Developing secondary students' understanding of the generality and purpose of proof

In spite of their fundamental role in mathematics, prior studies have found that high school students struggle to develop conceptual understanding of proof, and in particular, understand the generality requirement for proofs. Using design research methodology, I investigated an alternative method of introducing ten advanced 9th grade students to proof using problems and instructional strategies that emphasized the generality and purpose of proofs. The design study consisted of 14 sessions, each lasting approximately 30 minutes, that were held twice a week for seven weeks. Semistructured interviews were also conducted with students at the beginning and end of the study to track changes in students' understanding of proof. Findings from this dissertation study are presented as three stand-alone articles. In the first article, I present a framework for assessing students' understanding of different proof components based on Stylianides' (2007) definition of proof. I argue that the proposed framework provides a more nuanced perspective of students' understanding of proof and allows the researcher or teacher to identify aspects of proof that students understand as well as ways that they can improve their argument. In the second article, I analyze my design conjecture that using universal claims, or claims involving the universal quantifier "all" or "any", would afford opportunities for students to 1) engage in the reasoning-and-proving process, 2) talk about reasoning-and-proving, and 3) develop an intellectual necessity for proof. In the third article, I trace one student's successes and challenges in constructing proofs that adhered to the generality requirement over the course of the study. In this article, I demonstrate how her understanding of the purpose of proof supported her transition from empirical to deductive arguments on proof tasks. I conclude in chapter 5 with implications for research and teaching.Includes bibliographical reference

Technology-Mediated Dot Talks

Visual Dot Card Number Talks are a classroom routine which allow students to recognize patterns, build number fluency, and consider others’ thinking. We explicate the use of technologies—Nearpod and Desmos—for Dot Talks reimagined for use in virtual, hybrid, or in-person settings

South Africa\u27s Forgotten Freedom Fighters

More than two decades have passed since the end of apartheid, yet freedom is elusive for those who battled to bring democracy to South Africa. There are more than 16,000 military veterans living in South Africa, according to the Department of Military Veterans database. The real figure is likely much larger since the database is still under development. Many of these veterans are caged by the psychological, social and financial consequences of their past

Resin Production in Natural and Artificial Red-Cockaded Woodpecker Cavity Trees

Resin flow was measured in red-cockaded woodpecker (Picoides borealis Vieillot) clusters in longleaf pine (Pinus palustris Mill.) in the southern region of the Angelina National Forest, Texas. Resin flow (ml) at 1.4 m height over 24 hrs was measured from one 2.5 cm punch through the phloem between 0700 and 1000 hrs from March 1999 to September 2000, for a total of 9 measurements per tree. Resin was sampled in naturally active cavity trees, artificial (insert) active, natural inactive, artificial inactive and control pines (84 sample trees). Resin flow pattern was significantly different during the year, but not significantly different in the cavity tree type resin flow. Cavity trees in the 90th percentile (\u3e33.0 ml resin in 24 hrs) were defined as super resin producing. High average resin flows in August 1999 and September 2000 indicate when to sample resin for potential cavity trees. Regression equations were produced to estimate future resin production

Explaining differences in one teacher’s instruction across multiple tracked fifth-grade classes

In this article, we describe the case of “Keri,” a fifth-grade teacher who had completed an Elementary Mathematics Specialist (EMS) certification program. Drawn from a larger study investigating the knowledge, beliefs, and practices of EMSs, Keri\u27s case was unique in that she was teaching mathematics to four classes in a departmentalized structure, where students were placed into different classes according to perceived mathematics ability. Observations from the larger study revealed that Keri\u27s instructional practices did not align with her reported beliefs and knowledge. To explore this deviation, we conducted a case study where we observed Keri\u27s instruction across multiple classes and used interviews to explore reasons for Keri\u27s instructional decisions in terms of her perceived professional obligations. We found that Keri did employ practices that were aligned with her reported beliefs and knowledge such as pressing students for mathematical justifications, but only in her “higher ability” classes. Interview data suggested that Keri\u27s decisions were driven by a strong obligation to individual students, overriding other obligations. We describe implications of these findings, including the limitations of teacher assessments and surveys as proxies for teaching quality, and discuss recommendations for approaches to teacher development that account for teachers’ perceived obligations

School Inequality: Challenges and Solutions

“Welcome Remarks” were given by Wendy Collins Perdue, Dean of the University of Richmond School of Law, and Dr. Ronald Crutcher, President of the University of Richmond. (9:05 a.m.–9:15 a.m.) “Keynote” by Catherine Lhamon, U.S. Department of Education Assistant Secretary for Civil Rights. (9:15 a.m.–10:00 a.m.) “State-Level School Finance Panel” by Kimberly Robinson, Professor of Law at the University of Richmond School of Law; David Hinojosa, National Director of Policy with the Intercultural Development Research Association (IDRA); and Molly Hunter, Director of the Education Law Center’s Educational Justice program. (10:00 a.m.–11:10 a.m.) “Innovation in Addressing School Inequality Panel” by Lisa Scruggs, Partner with Duane Morris, LLP; Javaid Siddiqi, Director of Hunt-Kean Leadership Fellows at the Hunt Institute; and Jerusha Conner, Associate Professor and Graduate Education Program Coordinator at Villanova University, College of Liberal Arts and Sciences. (11:10 a.m.–12:20 p.m.) “School Discipline Panel” by Jason Nance, Associate Professor of Law and the Associate Director for Education Law and Policy at the Center on Children and Families at the University of Florida Levin College of Law; Marilyn Armour, a University Distinguished Teaching Professor and Director of the Institute for Restorative Justice and Restorative Dialogue at the University of Texas at Austin School of Social Work; Pamela Meanes, Partner at Thompson Coburn LLP and former President of the National Bar Association; and Meredith Harbach, Professor of Law at the University of Richmond School of Law. (1:20 p.m.–2:30 p.m.) Finally, a debate on the role the federal government should hold in the governance of public education featured Preston Green III, the John and Carla Klein Professor of Urban Education and Professor of Educational Leadership and Law at the University of Connecticut (arguing in favor of the federal government having an increased role) and Gerard Robinson, Resident Fellow of Educational Policy Studies at the American Enterprise Institute (arguing against). (2:30 p.m.–3:30 p.m.

A Systematic Review of Social Factors and Suicidal Behavior in Older Adulthood

Suicide in later life is a global public health problem. The aim of this review was to conduct a systematic analysis of studies with comparison groups that examined the associations between social factors and suicidal behavior (including ideation, non-fatal suicidal behavior, or deaths) among individuals aged 65 and older. Our search identified only 16 articles (across 14 independent samples) that met inclusion criteria. The limited number of studies points to the need for further research. Included studies were conducted in Canada (n = 2), Germany (n = 1), Hong Kong (n = 1), Japan (n = 1), Singapore (n = 1), Sweden (n = 2), Taiwan (n = 1), the U.K. (n = 2), and the U.S. (n = 3). The majority of the social factors examined in this review can be conceptualized as indices of positive social connectedness—the degree of positive involvement with family, friends, and social groups. Findings indicated that at least in industrialized countries, limited social connectedness is associated with suicidal ideation, non-fatal suicidal behavior, and suicide in later life. Primary prevention programs designed to enhance social connections as well as a sense of community could potentially decrease suicide risk, especially among men

Identification of 12 new susceptibility loci for different histotypes of epithelial ovarian cancer.

To identify common alleles associated with different histotypes of epithelial ovarian cancer (EOC), we pooled data from multiple genome-wide genotyping projects totaling 25,509 EOC cases and 40,941 controls. We identified nine new susceptibility loci for different EOC histotypes: six for serous EOC histotypes (3q28, 4q32.3, 8q21.11, 10q24.33, 18q11.2 and 22q12.1), two for mucinous EOC (3q22.3 and 9q31.1) and one for endometrioid EOC (5q12.3). We then performed meta-analysis on the results for high-grade serous ovarian cancer with the results from analysis of 31,448 BRCA1 and BRCA2 mutation carriers, including 3,887 mutation carriers with EOC. This identified three additional susceptibility loci at 2q13, 8q24.1 and 12q24.31. Integrated analyses of genes and regulatory biofeatures at each locus predicted candidate susceptibility genes, including OBFC1, a new candidate susceptibility gene for low-grade and borderline serous EOC