Semester

Summer

Date of Graduation

2021

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Vicki Sealey

Committee Member

Harvey Diamond

Committee Member

Erin Goodykoontz

Committee Member

Nicole Engelke Infante

Committee Member

Virginia Kleist

Abstract

An individual’s knowledge of definite integrals can range from rote memorization to a strong foundational connection harkening back to its Riemann sum limit definition. In my research, I conducted seven task-based face-to-face interviews with Applied Calculus students. Through the use of real-life examples and guided reinvention, I analyzed ways in which these students, who all initially demonstrated rote memorization, could exhibit a Riemann sum based level of comprehension. This research was conducted in the confines of a student population with definite integral experience, but no formal instruction on limits. My results show that the lack of computational emphasis in class produced little to no restrictions in student development of a Riemann sum based understanding of the definite integral.

Emphasis on units facilitate this evocation of a Riemann sum definition while attempting to rationalize the use of a definite integral in an application context. The use of units also aided in the assigning of meaning to the individual symbols that construct the definite integral. By applying meaning to each symbol in the decomposition of the definite integral, students were able to better rationalize the connections the integrand, differential, and their multiplicative underpinnings have to a Riemann sum. I also demonstrate that the limit’s role in student understanding of the definite integral should be dichotomized into approximation and exact sublayers. My work culminates with a sample lesson on definite integrals, which places primary focus on using population growth to elicit connections between the definite integral and its Riemann sum definition through the use of unit transformation and graphical representations.

Download

Share

COinS