Abstract

Solving right-triangle problems as well as integrating right-triangle and circular trigonometry into a coherent understanding are documented challenges for students. Prior research has deeply investigated students constructing ratio meanings for sine and cosine using a circle-first approach, which then can be extended to right triangles. Also, past work has separately documented distinct meanings that are useful for sine and cosine, in addition to the ratio meaning, including as lengths in a unit circle and coordinate points. This study took these existing ideas and put them together into one possible developmental progression, based on APOS theory, where students can learn length, coordinate value, and ratio meanings in a circle-first approach, and then extend these understandings to right triangles. This study developed a genetic decomposition and subsequent hypothetical learning trajectory (HLT) grounded in APOS theory and prior research recommendations to support the development of an object level understanding of sine and cosine as ratios. This HLT was implemented with a group of three college algebra students through six lesson interviews. The results suggest that the proposed genetic decomposition and HLT successively supported participants in recognizing sine and cosine as ratio objects and their coherence across right-triangle and circular contexts. These findings have implications for curriculum design in introductory trigonometry courses such as the inclusion and order of certain concepts taught to students.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics Education

Rights

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