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

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

Students' 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 re�finements to our genetic decomposition and o�ffer 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.</p

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

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