| Continuity is a central concept in calculus. Yet very few students seem to understand the nature of continuity. The research described was conducted in two stages. Students were asked questions in multiple choice and true/false format regarding function, limit and continuity. These results were used to identify participants as strong, weak or average. Eight participants were later interviewed for the second stage of the study. The interview questions were designed to explore how students thought of infinity, function, limit and continuity. In addition there were questions regarding real-world problems.;Data were gathered through administration of the above instruments and one-on-one interviews. Responses on the written instruments were coded as right or wrong. Interview responses were interpreted according to the framework of Action Process Object Schema (APOS) Theory. This framework provides a tool for classifying the knowledge development indicated by students' responses. At times students characterized as strong gave responses at the process or object level. None of the weak or average students gave responses at the object level and few if any were identified as process.;As recent research describes, this study shows that most calculus students only think of functions as chunky, not smooth, when reflecting on change. Weak students had significant difficulty finding the domain of a function. Average students were inconsistent in their approach to solving problems, and were very dependent on graphical representation of a function. Strong students had a large example space, were able to reason with properties of function, recognized the necessity of reasoning consistently with the algebraic and graphical forms of the same function, and tended to be "smooth" thinkers.;This study concludes with directions for future research and implications for teaching. Recommendations for teaching include the use both algebraic and graphical forms of function and emphasizing strengths and weakness of each in the context of the problem being solved. I also recommend that instructors use a variety of functions when showing students examples to help students develop their own examples to make sense of concepts, and challenges potential contradictory concept images. Furthermore, this study points to the need of instructors placing strong emphasis on the interrelationship between limit, continuity and differentiability. |
