Semester

Fall

Date of Graduation

2022

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Vicki Sealey

Committee Co-Chair

Jessica Deshler

Committee Member

Jessica Deshler

Committee Member

David Miller

Committee Member

John Goldwasser

Committee Member

Kasi Jackson

Abstract

The concept of integration appears in many different scientific fields, and students’ understanding of and ability to use the definite integral in applications is important to success in their STEM (science, technology, engineering, and mathematics) classes. One of the first types of application problems that students encounter is finding the volume of a solid using the definite integral. How students approach these problems and how they use the definite integral to find volumes can have an impact on their future use and understanding of the definite integral.

This study involves a deep and thorough investigation of how ten students understand the definite integral when solving two types of volume problems: revolution volume problems and non-revolution volume problems. First, using the Riemann Integral Framework (Sealey, 2014), I analyzed how students understood the underlying structure of the definite integral when solving revolution volume problems. Using Piaget’s (1971) learning theory of structuralism, I then examined how students’ understanding of the familiar revolution volume problems affected and influenced their solving of novel non-revolution volume problems. The data was collected via one-on-one interviews where students worked through three different volume problems and discussed their thoughts and work.

The findings of this study can be summarized in three parts. First, students can build symbolically correct revolution volume problem integrals without understanding conceptually why their integral is correct. These students relied on memorized formulas without understanding why the formulas worked. Second, students’ memorized formulas for revolution volume problems break down when attempting to apply them to non-revolution volume problems. Third, display of or development of conceptual understanding emerged either when being asked deliberate and probing questions about their revolution volume integrals or separately while solving the non-revolution volume problems. The students who were able to discuss their revolution volume problem integrals conceptually accurately had continued success throughout the interview.

Revolution volume problems are a standard application of the definite integral and many textbooks spend a lot of time and pages on them, but as this study has shown, using revolution volume problems alone or without asking conceptual questions is not enough to ensure understanding of how definite integrals work to solve volume problems. Non-revolution volume problems provide an environment that is resistant to students’ inclinations to memorize formulas and provides a greater opportunity for students to attend to the underlying structure of the definite integral.

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