Date of Award

Winter 2020

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Orly Buchbinder

Second Advisor

Karen J Graham

Third Advisor

Rita Hibschweiler

Abstract

Traditionally, post-secondary developmental mathematics courses aspire to equip students with mathematical content knowledge needed to succeed in calculus and subsequent STEM courses. The literature shows that this goal alone is insufficient, as the emphasis on content acquisition often comes at the expense of developing higher-order skills such as argumentation, reasoning, and flexibility in mathematics problem solving (Chiaravalloti, 2009; Partanen & Kaasila, 2014; Star et al., 2015). Redesigning curricula with these additional objectives in mind requires providing students with opportunities to engage with mathematics in ways that may contrast with their past experiences or expectations. It requires changing patterns of classroom engagement and development of different classroom norms. This mixed methods research study incorporated a semester-long teaching experiment that aimed to support students' development of higher-order skills by negotiating productive classroom norms. One of the primary interventions was a sequence of "Multiple Solutions Activities" that required groups of students to analyze and critique unfamiliar or erroneous mathematical solutions. The overarching goal of the research was to study students' engagement during these activities across the semester by characterizing the nature of specific types of classroom norms. Social norms describe the classroom participation structure, while sociomathematical norms focus on aspects of student activity that are inherently mathematical, such as what constitutes an acceptable mathematical solution (Yackel & Cobb, 1996). Because of a reflexive relationship between norms and beliefs, students' social and mathematical beliefs were also of interest to characterize the influence of the teaching experiment; these beliefs were assessed by a pre- and post-course questionnaire. The results paint a complex picture of student engagement and values. Despite quantitative analysis suggesting encouraging improvements in students’ mathematical engagement, qualitative analysis highlighted that this change was not homogenous. In particular, the analysis revealed variations in students’ perceptions of the value of multiple solutions and in the nature of the norms developed in student groups. Consequently, the study highlights the lasting impact of classroom norms on students' beliefs, and vice versa, which may hinder the development of alternative norms in subsequent classes. The results of the project also expand upon Yackel and Cobb's (1996) Interpretive Framework for characterizing classroom engagement by suggesting a reflexive relationship exists between social and sociomathematical norms. The data analysis describes concurrent development and mutual influence between the participation structure of a group and their taken-as-shared mathematical beliefs. In all, the project shows that deliberate attention towards negotiating productive classroom norms and students’ in-class engagement can positively affect students’ attitudes towards multiple solutions.

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