowledge of and need to further investigate CSMP. Limitations include a small sample size and the virtual nature of data collection due to COVID19, but such limitations are easily addressed with continued research.