Investigating the Development of Proof Comprehension: The Case of Proof by Contradiction

This dissertation reports on an investigation of transition-to-proof students\u27 understanding of proof by contradiction. A plethora of research on the construction aspect of proof by contradiction is available and suggests that the method is one of the most difficult for students to construct and comprehend. However, there is little research on the students\u27 comprehension of proofs and, in particular, proofs by contradiction. This study aims to fill this gap in the literature. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, this study proposes a preliminary genetic decomposition for how a student might construct the concept `proof by contradiction\u27 and a series of five teaching interventions based on this preliminary genetic decomposition. Data was analyzed in two ways: (1) group analysis of the first two teaching interventions to consider students\u27 initial conceptions of the proof method and (2) case study analysis of two individuals to consider how students\u27 understanding developed over time. The genetic decomposition and teaching interventions were then revised based on the results of the data analysis. This study concludes with implications for teaching the concept of proof by contradiction and suggestions for further research on the topic

A Case Study of Community of Inquiry Presences and Cognitive Load in Asynchronous Online STEM Courses

The design and facilitation of asynchronous online courses can have notable impacts on students related to persistence, performance, and perspectives. This case study presents current conditions for cognitive load and Community of Inquiry (CoI) presences in an asynchronous online introductory undergraduate STEM course. Researchers present the novel use of Python script to clean and organize data and a simplification of the instructional efficiency calculation for use of anonymous data. Key relationships between cognitive load and CoI presences are found through validated use of NASA-TLX instrument and transcript analysis of discussion posts. The data show that student presences are not consistent throughout a course but are consistent across sections. Instructor presences are not consistent throughout a course or across sections. The study also explored predominant factors within each presence, confirming previous reports of low cognitive presence in discussions. The highest extraneous cognitive load was reported for understanding expectations and preparing an initial post. These results provide support for improvements to course design and instructor professional development to promote Community of Inquiry and reduce extraneous cognitive load

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

Phase 3 trials of ixekizumab in moderate-to-severe plaque psoriasis

BACKGROUND Two phase 3 trials (UNCOVER-2 and UNCOVER-3) showed that at 12 weeks of treatment, ixekizumab, a monoclonal antibody against interleukin-17A, was superior to placebo and etanercept in the treatment of moderate-to-severe psoriasis. We report the 60-week data from the UNCOVER-2 and UNCOVER-3 trials, as well as 12-week and 60-week data from a third phase 3 trial, UNCOVER-1. METHODS We randomly assigned 1296 patients in the UNCOVER-1 trial, 1224 patients in the UNCOVER-2 trial, and 1346 patients in the UNCOVER-3 trial to receive subcutaneous injections of placebo (placebo group), 80 mg of ixekizumab every 2 weeks after a starting dose of 160 mg (2-wk dosing group), or 80 mg of ixekizumab every 4 weeks after a starting dose of 160 mg (4-wk dosing group). Additional cohorts in the UNCOVER-2 and UNCOVER-3 trials were randomly assigned to receive 50 mg of etanercept twice weekly. At week 12 in the UNCOVER-3 trial, the patients entered a long-term extension period during which they received 80 mg of ixekizumab every 4 weeks through week 60; at week 12 in the UNCOVER-1 and UNCOVER-2 trials, the patients who had a response to ixekizumab (defined as a static Physicians Global Assessment [sPGA] score of 0 [clear] or 1 [minimal psoriasis]) were randomly reassigned to receive placebo, 80 mg of ixekizumab every 4 weeks, or 80 mg of ixekizumab every 12 weeks through week 60. Coprimary end points were the percentage of patients who had a score on the sPGA of 0 or 1 and a 75% or greater reduction from baseline in Psoriasis Area and Severity Index (PASI 75) at week 12. RESULTS In the UNCOVER-1 trial, at week 12, the patients had better responses to ixekizumab than to placebo; in the 2-wk dosing group, 81.8% had an sPGA score of 0 or 1 and 89.1% had a PASI 75 response; in the 4-wk dosing group, the respective rates were 76.4% and 82.6%; and in the placebo group, the rates were 3.2% and 3.9% (P<0.001 for all comparisons of ixekizumab with placebo). In the UNCOVER-1 and UNCOVER-2 trials, among the patients who were randomly reassigned at week 12 to receive 80 mg of ixekizumab every 4 weeks, 80 mg of ixekizumab every 12 weeks, or placebo, an sPGA score of 0 or 1 was maintained by 73.8%, 39.0%, and 7.0% of the patients, respectively. Patients in the UNCOVER-3 trial received continuous treatment of ixekizumab from weeks 0 through 60, and at week 60, at least 73% had an sPGA score of 0 or 1 and at least 80% had a PASI 75 response. Adverse events reported during ixekizumab use included neutropenia, candidal infections, and inflammatory bowel disease. CONCLUSIONS In three phase 3 trials involving patients with psoriasis, ixekizumab was effective through 60 weeks of treatment. As with any treatment, the benefits need to be weighed against the risks of adverse events. The efficacy and safety of ixekizumab beyond 60 weeks of treatment are not yet known

Community of Inquiry and Cognitive Load in Online STEM: Persistence, Performance, and Perspectives

Overview:This submission is for the Engaged Student Learning Track of the Development and Implementation, Level I program for NSF IUSE. Our goal is to support Community of Inquiry in asynchronous online STEM learning while mitigating impacts to cognitive load, with the goal of positively impacting persistence, performance, and perspectives. Online courses in general face large withdrawal rates; online STEM courses are no exception. In this study, we will design and test a pilot program for infusing Community of Inquiry into the asynchronous course discussions, including a best practices redesign of the discussion prompts, rubrics, and instructions by subject matter experts and instructional designers as well as professional development for instructors teaching the courses. This study will be implemented at Embry-Riddle Aeronautical University. The pilot program will utilize existing institutional frameworks to support the redesign, with scaffolding to support the redesign and professional development. In our mixed-methods study, we will collect student information related to academic success (course grades, discussion transcript analysis), persistence (withdrawal rate), and perspectives (survey – STEM attitudes(1), cognitive load (2), community of inquiry (3)), complemented by focus group interviews.Intellectual Merit:The key deliverable will be the pilot discussion framework designed from modern best practices, which can be transferred and scaled to other asynchronous online STEM courses. Another key contribution of this study will be the data generated regarding community of inquiry and cognitive load versus STEM attitudes. This data can be disseminated for a broad audience through manuscript publication in reliable open access venues.Broader Impacts:This project will generate a Community of Inquiry – Cognitive Load framework for asynchronous online STEM courses that supports persistence, performance, and positive attitudes through promotion of social, teaching, and cognitive presences while mitigating cognitive load impacts. This project offers a benefit to society through research disseminated through open access, high impact venues aimed at practitioners and administrators. This framework can be used by course designers and educators for both design and facilitation of asynchronous online discussions. The framework efficacy will be thoroughly evaluated on multiple criteria through this project. The research generated will advance discovery and promote learning through:• establishing the influence of community of inquiry and cognitive load on persistence, performance, and perspectives;• clearly identifying withdrawal reasons from asynchronous online STEM courses, allowing for targeted interventions beyond the framework

Asynchronous Discovery Activity: Learning to Fly with the Wind

This project focuses on the design and impact of an asynchronous, discovery-based activity in an asynchronous College Algebra course. The activity will be written and administered in Doenet. Generally, the activity would walk students through multiple stages of solving a traditional precalculus problem: A plane is going from city A to city B. There is a wind of X mph blowing from XX direction. What direction does the pilot need to make to adjust for the wind? The Doenet activity would provide feedback as the student investigates what additional information they may need to solve the question, potential concepts they could rely on, application of one concept, and then validation with a visual of a plane flying between the two cities (with loop to return if they weren’t correct). After creating the activity, it would be included as a mandatory assignment within a course I would teach in Spring 2024. While data would be collected at this point, money from the mini-grant would be used to solicit students to participate in a survey on their experiences with the activity and record their thoughts as they interact with the activity

A Case Study of Community of Inquiry Presences and Cognitive Load in Asynchronous Online STEM Courses

The design and facilitation of asynchronous online courses can have notable impacts on students related to persistence, performance, and perspectives. This case study presents current conditions for cognitive load and Community of Inquiry (CoI) presences in an asynchronous online introductory undergraduate STEM course. Researchers present the novel use of Python script to clean and organize data and a simplification of the instructional efficiency calculation for use of anonymous data. Key relationships between cognitive load and CoI presences are found through validated use of NASA-TLX instrument and transcript analysis of discussion posts. The data show that student presences are not consistent throughout a course but are consistent across sections. Instructor presences are not consistent throughout a course or across sections. The study also explored predominant factors within each presence, confirming previous reports of low cognitive presence in discussions. The highest extraneous cognitive load was reported for understanding expectations and preparing an initial post. These results provide support for improvements to course design and instructor professional development to promote Community of Inquiry and reduce extraneous cognitive load

Argumentative knowledge construction in asynchronous calculus discussion boards

Classroom discourse constitutes a fundamental activity in which learners can acquire knowledge. A multifaceted phenomenon, any enacted classroom discussion entails the enmeshment of social, cultural, curricular, and modality factors. Focusing specifically on discourse in the context of mathematical discussion activities in the asynchronous online modality, we propose use of Weinberger and Fischer’s (2006) Argumentative Knowledge Construction framework for design research. We contend that this framework, suitably amended to meet the particular needs of mathematics courses, may enable in-depth analysis of major dimensions of students’ knowledge construction as they engage in activities in an asynchronous modality. Research using this framework in the context of face-to-face mathematical learning (Keene, Williams, & McNeil, 2016) and in online settings in other disciplines (Schrire, 2006; Clark & Sampson, 2008; Dubovi & Tabak. 2020) has been reported