Investigating high school students' understandings of angle measure
Hardison, Hamilton Lee
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The purpose of this study was to understand how ninth-grade students quantify angularity. Angle and angle measure are critical topics in mathematics; however, students’ difficulties with these topics are well documented in research literature. In contrast, little is known about how students develop propitious quantifications of angularity, and researchers have critiqued previous curricular approaches. Although some scholars have identified quantifications of angularity beneficial for the study trigonometry, these quantifications leverage multiplicative comparisons of circular quantities (e.g. arc length and circumference) and, as such, are an unlikely starting point for students’ initial quantifications. Previous studies examining how students reason with angles have obfuscated students’ quantifications by failing to emphasize the attribute to be measured or providing measurement tools like protractors. This dissertation research is a response to calls for studies investigating students’ quantifications of angularity, particularly at the high school level. As teacher-researcher, I taught a pair of ninth-grade students at a rural high school in the southeastern U.S. in a constructivist teaching experiment from October 2015 to April 2016. Teaching practices included creating and posing problems involving angle models, interacting with students to understand their ways of reasoning, and reflecting on students’ mathematical activities. From video records of these sessions, I constructed second-order models accounting for students’ initial quantifications of angularity and modifications to these quantifications occurring throughout the teaching experiment. From the students’ activities, I abstracted three motions involved in the construction of angularity and five operations used to alter the side lengths of angle models while preserving angularity. During the teaching experiment, both students constructed quantifications of angularity involving extensive quantitative operations (e.g., segmenting, iterating); these extensive quantifications have not been discussed in previous empirical literature. Students’ constructions of right-angle and full-angle templates were a significant resource for students’ understandings of the degree as a standard unit of angular measure. Students developed conceptions of degrees that involved positing familiar angular templates as composite units (e.g., a right angle as a 90-unit composite or a full angle as a 360-unit composite), and their quantifications differed across rotational and non-rotational angle contexts.