A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

ARIZONA STATE UNIVERSITY August 2010

©2010 Kevin Charles Moore All Rights Reserved

The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry by Kevin C. Moore

has been approved July 2010

Graduate Supervisory Committee: Marilyn Carlson, Chair Patrick Thompson Michael Oehrtman Luis Saldanha Kyeong Hah Roh Matthias Kawski

ACCEPTED BY THE GRADUATE COLLEGE

ABSTRACT Past research has revealed that both students and teachers have difficulty understanding and using the sine and cosine functions. They also hold weak understandings of ideas foundational for learning trigonometry (e.g., angle measure and the unit circle) and disconnected conceptions of the various contexts of trigonometry (e.g., unit circle and right triangle). This dissertation reports results of an investigation into the understandings and reasoning abilities involved in learning ideas of angle measure and the sine and cosine functions. The data was collected using a teaching experiment methodology. The instructional sequence was designed to support precalculus students in constructing understandings of angle measure and the radius as a unit for measuring an angle. Students were then supported in reasoning about how an angle measure and a distance vary in tandem. The instruction leveraged these reasoning abilities to introduce the sine and cosine functions in a unit circle context. Findings from the investigation revealed the importance of students’ conceptualizing measurable (and varying) attributes of a situation (quantities) when conceptualizing angles and their measures. The idea of angle measure, and particularly the radius as a unit for measuring an angle, was also found to be foundational for learning and using the sine and cosine functions. When conceptualizing the sine and cosine functions, students needed to reason about how an angle measure and a varying distance change in tandem to model the periodic behavior between these two quantities. A process conception of function was also necessary for understanding and using the sine and cosine functions. This study’s findings characterized the critical role that quantitative and covariational reasoning played in students developing the dynamic imagery needed to generate a sine or cosine iii

graph representing periodic motion. Finally, there was a wide variation in the students’ willingness to engage in making meaning of the context of a problem. The findings revealed that if a student relies on imitating others’ actions and carrying out nonquantitative procedures, and is not willing, curious, or confident enough to engage in meaning making, the student will likely have difficulty understanding new ideas.

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ACKNOWLEDGMENTS I would like to first thank my wife, Joy, for all of her love and support during my continued education. I can’t thank her enough for everything she’s done for me, and without her I would have never been able to accomplish this. I will never know how she put up with me during this process, and I can’t wait to begin our post-graduation life. Also, I would like to thank my mother, father, and sister. They have been instrumental in everything I have done and accomplished in my life. I couldn’t ask for a better family and I will always be grateful for the love and opportunities they have given me. Next, I would like to thank my advisor, Dr. Marilyn Carlson, for all of the opportunities she has offered and her continual support and guidance throughout my time at Arizona State University. She is an amazing mathematics educator, thinker, and person, and she continually generates experiences for me to grow and learn. I am truly blessed to have had the opportunity to be mentored and influenced by