The Teaching and Learning of Parametric Functions: A Baseline Study

This dissertation reports on an investigation of fifteen second-semester calculus students\u27 understanding of the concept of parametric function, as a special relation from a subset of R to a subset of R2. A substantial amount of research has revealed that the concept of function, in general, is very difficult for students to understand. Furthermore, several studies have investigated students\u27 understanding of various types of functions. However, very little is known about how students reason about parametric functions. This study aims to fill this gap in the literature. Employing Action--Process--Object--Schema (APOS) theory as the guiding theoretical perspective, this study proposes a preliminary genetic decomposition for how a student might construct the concept of parametric function. To determine whether the students in this study made the constructions called for by the preliminary genetic decomposition or other constructions not considered in the preliminary genetic decomposition, data is analyzed regarding students\u27 reasoning about parametric functions. In particular, this study explores (1) students\u27 personal definitions of parametric function; (2) students\u27 reasoning about parametric functions given in the form p(t)=(f(t),g(t)); (3) students\u27 reasoning about parametric functions on a variety of tasks, such as converting from parametric to standard form, sketching a plane curve defined parametrically, and constructing a parametric function to describe a real-world situation; and (4) students\u27 reasoning about the invariant relationship between two quantities varying simultaneously when described in both a graph and a real-world problem. Then the genetic decomposition is revised based on the results of the data analysis. This study concludes with implications for teaching the concept of parametric function and suggestions for further research on this topic

Weak Primary Decomposition of Modules Over a Commutative Ring

This paper presents the theory of weak primary decomposition of modules over a commutative ring. A generalization of the classic well-known theory of primary decomposition, weak primary decomposition is a consequence of the notions of weakly associated prime ideals and nearly nilpotent elements, which were introduced by N. Bourbaki. We begin by discussing basic facts about classic primary decomposition. Then we prove the results on weak primary decomposition, which are parallel to the classic case. Lastly, we define and generalize the Compatibility property of primary decomposition

Use of Inherited Differences among Strains of Inbred Mice to Study Genetic Determinants of Steroid Biosynthesis a

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72865/1/j.1749-6632.1984.tb38364.x.pd

Motility of select ovarian cancer cell lines: Effect of extracellular matrix proteins and the involvement of PAK2

The interaction between tumor cells and extracellular matrix (ECM) proteins influences cell migration and the invasive behavior of cancer cells. In this study, we provide experimental evidence that collagen I and fibronectin affect ovarian cancer cell migration. In vitro wound healing assays and transwell migration assays were used to measure both total wound healing and directionality of individually migrating OV2008 and C13 ovarian cancer cells on glass, collagen I and fibronectin. Involvement of p21-activated kinase 2 (Pak2) in the motility of these cell lines was investigated using a chemical inhibitor as well as siRNA transfection. Culturing ovarian cancer cells on collagen type I (COLL) increased the migratory ability of OV2008 and C13 cells by increasing the directional migration of cells. In contrast, fibronectin (FN) decreased the migratory ability of OV2008 cells by reducing their ability to migrate directionally. When both cell lines are cultured on COLL and treated with increasing concentrations of a PAK inhibitor (IPA-3), there is a dose-dependent response such that there is a decrease in migration with an increase in inhibitor concentration. Further experiments utilizing PAK2 knockdown via siRNA transfection demonstrated significantly reduced migration of OV2008 cells on COLL as compared to those receiving control siRNA. In conclusion, our results indicate that FN and COLL affect the motility of the selected ovarian cancer cells lines and the effect of COLL is likely mediated, at least in part, by PAK2. A better understanding of the molecular events that contribute to tumor invasion and metastasis is crucial for developing novel treatment strategies to improve the long-term survival of women with ovarian cancer. As PAK2 is involved in motility, it should be further explored as a pro-metastatic gene in ovarian cancer

Students\u27 Understanding of the Concepts Involved in One-Sample Hypothesis Testing

Hypothesis testing is a prevalent method of inference used to test a claim about a population parameter based on sample data, and it is a central concept in many introductory statistics courses. At the same time, the use of hypothesis testing to interpret experimental data has raised concerns due to common misunderstandings by both scientists and students. With statistics education reform on the rise, as well as an increasing number of students enrolling in introductory statistics courses each year, there is a need for research to investigate students’ understanding of hypothesis testing. In this study we used APOS Theory to investigate twelve introductory statistics students’ reasoning about one-sample population hypothesis testing while working two real-world problems. Data were analyzed and compared against a preliminary genetic decomposition, which is a conjecture for how an individual might construct an understanding of a concept. This report presents examples of Actions, Processes, and Objects in the context of one-sample hypothesis testing as exhibited through students’ reasoning. Our results suggest that the concepts involved in hypothesis testing are related through the construction of higher-order, coordinated Processes operating on Objects. As a result of our data analysis, we propose refinements to our genetic decomposition and offer suggestions for instruction of one-sample population hypothesis testing. We conclude with appendices containing a comprehensive revised genetic decomposition along with a set of guided questions that are designed to help students make the constructions called for by the genetic decomposition

Preliminary Evaluation of the InCHARGE Program Among Older African Americans in Rural Alabama

Objective: Blindness rates among older African Americans are two times higher than for older whites. Our purpose was to understand attitudes about eye care and perceived barriers to care among older African Americans living in rural Alabama and to determine whether an educational program reduced perceived barriers to care. InCHARGE, an eye health education program for older African Americans, promotes eye disease prevention by conveying the personal benefits of annual dilated comprehensive eye care and by teaching strategies to minimize barriers to eye care. Design and Participants: InCHARGE was presented in five senior centers to 111 individuals. Using a questionnaire before and three months after InCHARGE, we evaluated what impact InCHARGE had on attitudes and knowledge about prevention and strategies for reducing barriers. Results: Before InCHARGE, 52.3% reported receiving an eye examination in the past year. Almost all indicated that they felt finding, getting to, and communicating with a doctor were not problems yet about one-quarter indicated that the cost of an examination and/or eyeglasses were problems. After InCHARGE the percentage saying that cost was a problem increased to almost half. Conclusions: Older African Americans in rural Alabama have positive attitudes about comprehensive eye care, yet only about half reported receiving an exam by an eye care provider in the past year. The cost of care is a barrier for many, a problem that was not mitigated by InCHARGE. In order to improve eye health in this population, eye health education initiatives are not enough; economic strategies must be implemented to address the cost barrier