Date of Award

6-2021

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Mariana Levin

Second Advisor

Dr. Tabitha Y. Mingus

Third Advisor

Dr. Laura R. Van Zoest

Fourth Advisor

Dr. Shiv S. Karunakaran

Keywords

Proving process, intro to proof, stuck points, productive struggle, collegiate mathematics

Abstract

Learning to prove mathematical propositions is a cornerstone of mathematics as a discipline (de Villiers, 1990). However, since proving is a different mathematical activity as compared to students’ prior experience, research has also shown that many undergraduate students struggle to learn to prove, including those who major in mathematics (Moore, 1994; Selden, 2012). While the field has generated research that has analyzed the final products of proof (Selden & Selden, 2009) and there are frameworks for analyzing problem-solving processes (e.g., Carlson & Bloom, 2005; Schoenfeld, 1985, 2010), much remains to be known about analyzing undergraduate students’ proving processes. With a focus on impasses in the proving processes, this dissertation study provides a more fine-grained account by characterizing both students’ overall proving process and their navigating actions. This study explores (a) undergraduates’ proving processes, (b) where students get stuck during the process of constructing proofs, and (c) how students navigate out of their stuck points. In particular, the results of this study can be interpreted as providing information about how undergraduate students engage in productive struggle as they attempt to prove mathematical statements (Hiebert & Grouws, 2007). Given the difficulty undergraduate students face in higher-level math courses, understanding the ways in which they struggle is important for building more inclusive classroom environments.

The data for this study consisted of semi-structured task-based interviews with 10 undergraduates enrolled in a transition-to-proof course. Interviews were video-recorded and students’ real-time proof work was captured using a Livescribe™ pen. Through my data analysis, I created an analytical framework for classifying students’ stuck points and their navigating actions, grounded in the proving processes maps that I generated for each participant. The results of this study indicated that undergraduate students do not engage in strictly linear or sequential proving processes, and they often encounter multiple stuck points when engaging in proving activities. Undergraduate students’ proving processes around stuck points can be categorized into three main types: Type I (no related outcome produced), Type II (related outcome produced but not linked to the main argument), and Type III (at least one related outcome produced and linked with the main argument). Although navigating actions or attempts were observed in all cases analyzed, not all actions led to making productive progress or led students to successfully complete proof tasks. With a deeper look at the navigating actions in two comparative cases, I observed that certain actions, such as setting a goal for their work, producing related results, and linking these results back to the main argument, occurred only in productive progress.

This study provides both theoretical and pedagogical tools for unpacking specific moments of students’ struggles, which is important for understanding and supporting students’ growth with respect to the discipline of mathematics. The different proving processes and navigation actions characterized in this study can help instructors to have a better understanding about productive struggle and to support students in engaging in more productive struggle during their proving practices.

Access Setting

Dissertation-Open Access

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