Date of Award

8-23-2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Teaching and Leadership

Advisor(s)

Duane Graysay

Keywords

Mathematical literacy;Proof comprehension;self-explaining;self-explanation;Variation Theory

Abstract

In this dissertation, I report results from a study that investigated the impact of students’ engagement in a series of tasks in an introductory proof course that was designed to stimulate their learning of reading strategies to support their comprehension of mathematics proofs. Specifically, I investigated the impact of the design of five self-explanation tasks, designed through principles of variation, on undergraduates’ discernment of reading strategies; and if students’ learning of specific reading strategies influenced their comprehension of the proofs they read. The goal of this study was to answer two questions: How do undergraduate students in an introductory proof course who engage in self-explanation training tasks, designed and implemented using principles from Variation Theory, develop the capacity to use effective reading comprehension strategies to read proof? How does using Variation Theory as a design principle in self-explaining tasks support undergraduates’ learning of proof reading strategies and their proof comprehension? The study reports on data collected from 8 undergraduate students enrolled in an introductory proof course across a 15-week semester. I conducted this study to investigate and model students’ learning and development of four strategies for reading mathematics proofs across a semester, which used qualitative methods employing six main data-collection methods. The six qualitative methods I used included administering a pre-post comprehension assessment, administering a demographic survey, conducting participant observations, conducting a verbal think-aloud protocol (Cho, 2020) and semi-structured interviews; and collecting written work from students. I then used deductive coding approaches (i.e., a priori coding) to analyze students’ responses to assessments, inductive coding approaches (i.e., open, axial, and selective coding) to analyze student responses to self-explanation task assignments, and a combination of both coding approaches for data collected about students’ engagement in verbal think-aloud protocols, semi-structed interviews, and field observations. However, students responses to the demographic survey was used to introduce participants in each of their cases and to contextualize findings obtained in other data sets. In response to the first research question, the findings of this study suggest that undergraduate students in the chosen introductory proof course who engaged with the self-explanation tasks, designed and implemented using principles of Variation Theory developed an understanding of most characteristics and features associated to the four chosen reading strategies. Though students developed an understanding of features and characteristics associated to each of the four reading strategies that were learning objectives of the study, they did not use the reading strategies in their reading of subsequent proofs in the study. Research on the use of Variation Theory (see Kullberg et al., 2017) have argued that well-designed tasks are not sufficient for learning to occur because students should also engage in instructional environments that promote their discernment of features. However, findings from this study suggest that students’ were immersed in an instructional environment intended to be entirely student-centered, but the instruction was enacted in an instructor-centered way, which was opposite to the active involvement of students in their own learning promoted by the designed tasks. In students’ instructional environment, their understanding of the reading strategies also did not transfer into them using the strategies since they also heavily read their proofs verbatim during their class presentations, despite being instructed to do otherwise. However, findings from the pre and post assessments also suggest that students experienced an overall positive change in their reading comprehension after engaging with the tasks (i.e., the start of the semester to the end of the semester). Most of the positive change that occurred in students’ reading comprehension was in their holistic comprehension. Specifically, those students experienced an increase in their ability to understand a proven statement, comprehend the purpose of a sentence within a proof framework, identifying a good summary for a proof, identifying the structure of a proof, and transferring the general ideas or methods to another context. Another purpose of this study was to investigate how using Variation Theory as a design principle in self-explanation tasks would support undergraduate students’ learning of reading strategies to comprehend mathematics proofs. In response to my second question, the findings of this study suggest that using Variation Theory as a design principle for the design and implementation of tasks were effective at creating learning opportunities to promote students’ discernment of the intended variation in each task. Specifically, findings suggest that these learning opportunities promoted students’ discernment of distinguishing features associated to each reading strategy. However, the design of the tasks was insufficient to promote students’ use of the strategies in their reading of subsequent proofs. This study presents several implications for different stakeholders that either have direct or indirect affiliations associated to undergraduate introductory proof courses. Specifically, the findings of this study may be of relevance for instructors of introductory proof courses, mathematics departments, organizations dedicated to undergraduate program design, community of reading and literacy research, and the community of equity and social justice researchers. The implications presented in this study suggest integrating the designed tasks into introductory, advanced mathematics courses, design modifications that should be made to the study, and disparities that were brought to the fore related to some students’ diverse linguistic backgrounds. Future research should focus on replicating the study with larger sample sizes and investigating the effect of complementing the design of the tasks with instruction that calls attention to the variation present in each task.

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