Trigonometry is an essential part of mathematics education (NCTM, 2000; NGA, 2010). Trigonometry is prevalent in studies of pure mathematics as well as physical applications. Trigonometric identities and transformations are particularly important. However, students and even teachers have struggled to articulate and justify trigonometric concepts (Moore, 2013; Tuna, 2013). Students have also struggled with identities and transformations in non-trigonometric contexts (Borba & Confrey, 1996; Tsai & Chang, 2009). This paper will describe a research project which articulates the critical stages through which students must pass to understand trigonometric identities and transformations. These critical stages were first hypothesized based on a review of the literature. Then undergraduate precalculus students were recruited to participate in a series of task-based interviews in order to examine the process by which students come to understand and justify trigonometric identities and transformations. The critical stages were revised based on the results of these interviews. Following the interviews, hypothesized lesson plans for the subjects were revised and implemented. The implementation of the lesson plans did not collect enough information to draw any conclusions, but the critical stages underscore the importance of students being able to move fluidly among representations.
","abstract_html":"<p>Trigonometry is an essential part of mathematics education (NCTM, 2000; NGA, 2010). Trigonometry is prevalent in studies of pure mathematics as well as physical applications. Trigonometric identities and transformations are particularly important. However, students and even teachers have struggled to articulate and justify trigonometric concepts (Moore, 2013; Tuna, 2013). Students have also struggled with identities and transformations in non-trigonometric contexts (Borba & Confrey, 1996; Tsai & Chang, 2009). This paper will describe a research project which articulates the critical stages through which students must pass to understand trigonometric identities and transformations. These critical stages were first hypothesized based on a review of the literature. Then undergraduate precalculus students were recruited to participate in a series of task-based interviews in order to examine the process by which students come to understand and justify trigonometric identities and transformations. The critical stages were revised based on the results of these interviews. Following the interviews, hypothesized lesson plans for the subjects were revised and implemented. The implementation of the lesson plans did not collect enough information to draw any conclusions, but the critical stages underscore the importance of students being able to move fluidly among representations.</p>","abstract_has_math":false,"creators":["Bornstein, Neil Moshe"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Sharon McCrone","Karen Graham","Orly Buchbinder"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-01-01T08:00:00Z","date_published":"2017-01-01T08:00:00Z","updated_at":"2026-07-24T05:22:01Z","subjects":["Education","Identities","Local Instruction Theory","Representations","Transformations","Trigonometry","Mathematics education"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholars.unh.edu/dissertation/154","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sharon McCrone","Karen Graham","Orly Buchbinder"]},{"key":"dc:creator","label":"Author","values":["Bornstein, Neil Moshe"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Education","Identities","Local Instruction Theory","Representations","Transformations","Trigonometry","Mathematics education"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholars.unh.edu/dissertation/154"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Trigonometry is an essential part of mathematics education (NCTM, 2000; NGA, 2010). Trigonometry is prevalent in studies of pure mathematics as well as physical applications. Trigonometric identities and transformations are particularly important. However, students and even teachers have struggled to articulate and justify trigonometric concepts (Moore, 2013; Tuna, 2013). Students have also struggled with identities and transformations in non-trigonometric contexts (Borba & Confrey, 1996; Tsai & Chang, 2009). This paper will describe a research project which articulates the critical stages through which students must pass to understand trigonometric identities and transformations. These critical stages were first hypothesized based on a review of the literature. Then undergraduate precalculus students were recruited to participate in a series of task-based interviews in order to examine the process by which students come to understand and justify trigonometric identities and transformations. The critical stages were revised based on the results of these interviews. Following the interviews, hypothesized lesson plans for the subjects were revised and implemented. The implementation of the lesson plans did not collect enough information to draw any conclusions, but the critical stages underscore the importance of students being able to move fluidly among representations.
"]},{"key":"dc:title","label":"Title","values":["Music of the triangles: How students come to understand trigonometric identities and transformations"]}]}],"canonical_facts":{"dc:contributor":["Sharon McCrone","Karen Graham","Orly Buchbinder"],"dc:creator":["Bornstein, Neil Moshe"],"dc:description.abstract":["Trigonometry is an essential part of mathematics education (NCTM, 2000; NGA, 2010). Trigonometry is prevalent in studies of pure mathematics as well as physical applications. Trigonometric identities and transformations are particularly important. However, students and even teachers have struggled to articulate and justify trigonometric concepts (Moore, 2013; Tuna, 2013). Students have also struggled with identities and transformations in non-trigonometric contexts (Borba & Confrey, 1996; Tsai & Chang, 2009). This paper will describe a research project which articulates the critical stages through which students must pass to understand trigonometric identities and transformations. These critical stages were first hypothesized based on a review of the literature. Then undergraduate precalculus students were recruited to participate in a series of task-based interviews in order to examine the process by which students come to understand and justify trigonometric identities and transformations. The critical stages were revised based on the results of these interviews. Following the interviews, hypothesized lesson plans for the subjects were revised and implemented. The implementation of the lesson plans did not collect enough information to draw any conclusions, but the critical stages underscore the importance of students being able to move fluidly among representations.
"],"dc:identifier":["https://scholars.unh.edu/dissertation/154"],"dc:subject":["Education","Identities","Local Instruction Theory","Representations","Transformations","Trigonometry","Mathematics education"],"dc:title":["Music of the triangles: How students come to understand trigonometric identities and transformations"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T05:22:01Z"}