In the first part, we investigate Birman, Ko, and Lee's left canonical form of a braid and give a new diagrammatic approach. We use the left canonical form to characterize almost strongly quasipositive braids. In the second part, we investigate students' confidence in mathematics and problem solving skills. Every math instructor has heard students…

To make sense of our interconnected and algorithm driven world, students increasingly need proficiency with computational thinking (CT), systems thinking (ST), and computational modeling. One aspect of computational modeling that can support students with CT, ST, and modeling is testing and debugging. Testing and debugging enables students to…

This dissertation examines the development of combinatorial reasoning in high school students through an in-depth analysis of a problem-solving session involving the Pizza Problem. The study, part of the long-term Rutgers-Kenilworth longitudinal research project, focuses on four 11th-grade students as they explore and connect concepts related to…

As students with autism spectrum disorder (ASD) participate in the general education setting at greater rates, they are expected to meet rigorous grade-level academic standards. Students with ASD typically struggle with problem solving, higher-ordered thinking, and mathematical reasoning. When students enter middle school, mathematics becomes more…

This qualitative study was conducted to understand the instructional strategies used by high school integrated STEM (iSTEM) teachers to apply real-world problem solving in their classrooms in the state of Indiana. The problem addressed by this study was the need to understand the instructional strategies employed by iSTEM teachers in their…

This study investigated evidence of how students' mental models of fundamental kinematic relations evolved (i.e., developed cognitively over time) as observed during an introductory course in calculus-based classical mechanics. The core of the curriculum is based on a claim known as Galileo's principle of relativity. The course material comprised…

Through instructional tasks, teachers can present content in ways that require students to apply critical thinking, problem-solving, and other cognitive skills. The purpose of this mixed methods collective case study was to determine how teacher-assigned instructional tasks in mathematics impact students' learning of mathematical concepts. In this…

Uncertainties are inherent in education as learners encounter gaps in their awareness of topics, receive new information that potentially conflicts with prior understandings, and grapple with known unknowns in order to construct new knowledge. While researchers have shown that students productively struggling with problematized and uncertain…

The purpose of this study was to evaluate the effect of an intervention utilizing modified schema-based instruction (MSBI) on the mathematical problem-solving ability of elementary school students with mild to moderate disabilities. Four participants were taught to solve percent of change word problems, which required them to calculate the…

This study examined the engagement of preservice elementary teachers (PSTs) with mathematics tasks that were open in various ways and to varying extents. In K-12 settings, practitioners who have implemented recently popular open tasks report increased engagement and from a wider variety of students. While these anecdotal reports and related…

In 2 experiments, I studied the effects of an Equivalence Based Instruction (EBI) math intervention on the emergence of untaught selection responses and abstraction to production responses. In Experiment I, using a multiple baseline design, I implemented the EBI intervention among a group of 17 first grade participants with varying levels of math…

In this dissertation I explore instructors' work with instructional tasks as they plan to introduce various representations of derivatives. Drawing on the notion of "framing" from sociology, I adopt a situative lens to study calculus instructors' planning of instructional tasks. I rely on Herbst and Chazan's (2012) notion of…

This quasi-experimental design study aimed to examine how exposure to supporting productive struggle teaching practice impacts students' conceptual understanding, procedural fluency, and strategic competence when solving high-demanding quadratic functions problem solving tasks. Results suggested that students in the supporting productive struggle…

Mathematics plays a crucial role in engineering education. Students who are good at math are often advised to consider engineering as a career. In this study, engineers, mathematics teachers, and engineering professors were asked about the type of mathematics representative of engineering, whether the engineering problems created by math…

The study examines the perspectives of using the Desmos calculator of Algebra I students' conceptual understanding and procedural fluency to write, graph, and solve linear equations in Algebra I STAAR. While the students have continuously used technology for mathematics assessment, emergent bilingual students in South Texas still need help passing…