Correlations Between Students\u27 Multiple Ways of Thinking About the Derivative and their Abilities to Solve Applied Derivative Problems

Students’ understanding, thinking and difficulties with the derivative and their difficulties in solving applied problems have been the subject of rich research work. There is also extensive research on students’ thinking, understanding and difficulties with the derivative’s underlying concepts such as functions. In the Study I, the existing work on students’ ideas about the derivative was expanded by defining different categories of students’ ways of thinking about it. Polishing these categories, a possible database of students’ ways of thinking about the derivative was created. Then their difficulties with applied related-rate, optimization and graphing problems were analyzed and rubrics were created to quantify their answers. It was also shown that students’ difficulties with applied graphing problem of the derivative correlate to their static and dynamic understanding of the functions’ graphs. Very little research has examined students’ ways of thinking about derivative through the lens of their work on applied questions. The focus of the Study II was on whether relationships exist between students’ ways of thinking about derivative and their abilities in solving applied derivative problems. Survey data and clinical interviews were conducted to look at students’ multiple ways of thinking and their work on applied derivative problems. “Multiple ways of thinking” refers to two or more ways of thinking about derivative (e.g., slope of the tangent line at a point on a function or instantaneous rate of change). Fine-grained analysis of the students’ written surveys and clinical interviews revealed that students’ having more than two ways of thinking about the derivative correlate to their higher abilities in solving applied derivative problems