Renee LaRue

Date of Graduation

2016

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Nicole Engelke Infante

Committee Co-Chair

Johnna Bolyard

Committee Member

Marjorie Darrah

Committee Member

John Goldwasser

Committee Member

Vicki Sealey

Abstract

The purpose of this dissertation is to investigate how students think about and understand optimization problems in first semester calculus. To solve an optimization problem, one must identify the quantity to be maximized or minimized and then construct an optimizing function modeling the scenario described in the problem, particularly paying attention to how the desired quantity varies under the given constraint. Once this optimizing function has been constructed, the problem solver uses calculus and algebra skills to analytically find the absolute maximum or minimum of the function in the realistic domain of the problem. These problems are notoriously difficult for students, but have been largely unexplored by the research community.;To examine how students think about and understand optimization problems, I interviewed seven first semester calculus students as they solved two optimization problems and answered questions related to the optimization problem-solving process. Analysis of this interview data revealed six mathematical concepts that play a key role in students' concept images of the optimizing function. In Paper 1, I describe how these mathematical concepts influence students' problem-solving activities and their construction of the optimizing function.;In this dissertation I also have created an Optimization Problem-Solving Framework that describes the desired conceptual and analytical thought processes students should engage in while solving an optimization problem. In Paper 2, I describe this framework and present the results of analyzing students' thought processes while solving a classic optimization problem. The students demonstrated evidence of engaging in pseudo-conceptual and pseudo-analytical thought processes as they solved the optimization problem, particularly in the orienting and planning phases.;Finally, in Paper 3, I describe students' responses while doing an activity designed to assess their ability to connect the optimizing function they constructed to the graphical representation of the optimizing function. This analysis is used to frame a discussion of suggested teaching interventions to help students develop a conceptual understanding of the optimization problem-solving process.

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