Identifying Structure in Introductory Topology: Diagrams, Examples, and Gestures

Abstract

Despite the prevalence of research in calculus, linear algebra, abstract algebra, and analysis in undergraduate mathematics, the teaching and learning of general topology is a largely unexplored area of research. Although enrollment in courses like linear algebra is often higher than that of topology, the study of students' learning and understanding of topology is of great significance to the Research in Undergraduate Mathematics Education (RUME) community. Courses in topology present many students with their first experience in axiomatic reasoning and explicit interactions with mathematical structure, itself. I present a thorough case study of Stacey, an undergraduate taking a first course in undergraduate topology. Through the lenses of mathematical structuralism, constructivism, embodied cognition, and commognition, I investigated Stacey's proving behaviors. Papers 1, 2, and 3 present a top-down description of Stacey's behaviors as she sought to identify the key ideas of proofs in general topology. In Paper 1, I described Stacey's proving behaviors using vocabulary borrowed from the literature on problem solving and showed that she used diagrams to arrive at the key idea. In Paper 2, I observed that Stacey seldom produced specific examples, but she reasoned about her diagrams as examples and manipulation of these examples led her to the key ideas of several proofs and to identify appropriate counterexamples when necessary. In Paper 3, I used the theories of embodied cognition and commognition to argue that Stacey's use of diagrams to ground abstract structures in the external world gave her the ability to manipulate those structures spatially, ultimately leading her to the key idea. These three papers are three perspectives on the same theme, each digging more deeply than the one before. Stacey's behaviors in Papers 1, 2, and 3 describe her search for and investigation of abstract mathematical structures. Stacey's recognition of structure and her ability to work with it helped her to succeed in writing proofs. I conclude this dissertation with suggestions for teaching, including incorporation of the theories of embodied cognition and commognition, as well as directions for future research. iii DEDICATION To my parents, John and Carmela Gallagher, for their love, support, and patience and to all my students, past, present, and future, may you always know you are capable of more than you think. v