TitleTo infinity and beyond
NameRadu, Iuliana (author), Weber, Keith (chair), Maher, Carolyn (internal member), Cohen-Corwin, Amy (internal member), Tirosh, Dina (outside member), Rutgers University, Graduate School - New Brunswick,
Infinite--Study and teaching,
Iterative methods (Mathematics)--Study and teaching,
Mathematics--Study and teaching
DescriptionThere is evidence that students’ and mathematicians’ images of and reasoning about concepts such as infinite unions and intersections of sets, limits, and convergent series often invoke the consideration of infinite iterative processes. Therefore, not reasoning normatively about infinite iteration may hinder students’ understanding of fundamental concepts in mathematics. Research suggests that the vast majority of college students are inclined to use non-normative reasoning when presented with tasks that challenge them to imagine an infinite iterative process as completed and define its outcome. However, there is almost no research on how students’ conceptions of completed infinite iteration can be refined in a normative direction.
Adopting a design research approach, the purpose of this thesis is to develop a local instruction theory for completed infinite iteration. This design entails multiple cycles through phases of development, implementation, and analysis. The study consisted of a sequence of two constructivist teaching experiments conducted with pairs of mathematics majors, during which the students worked collaboratively through a variety of infinite iteration tasks. Each cycle consisted of 6 or 7 sessions lasting approximately 2 hours, together with Pre-test and Post-Test sessions. The data analysis consisted of multiple phases of iterative analysis of the videotaped sessions and written work and was guided by situated learning transfer theories.
The participants in this study employed a variety of types of reasoning when solving completed infinite iteration tasks, some normative and some not. One way that the students refined their reasoning on infinite iteration was by making references to previously solved tasks (in the presence of a concern for consistent reasoning across tasks), although the refinement was not always in normative directions. The challenge that a researcher faces consists in helping students use such references to establish “anchors” in normatively solved tasks to which they can then contrast the solution paths proposed for other tasks. A variety of approaches to responding to this challenge are explored. The study culminates with the formulation of a local instruction theory for completed infinite iteration, containing among other things a sequence of instructional activities and a rationale for their appropriateness.
NoteIncludes bibliographical references (p. 469-473)
Noteby Iuliana Radu
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work