Insights from College Algebra Students' Reinvention of Limit at Infinity

Abstract

The limit concept in calculus has received a lot of attention from mathematics education researchers, partly due to its position in mathematics curricula as an entry point to calculus and partly due to its complexities that students often struggle to understand. Most of this research focuses on students who had previously studied calculus or were enrolled in a calculus course at the time of the study. In this study, I aimed to gain insights into how students with no prior experience with precalculus or calculus might think about limits via the concept of limit at infinity, with the goal of designing instructional tasks based on these students’ intuitive strategies and ways of reasoning. In particular, I designed a sequence of instructional tasks that starts with an experientially realistic starting point that involves describing the behavior of changing quantities in real-world physical situations. From there, the instructional tasks build on the students’ ways of reasoning through tasks involving making predictions about the values of the quantity and identifying characteristics associated with making good predictions. These instructional tasks were developed through three iterations of design research experimentation. Each iteration included a teaching experiment in which a pair of students engaged in the instructional tasks under my guidance. Through ongoing and reflective analysis, the instructional tasks were refined to evoke the students’ intuitive strategies and ways of thinking and to leverage these toward developing a definition for the concept of limit at infinity. The final, refined sequence of instructional tasks together with my rationale for each task and expected student responses provides insights into how students can come to understand the concept of limit at infinity in a way that is consistent with the formal definition prior to receiving formal instruction on limits. The results presented in this dissertation come from the third and final teaching experiment, in which Jon and Lexi engaged in the sequence of instructional tasks. Although Jon and Lexi did not construct a definition of limit at infinity consistent with a formal definition, they demonstrated many strategies and ways of reasoning that anticipate the formal definition of limit at infinity. These include identifying a limit candidate, defining the notion of closeness, describing the notion of sufficiently large, and coordinating the notion of closeness in the range with the notion of sufficiently large in the domain. On the other hand, Jon and Lexi demonstrated some strategies and ways of reasoning that potentially inhibited their development of a definition consistent with the formal definition. Pedagogical implications on instruction in calculus and its prerequisites are discussed as well as contributions to the field and potential directions for future research