**The Story of my Dissertation
**

I didn’t know it at the time, but when I taught Calculus in high school, I started to ask questions that my dissertation research would help me understand. I tried to present Calculus as a tool for understanding the rate of change of real quantities instead of a list of derivative rules to memorize. Some of my students raved about my teaching. One said “I love how you teach us why everything works-I don’t have to memorize anything and I can solve all the problems on the homework.” Other students who saw the same lecture and had the same homework would come back to class the next day complaining that I hadn’t taught them anything on the homework. All my students listened carefully and participated in class-why did some think my instruction was everything they needed to do the homework and others saw it as unrelated? It didn’t seem to be a matter of motivation or academic ability because some of the students who were the most perplexed by the homework had straight A transcripts.

When I began graduate school and helped Pat Thompson in his redesigned Calculus class he helped me understand how students’ foundational understandings of rate of change, variation and function impacted the Calculus ideas they were able to make sense of. My dissertation project investigates how Calculus students’ meanings for quotient, fraction and rate of change influence how they make sense of rate of change functions (derivatives) in Calculus. I’m finding that many smart and motivated students arrive to Calculus with such weak meanings for quotient, fraction and rate of change that making sense of conceptual Calculus instruction is very difficult for them. Of course, my research is only part of the answer to the questions I had as a teacher, but the things my students said in my classroom are starting to make sense to me. Even better, now that I know more about their thinking, I have more ideas about how to help them.

I wish I’d known more about my students’ mathematical thinking as a high school Calculus teacher and I’m inspired to help teachers learn more about student thinking. A component of my dissertation uses the diagnostic instrument developed at ASU to better understand secondary teachers’ meanings for slope, measure, and rate of change. Knowing how teachers’ convey ideas to students helps me hold discussions with teachers about the mathematical meanings for foundational ideas that best support students’ future learning.