The impact of students' understanding of derivatives on their performance while solving optimization problems
Swanagan, Brian Scott
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Students regularly struggle with applications, particularly those concerning optimization problems, in calculus. Although teachers would like their students to learn to transfer their knowledge to nonroutine and real-life situations, students run into a number of difficulties, including using their understanding of the derivative when finding optimal solutions. In this qualitative study, I interviewed five AP Calculus students and observed them while they solved three optimization problems. I focused on their problem solving, points of struggle, use of the derivative, and understanding of the derivative. Each student participated in a 2- to 3-hour interview, during which I asked about their background, gave them three problems to solve, questioned them about their solution attempts, and inquired into their understanding of the derivative. Only one of the students actually constructed a complete solution, whereas the others varied in their progress. Their limited use and understanding of the derivative was surprising, with the most capable student showing by far the greatest understanding. In their solutions, they struggled with variables, substitution, optimization, objective functions, progress assessment, verification of results, recall of previous learning, calculator use, word problems, and derivative use. Based on these findings, I make recommendations for future research regarding why these students struggle and some teaching methods that may be tried to help them.