- Generalization is a fundamental mathematical practice across all disciplines and content areas (Amit & Klass, 2005; Lannin, 2005; Pierce, 1902; Vygotsky, 1986; Ellis, Lockwood, Tillema & Moore, 2017). While a considerable amount of research has been conducted on students' generalizing activity in algebraic contexts (Amit & Klass 2005; Becker & Rivera, 2006; Carpenter & Franke, 2001; Cooper & Warren, 2008), recently more attention has been paid to understanding the ways that undergraduate students generalize (Dorko, 2016; Dorko & Lockwood, 2016; Dorko & Weber, 2014; Kabael, 2011; Jones & Dorko, 2015; Fisher 2007; Lockwood, 2011, Lockwood & Reed, 2016). There remains much to be investigated about ways in which postsecondary students generalize within formal mathematics.
In addition, the topic of real analysis is foundational to undergraduate and graduate mathematics programs. There is relatively little known in the field about student understanding in real analysis (aside from students' constructions of formal limits). An important step toward helping students be successful in mathematics programs is to better understand student reasoning about challenging and abstract concepts in real analysis.
This dissertation explores the ways in which undergraduate students generalize their understanding of core concepts in real analysis. By exploring student generalization in real analysis, I am able to investigate how students generalize mathematics that has already been formalized. In this dissertation, I seek to answer the following research questions: 1) What do students attend to as they generalize formal mathematical knowledge? and 2) In what ways do students' understandings of real analysis on R! influence their understandings or real analysis in more abstract settings?
To gather data, I conducted two 15-hour teaching experiments with undergraduate mathematics students that had completed the introductory sequence in real analysis. One teaching experiment was with a pair of students, and the other was with a single student. Over the course of these teaching experiments, each student group engaged in a reinvention of the formal definition of a metric space, generalizing from their knowledge of distance measurement, sequential convergence, and other topological properties of !R. By conducting these reinventions, I both learned about the ways in which the students generalized their knowledge and about the ways that they fundamentally understood the guiding structure of the metric spaces they explored.
Analysis revealed theoretical constructs that can be applied to further research and teaching. These constructs describe the nuances of the students' formal mathematical generalizations, as well as the structures they attend to when engaging with the concepts in real analysis. In particular, findings revealed two new ways that students can generalize, those of mapping and pattern matching. The findings also allowed for a characterization a natural way for students to reason about real vector spaces in relation to its building blocks, called component-wise reasoning. Findings such as these can inform future research on the ways students generalize formal mathematics, as well as on the ways that they understand concepts in real analysis.