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Explanation and Reflection in Calculus Learning


This dissertation focuses on explanation and reflection in mathematics and offers an instructional intervention for supporting student learning. Explanation and reflection are considered hallmarks of deep understanding; they are also tools for promoting learning. The primary context of study was calculus, because it is often considered a gatekeeper that prevents students from accessing higher-level mathematics and Science, Technology, Engineering, and Mathematics (STEM) careers. Simultaneously, there is concern that even students who are successful (in the conventional sense) lack deep mathematical understanding. I collected both conventional and deeper performance measures, because both have implications for learning and performance.

Over three semesters of instruction I used design-based research methods to iteratively develop and refine an intervention. The intervention, Peer-Assisted Reflection (PAR), used peer-review and self-reflection as means to promote explanations. The core PAR activities required students to: (1) engage in meaningful problems, (2) reflect on their own work, (3) analyze a peer's work and give and receive feedback, and finally (4) revise their own work based on insights gained throughout this cycle. PAR was supported by other aspects of the instructional environment, such as adequate training and opportunities for students to explain their ideas regularly during class sessions.

The first semester (pilot study) took place in an introductory algebra classroom in a community college; the second and third semesters (Phases I and II) took place in an introductory college calculus course in a research university. The purpose of the pilot study was to refine theoretical principles to create an effective instructional intervention. Phase I was the first full implementation of the intervention, and Phase II served to replicate and extend the findings from Phase I. During Phases I and II quasi-experimental methods were used to compare students in an experimental section to students in parallel sections of the same course. I collected the following data: students' common exams, pre- and post-surveys about beliefs, student interviews, video observations of class sessions, copies of students' PAR assignments, and audio records of student conversations.

Students in the experimental sections were more successful in calculus than their counterparts in the comparison sections. During Phase I the experimental success rate (A's, B's, and C's in the course) was 13% higher than the comparison section. The difference in success was 23% during Phase II, due to iterative refinements of the intervention. These improvements were statistically significant on common department exams in Phase I (average exam score 73.03% vs. 66.84% and 67.32% in the two comparison conditions) and Phase II (average exam score 75.20% vs. 64.17%). Improvements were evident on both conceptual and procedural problems. Students in the experimental sections also improved their written explanations and demonstrated improved persistence in problem solving.

In sum, engaging in PAR helped students improve their: success rates, exam scores, written explanations, and persistence. Because PAR is not grounded in specific mathematical (or formal) content, it should be a broadly applicable intervention for improving student learning in a variety of contexts. The findings of this dissertation also contribute to a theoretical understanding of peer-analysis and self-reflection as tools to promote explanations. For example, explaining to an authentic peer audience helped students focus on their communication, rather than just finding the correct answer. The findings also provide further evidence of the importance of conceptual understanding; by engaging in PAR, an intervention specifically targeted at improving explanations, students also improved their success on measures of procedural understanding.

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