Date of Award


Level of Access

Campus-Only Thesis

Degree Name

Master of Science (MS)




John R. Thompson

Second Committee Member

Michael C. Wittmann

Third Committee Member

Natasha M. Speer


Learning physics concepts often requires fluency with the underlying mathematics concepts. Mathematics not only serves as a representational tool in physics (e.g. equations, graphs and diagrams), but it also provides logical paths to solve complex physics problems. The broad scope of this research is to understand the extent to which students’ mathematical knowledge and understanding influence their responses to physics questions. Only a few studies in physics education research (PER) have investigated connections between student difficulties with physics concepts and those with either the mathematics concepts, application of those concepts, or the representations used. One mathematical concept that is widely used across a broad spectrum of disciplines such as physics, chemistry, biology, economics, etc., is the definite integral. Particularly in physics, where most phenomena are expressed in the language of mathematics, an in-depth knowledge of definite integrals is extremely important to understanding the phenomena. Studies in mathematics education have shown student difficulties with conceptual understanding of definite integrals. We studied students’ conceptual understanding of definite integrals that are relevant in physics contexts. We also identified specific difficulties that students have with definite integrals, particularly with graphical representations. One strong focus of this work was how students reasoned about integrals that yield a negative result. In this study, two written surveys were administered in introductory calculusbased physics and multivariable calculus classes, and seven individual interviews were conducted with students from the physics survey population. In the first survey, students were asked to determine the signs and compare the magnitudes of two integrals. In interviews, students' deep understanding of integrals was probed by varying representational features of the graphs from the first written survey. The second survey was administered in the following semester to test the reproducibility of some of the interview results in a larger population. Many of our findings corroborate previous results reported in the literature, including students’ using the area under the curve to reason about definite integrals, and ensuing difficulties generalizing area as always being a positive quantity. Additionally, novel results in this work include: multiple student difficulties in applying the Fundamental Theorem of Calculus in graphical situations; difficulties determining the signs of integrals that are carried out in the “negative direction” (i.e., from a larger to a smaller value of the independent variable); and student success invoking physical contexts to interpret certain aspects of definite integrals. Furthermore, we find that although students dominantly use area under the curve reasoning, including unprompted invocation of the Riemann sum, when contemplating definite integrals, their reasoning is often not sufficiently deep to help think about negative definite integrals. Overall, our results serve as one example that the connections between mathematics and physics are not trivial for students to make, and need to be explicitly pointed out. Implications for additional research as well as for instruction are discussed.