On the Axiomatization of Mathematical Understanding: Continuous Functions in the Transition to Topology




Cheshire, Daniel C.

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<p>The introduction to general topology represents a challenging transition for students of advanced mathematics. It requires the generalization of their previous understanding of ideas from fields like geometry, linear algebra, and real or complex analysis to fit within a more abstract conceptual system. Students must adopt a new lexicon of topological terms accompanied by a multitude of relationships among their underlying mathematical ideas. While some students are successful in coordinating these two related strands of understanding, many others encounter challenges as they attempt to accommodate their prior conceptual schemas within the context of the axiomatic system of topology. Although there has been increasing interest in studying students’ understanding of axiomatic systems, few researchers in the field of mathematics education have explored the ways that students think about and reason with the axioms of topology. I claim that distinctions between individual cases of student reasoning in topology can offer new insights into advanced mathematical thinking and learning, both in topology and other mathematical fields of study.</p> <p>To advance the research on students’ reasoning in advanced mathematics, I conducted a semester-long qualitative study to illuminate how six undergraduate mathematics majors approached the transition to axiom-based reasoning in their introductory topology course. Through a series of individual clinical interviews I observed and interpreted their mathematical activities as they completed proof tasks in the context of topologies with which they were unfamiliar. I found that they employed diverse strategies and reasoned with multiple conceptions of the open set and continuous function ideas as they embedded their formal and informal ways of understanding into schemas that would reflect the axiomatic system of topology. By exploring these participants’ transformative uses of properties during the accommodation of their schemas to axiomatic contexts, this study contributes to an emerging perspective on the construction of axiomatic mathematical understanding in general.</p>



Continuous functions, Topology


Cheshire, D. C. (2017). <i>On the axiomatization of mathematical understanding: Continuous functions in the transition to topology</i> (Unpublished dissertation). Texas State University, San Marcos, Texas.


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