Dynamic Geometry Environments (DGEs) are popular tools in the exploration of geometry. This research is designed to explore the confidence of undergraduate mathematics students as they make mathematical statements when completing geometric tasks using DGEs. Students completed two series of tasks in both Euclidean and hyperbolic geometry. The first series of tasks asked students about properties of parallel transports and the second series of tasks asked students about the existence of regular polygons. The ten students in this research used Geometry Explorer, a DGE which they had previous experience using in Euclidean geometry, but minimal experience using in hyperbolic geometry. Hyperbolic geometry tasks were included in this study because features of that geometry (e.g. curved lines and unexpected length measure) were expected to pose challenges for students' intuitive expectations. Because of this lack of intuition, students may use the features of DGEs (e.g. dragging and measurement) to make various justifications (e.g. authoritative, inductive, and deductive) of the mathematical claims they are making. Both the features of the DGE and students' justifications affect their confidence in the claims they make. This research explored the interaction between these three factors. Analysis of the data showed that these two series of tasks elicited both dragging and measurement tool usage. During the parallel transport tasks, students used these tools in both in an exploratory mode looking for relationships and a validation mode confirming previous conjectures. During the regular polygon construction tasks, students mainly used the tools in a validation mode. Additionally, many students waited until the hyperbolic portion of the tasks to begin using these tools. The tasks elicited a range of justifications, though students generally used inductive arguments. Deductive justifications, when used, were mainly for familiar tasks that took place within Euclidean geometry. Reported confidence was high across both series of tasks as well as across both Euclidean and hyperbolic geometry when working with the DGE. Reported confidence dropped when working on conjecturing or proof validation prompts that did not use the DGE. This research suggests there is still much work to be done investigating how students use tools, make justifications, and report confidence when using DGEs in both Euclidean and non-Euclidean geometries. The researcher recommends further study including the exploration of additional tools within DGEs, the dynamics of working in partners within DGEs, and how students' expectations of justification affect their responses.
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