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Leveraging Students' Intuitive Knowledge About the Formal Definition of a Limit


This dissertation explores the roles of students' intuitive knowledge in learning formal mathematics. The formal definition of a limit, or the epsilon-delta definition, is a critical topic in calculus for mathematics majors' development. It is typically the first occasion when students engage with rigorous, formal mathematics. Research has documented that the formal definition is a roadblock for most students in calculus, but has also de-emphasized the productive role of their prior knowledge and sense making processes. The temporal order of delta and epsilon has been suggested as a conceptual obstacle for students in understanding the structure of the formal definition. The dissertation investigates the nature of and the degree to which the temporal order of delta and epsilon is a difficulty for students. The fine-grained analysis of semi-structured interviews with elementary calculus students reveals a large repertoire of reasoning patterns about the temporal order. A microgenetic study of one student shows the diversity of knowledge resources and the complex process of reasoning. Knowledge in Pieces (diSessa, 1993) and Microgenetic Learning Analysis (Parnafes & diSessa, 2013, Schoenfeld, Smith & Arcavi, 1993) provide frameworks to explore the details of the structure of students' prior knowledge and their role in learning the topic. The study offers and examines the impact of an instructional treatment called the Pancake Story, designed specifically to productively link to students' intuitive knowledge. Leveraging the notion of quality control, the instructional treatment offers an alternative to the idea of functional dependence in reasoning about the temporal order of delta and epsilon. A detailed case study shows the process by which a student incorporates resources from the story into her existing knowledge about the temporal order. The findings in this dissertation support the claim that understanding the process of learning requires serious accounting for student's prior knowledge.

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