Date of Award

8-7-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Draga Vidakovic

Second Advisor

Valerie Miller

Third Advisor

Mariana Montiel

Fourth Advisor

Alexandra Smirnova

Abstract

Research shows that by observing properties of figures and making conjectures in non-Euclidean geometries, students can better develop their understanding of concepts in Euclidean geometry. It is also known that definitions in mathematics are an integral part of understanding concepts and are often not used correctly in proof or logic courses by students. To further investigate student understanding of mathematical definitions, this dissertation studied students’ uses of dynamic geometry software and group work to generalize their understanding of definitions as they completed activities in Taxicab geometry. As a result of the analysis from the group work and use of Geometer’s Sketchpad by 18 students in a College Geometry class, suggestions are provided to implement cooperative learning and technology in the classroom. In addition, results are provided from the data analysis of responses to questions pertaining to the definition of circle (and its relevant concepts) of 15 students enrolled in the course who volunteered to participate in semi-structured interviews. This dissertation specifically utilizes APOS Theory (Arnon et al., 2014) and the interaction of schema framework provided by Baker et al. (2000) to determine what components of the circle schema were evoked by these participating students during these interviews. By adapting and transferring their knowledge of concepts back and forth between Euclidean and Taxicab geometry, these students provided evidence for the relationships they had formed between the components of their circle schema. Further, they demonstrated a variety of levels of schema interaction of their evoked Euclidean geometry schema and Taxicab geometry schema. As a result, a model of schema interaction and suggested pedagogical activities were developed to help facilitate student understanding of the definition of a circle and other relevant concepts.

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