Structural Reasoning

Harel, Guershon; Soto, Osvaldo D.

doi:10.1007/s40753-016-0041-2

Cited by 12 publications

(9 citation statements)

References 18 publications

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“…In this paper, we discuss instances of the generation of abstract examples in response to proof-related prompts in general topology. The theory of constructivism proposes that knowledge and understanding are created in response to an intellectual need; in other words, learning is stimulated by the need to solve a problem that cannot be solved with the knowledge one currently possesses (Harel & Soto, 2017). According to constructivism, knowledge is constructed by the learner.…”

Section: Theoretical Perspectivementioning

confidence: 99%

“…The ability to reason structurally (Harel & Soto, 2017) is essential in advanced mathematics. Structural reasoning "refers to one's ability to reason in terms of general structures, not only in terms of their instances" (Harel & Soto, 2017, p. 231).…”

Section: Theoretical Perspectivementioning

confidence: 99%

“…Necessarily, the student's concept image (Tall & Vinner, 1981) directly influences the student's representation(s) of the structure in question. Harel and Soto (2017) reported on "categories of structural reasoning" (p.…”

Section: Theoretical Perspectivementioning

confidence: 99%

“…In Session 5, for example, Stacey encountered the definition of a dense subset for (we believe) the first time. It is not unreasonable to think that finding an explicit example might be challenging for her since she has never had the intellectual need to explore the dimensions of possible variation (Harel & Soto, 2017;Marton & Tsui, 2004). Another possibility is that Stacey realizes that, for the Awareness of what is invariant in the midst of change is exactly what is needed for proof construction.…”

Section: Dimensions Of Possible Variation and Range Of Permissible Chmentioning

confidence: 99%

“…In a study of when and how university faculty teaching proof-based, upper-division mathematics courses used examples when presenting proofs, Mills (2014) Structural examples represent a form of structural reasoning (Dorier & Sierpinska, 2001;Harel & Soto, 2017;Leron, 1983Leron, , 1985. Harel and Soto (2017) act upon them in order to see how the structure she has generated resolves the perturbation caused by encountering the mathematical conjecture.…”

Section: Introductionmentioning

confidence: 99%

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

Gallagher¹

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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

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Section: Theoretical Perspectivementioning

confidence: 99%

Section: Theoretical Perspectivementioning

confidence: 99%