Associative and reflective connections between the limit of the difference quotient and limiting process

This paper reports a study of how students may connect the limiting process inherent in the derivative to the limit of the difference quotient (LDQ) when solving problems. The data was collected mainly through task-based interviews with five eleventh grade students. It was found that the students used various kinds of limiting processes and connected them in different ways to LDQ. Some of them changed between these two representations, and some students explained one with the other. The two kinds of connections were, respectively, named as associative and reflective connections. One of the students, who made the associative connection, used LDQ skillfully. On the contrary, a student, who made the reflective connection, had major difficulties using LDQ. Therefore, students may at the early stage of their learning process of the derivative use different kinds of procedural and conceptual knowledge of LDQ.