Students' Errors in Setting up Difference Quotients and Connections to Their Conceptions of Function

Although finding the limits of the difference quotients in the definitions of the derivative is troubling for many students, a difficulty that preceded this confusion was observed: students were not able to correctly set up the difference quotients as required in the definitions. The purpose of this study is to uncover student errors in setting up the difference quotients and to discuss what these errors reveal about students' thinking of functions, and evaluations of the difference quotients. At the end of the study, a framework that aggregates criteria used (by past studies and this study) to assign student membership into a function conception category will be produced in an attempt to move towards a systematic classification of students' cognitive processes. Implications from this study can inform teaching practices and curriculum development, by helping students connect difference quotient evaluation with function composition.