Department of Mathematics
http://hdl.handle.net/1957/13737
Fri, 12 Feb 2016 03:51:44 GMT2016-02-12T03:51:44ZSome Results in Single-Scattering Tomography
http://hdl.handle.net/1957/58015
Some Results in Single-Scattering Tomography
Sherson, Brian
Single-scattering tomography describes a model of photon transfer through a object in which photons are assumed to scatter at most once. The Broken Ray transform arises from this model, and was first investigated by Lucia Florescu, Vadim A. Markel, and John C. Schotland, [2], in 2010, followed by an inversion, [1], in the case of fixed initial and terminal directions. Later, in 2013, Katsevich and Krylov, [6], investigated settings where terminal rays were permitted to vary, either heading towards or away from a focal point, providing inversion formulas in two- and three-detector settings.
In this thesis, we will explore these transforms, give them distributional meaning, and analyze how these transforms propagate singularities. This requires analysis of the relationships between a function and its antiderivative obtained from integration over a ray.
This thesis also introduces the Polar Broken Ray transform, in which the source position is fixed, initial direction varies, and the scattering angle is held constant. We will discover that the Polar Broken Ray transform is injective on spaces of functions supported in an annulus that is bounded away from the origin, and also derive distributional meaning to the Polar Broken Ray transform, and derive a relationship between a wavefront set of a function and that of its Polar Broken Ray transform.
Provided in the appendix are implementations of numerical inversions of the Broken Ray transform with fixed initial and terminal directions, as well as the Polar Broken Ray transform.
Graduation date: 2016
Thu, 10 Dec 2015 00:00:00 GMThttp://hdl.handle.net/1957/580152015-12-10T00:00:00ZListing as a Potential Connection between Sets of Outcomes and Counting Processes
http://hdl.handle.net/1957/57952
Listing as a Potential Connection between Sets of Outcomes and Counting Processes
Erickson, Sarah A.
Counting problems are rich in opportunities for students to make meaningful mathematical connections and develop non-algorithmic thinking; their accessible nature and applications to computer science make counting problems a valuable part of mathematics curricula. However, students struggle in various ways with counting, and while previous studies have indicated that listing may be a useful way to address student difficulties, little work has been done toward understanding exactly how students may connect lists of outcomes to their solutions to counting problems. To begin to address this, I conducted twenty task-based interviews with undergraduate students to probe the ways in which students conceptualize the relationship between sets of outcomes and counting processes. In this thesis, I describe the ways that students listed outcomes using an elaboration of English's (1991) solution strategies, and I frame my findings about their understanding using Lockwood's (2013) model of students' combinatorial reasoning. I discover that students reason about the relationship between lists of outcomes and counting processes with varying levels of sophistication, and I suggest that teachers could help students by making connections between sets of outcomes and counting processes more explicit.
Graduation date: 2016
Fri, 04 Dec 2015 00:00:00 GMThttp://hdl.handle.net/1957/579522015-12-04T00:00:00ZCollege Instructor Preparation : Enough to Feel Comfortable?
http://hdl.handle.net/1957/57950
College Instructor Preparation : Enough to Feel Comfortable?
Fleming, Eric (Eric Ryan)
Over several decades, much attention has been paid to the preparation of K-12 teachers. More recently, the body of literature on graduate teaching assistants' preparation for teaching has begun to increase. Since many graduate teaching assistants are hired as community college and university instructors, it is important to understand how they are prepared for teaching. The purpose of this thesis is to understand what newly hired instructors found helpful, and not helpful, about their education. A series of three interviews was conducted with four instructors over the course of one academic year. I share my findings from my investigation of the instructors' experiences during their first years on the job: what courses they draw on while teaching, what courses have influenced their teaching, and what courses they are unable to draw on while teaching. Lastly, I offer recommendations for what types of courses might be helpful in supplementing a prospective instructors' education based on the participants' experiences.
Graduation date: 2016
Fri, 20 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1957/579502015-11-20T00:00:00ZThe Mathematical Knowledge for Teaching of First Year Mathematics Graduate Teaching Assistants : The Case of Exponential Functions
http://hdl.handle.net/1957/57928
The Mathematical Knowledge for Teaching of First Year Mathematics Graduate Teaching Assistants : The Case of Exponential Functions
Keeling, Matthew (Matthew Jared)
A teacher's mathematical knowledge for teaching has been shown to have a positive correlation with their students' success (Monk, 1994). So, when half of the students that start out in a STEM major switch out to a non-STEM major before graduation to in large part to instructors pedagogical methods and inadequate teaching we must ask what does the mathematical knowledge for teaching look like at this level (Lowery, 2010; Seymour & Hewitt, 1997). To address this, there has been a lot of research in developing professional training activities for graduate teaching assistants that would build their mathematical knowledge for teaching. This research is being conducted without ever having looked at the mathematical knowledge for teaching that these graduate teaching assistants possess. In this research over a series of interviews with four graduate teaching assistants asking them questions about their mathematical knowledge for teaching. The results of this study are present here in this thesis.
Graduation date: 2016
Tue, 24 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1957/579282015-11-24T00:00:00Z