Investigation of misconceptions and mental models of mathematics major students about topological concepts

Date

2024-3-11

Author

Şahin, Şule

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The objective of this study is to investigate the misconceptions and mental models of cognitive processes of students in relation to topological concepts, such as topological space, metric space, Euclidean space, open set, closed set, closure of a set, ball, and neighborhood. This research mainly focuses its attention on mental models, with an emphasis on the construction of topological, geometric topological, and spatial mental models within the domains of topology and geometric topology. In our study, participants were third-year students in the mathematics department who enrolled in topology and geometric topology courses, respectively. The research used a qualitative method to reveal misconceptions and mental models of students within the field of topology, specifically in point set topology and geometric topology. The qualitative method was conducted following a case study methodology, which formed the design and implementation of the study. The context of the research interviews consisted of conceptual understanding and the definition of topological concepts, e.g., topological space, metric space, Euclidean space, open set, closed set, closure of a set, ball, and neighborhood. The primary sources of data for this study consisted of video-recorded interviews conducted to collect students' explanations and cognitive processes. During the course of the interviews, two specific techniques were employed: think-aloud protocol and interview-about-events. The data source for this study also included artifacts created by the students during the interviews. The data was analyzed utilizing the case study approach, as well as the procedure proposed by Miles and Huberman (1984), which consists of three stages: data reduction, data display, and conclusion or verification. In this study, I determined that there are various misconceptions and mental models related to topological concepts. I examined cognitive skills, e.g., topological reasoning, abstraction, visual representations, symbolic logic, set-theoretical explanation in point set topology, and dimensional comprehension. I also determined if there is a relation between these cognitive skills and the students’ mental models. According to these cognitive skills, as a result, we named the students’ mental models, e.g., topological mental models, topological set theoretical mental models, topological symbolic mental models, geometric topological mental models, Geometric mental models, and spatial mental models. Primitive mental models discovered led us to identify misconceptions about topological concepts, e.g., topological space, metric space, Euclidean space, neighborhood, open set, and ball concepts. In the results of this research, mental models, together with misconceptions, were revealed about topological concepts, e.g., topology, topological space, Euclidean space, metric space, open set, closed set, ball, and neighborhood.

Subject Keywords

Topological concepts, Mental models, Spatial mental models, Visuospatial thinking, Misconceptions in topology

URI

https://hdl.handle.net/11511/109793

Collections

Graduate School of Natural and Applied Sciences, Thesis