#### Date of Award

Spring 5-5-2023

#### Level of Access Assigned by Author

Open-Access Thesis

#### Degree Name

Master's of Science in Teaching (MST)

#### Department

Teaching

#### Advisor

Natasha Speer

#### Second Committee Member

Timothy Boester

#### Third Committee Member

Franziska Peterson

#### Abstract

Students find topics in Calculus to be challenging for various reasons. Existing research indicates that some of these challenges may not be related to the Calculus-specific content, but are instead associated with the prerequisite knowledge for that topic. Many topics in Calculus involve the application of prior knowledge that is not explicitly taught in a Calculus course, such as topics from algebra and geometry. Students’ thinking about algebra and geometry topics may influence their thinking related to Calculus-specific content. Second-semester Calculus includes topics such as Volume by Slicing and Volumes of Revolution which use integration to compute volumes defined by a function. These topics have connections to the prerequisite topic of volume which is first introduced in elementary grades. Better understanding how Calculus students conceptualize volume may help inform instructors as well as potential reforms to practices to better support student learning of these challenging Calculus topics. We collected survey responses from second-semester Calculus students with the purpose of identifying their conceptions of volume. Findings from the survey data suggest that some of the volume conceptions that are identified in the work of elementary school students are also identified in the work of Calculus students. Some elementary students encounter challenges related to conceptions of volume. Similar challenges related to these conceptions of volume are seen in the work of some Calculus students. We conducted task-based interviews with second- and third-semester Calculus students with the purpose of identifying their conceptions of integral and volume apparent from their work on a second-semester Calculus task. Findings from analysis of the interview data suggest that some integral conceptions that are apparent in student work on first- and third-semester Calculus tasks are also apparent in student work on second-semester Calculus tasks. We contend that these conceptions of the integral are significant ones throughout the Calculus sequence. Findings from the interview data also suggest that some of the conceptions of volume that were identified with elementary students with three-dimensional cube arrays were also identified in the work of Calculus students solving second-semester Calculus problems. This finding further indicates the significance of the development of these volume conceptions in elementary education, since those conceptions arise in some applications of volume. Lastly, our results indicate that Calculus students use their conceptions of volume to guide their conception of the integral and justify their conception of integral through their conception of volume. Instructors of second-semester Calculus should take note of how their students’ conceptions of volume influence their thinking in volume related Calculus tasks and design lessons centered around interpreting the volume aspects of these tasks. Additional findings, conclusions, and implications are presented in the final chapter.

#### Recommended Citation

West, Ian, "Calculus Students’ Conceptions of Integral and Volume in a Volumes of Revolution Task" (2023). *Electronic Theses and Dissertations*. 3806.

https://digitalcommons.library.umaine.edu/etd/3806