Insights from College Algebra Students' Reinvention of Limit at Infinity

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

GPU-Based Parallel Computing for Nanotechnology Research

An effective technology for parallel computing is the application of graphical processing units (GPU) to computationally intensive calculations. Present research in nanotechnology simulations requires intensive calculations that have the potential to be parallelized and may benefit greatly from GPU processing. These simulations involve eigenvalue calculations on matrices with sizes up to 7776 x 7776. GPU computing speeds up this core calculation by a factor of 2.5, saving hours of valuable research time. As the size of the matrix calculations increases, the speed up using GPU computing increases; however, at small matrix sizes the GPU actually takes longer to compute than the CPU

Insights from College Algebra Students' Reinvention of Limit at Infinity

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

Statistical Methods for Integrating Genomics Data

This dissertation focuses on methodology to integrate multiplatform genomic data with cancer applications. Such integration facilitates the discovery of biological information crucial to the development of targeted treatments. We present iBAG (integrative Bayesian Analysis of Genomics data), a two-step hierarchical Bayesian model that uses the known biological relationships between genetic platforms to integrate an arbitrary number of platforms in a single model. This method identifies genes important to a clinical outcome, such as survival, and the integration approach also allows us to identify which platforms are modulating the important gene effects. A glioblastoma multiforme (GBM) data set publicly available from The Cancer Genome Atlas (TCGA) is analyzed with iBAG. We flag several genes as important to survival time, and we include a discussion of these genes in a biological context. We then present a nonlinear formulation of iBAG, which increases the flexibility of the model to accommodate nonlinear relationships among the data platforms. The TCGA GBM data is again analyzed, and we carefully compare the results from both the linear and nonlinear formulation. Next we present a pathway iBAG model, piBAG, which includes gene pathway membership information and utilizes hierarchical shrinkage to simultaneously select important genes and assign pathway scores. The integration of multiple genomic platforms again allows us to determine which platform is regulating each important gene, and it also provides insight as to through which platform each pathway is taking effect. We apply this method to a different subset of the TCGA GBM data. Finally, we present integrative heatmaps, a novel visualization tool for illustrating integrated data. We use a TCGA colorectal cancer data set to demonstrate the integrative heatmaps. Through the various simulation studies and data applications in this dissertation, we conclude that the methods presented achieve their respective goals and outperform standard methods. We demonstrate that our methods provide many advantages, including increased estimation efficiency, increased power, lower false discovery rates, and deeper biological insight into the genetic mechanics of cancer development and progression

Praxair and the PTAB\u27s Shadow Over Biotechnology Patents

The biotechnology industry is one of the fastest growing fields in research and development. This may be attributed to the decision in Diamond v. Chakrabarty, where the Supreme Court held that a biotechnology invention was patent-eligible subject matter under 35 U.S.C. § 101. However, recent Supreme Court rulings have left the boundaries of § 101 uncertain, unworkable, and difficult for biotechnology industries to gain patent protections for their inventions. Before Congress enacted the AIA in 2011, the courts were the biggest influence on shaping the doctrine of patent eligible subject matter under § 101. But now with the new AIA post-grant proceedings, the PTAB plays an influential role in determining subject-matter eligibility. Through the new AIA post-grant proceedings, the PTAB has the ability to hear petitions that challenge the validity of a patent under §§ 101, 102, 103, or 112. But after the recent decision in Praxair Distribution., Inc. v. Mallinckrodt Hospital Products IP Ltd., the PTAB may now begin exerting too much influence over the doctrine of § 101. This decision, a case heard in inter partes review, threatens to stretch the PTAB’s power dangerously thin. Under the AIA, cases reviewed in inter partes review may not present challenges on patentable subject matter under § 101. However, in Praxair, the PTAB used parts of a § 101 analysis to determine that the claims were ineligible subject matter. The Federal Circuit affirmed the PTAB’s reasoning, suggesting that PTAB may be able to expand the reach of § 101 and allow petitioners to bring eligibility claims in inter partes review—where it is statutorily not allowed. Overall, the PTAB’s power over eligible subject matter makes it easier for applications and patents to be invalidated under §101. This could particularly harm biotechnology and bioscience industries where patent protection is at a disadvantage. This Note will discuss how the Supreme Court and PTAB have affected the subject-matter eligibility under § 101 and how this impacts patent rights for biotechnology innovation. Specifically, this Note will discuss how the PTAB’s decision in Praxair has expanded the scope of inter partes review and further added to the uncertainty of patentable subject matter

Log File-Based Dose Reconstruction to Moving Targets during Lung Stereotactic Body Radiation Therapy

Purpose: To perform film-based verification of 4D dose reconstruction to moving targets during lung stereotactic body radiation therapy (SBRT). Introduction: Current patient-specific quality assurance measures to test deliverability of plans with dynamic intensity modulation involve delivering beams to static measurement device and comparing the planned dose to measurement. However, motion-induced dose errors are not detected with static measurement. Previous studies have investigated combining machine log data with respiratory tracking to determine moving-target dose. By combining machine log data with anatomic and density information at each breathing phase from 4D-CT, intrafraction anatomical deformation due to respiration may be accounted for. However, to our knowledge, a film-based verification of dose reconstruction using machine log data, intrafraction respiratory tracking, and 4D-CT has yet to be performed. Methods: Lung SBRT plans were anonymized for 12 patients treated at our institution. Treatment plans were copied onto known geometry (programmable respiratory phantom) and dose was computed. Each SBRT plan was delivered to the phantom twice; first using 3 sec/breath (SPB), again at 6 SPB. Respiratory traces were acquired during treatment. Logfiles were acquired after treatment and partitioned according to breathing amplitude. Next, in-house code was used to import logfile beams into the treatment planning system. Dose was computed on each 4D-CT image using the imported beams and deformably accumulated. The accumulated, planned, and measured doses for each plan and breathing rate were compared using gamma analysis. Results: Gamma passing rates (GPR) (3%, 2mm, 10% threshold) of 4D dose reconstruction vs. planned dose were \u3e94% (mean 98.9% range 94.1%-100%) for all plans at each breathing rate. No significant difference was found between the 3 and 6 SPB GPRs (p=0.310). Overall, the 4D dose reconstructions were found to better agree with film measurement, within the tumor motion extent, than the treatment plan for both breathing rates (3 SPB: p=0.013, 6 SPB: p=0.017). Conclusions: Log file-based dose reconstruction was verified using film measurement for 12 lung SBRT plans delivered to a respiratory motion phantom. We showed that, given predictable phantom motion, 4D dose reconstruction resulted in significantly higher GPR compared with film than treatment plan to static geometry

Short-Range Sonar Development

Our project involves the development of a short-range detection sonar system using Matlab signal processing techniques. Currently, the necessary equipment has been researched and purchased, and the preliminary work on output signal creation, transmission, and reception is being conducted