The Maine RiSE Center
and the University of Maine
ORAL THESIS DEFENSE
Thesis Advisor: Dr. John R. Thompson
An abstract of the Thesis Presented
In Partial Fulfillment of the Requirements for the
Degree of Master of Science (in Teaching)
STUDENT UNDERSTANDING OF DEFINITE INTEGRAL
USING GRAPHICAL REPRESENTATIONS
Learning of physics concepts often requires fluency with the underlying mathematics concepts. 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. We studied the extent to which the conceptual understanding of definite integrals affects the understanding of physics concepts that involve definite integrals. 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.
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 context 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.
375 Stevens Hall
Friday, December 7, 2012
10:00 am – 12:00 pm