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…

The purpose of this mixed methods study was to investigate the implementation of a researcher-created numberless word problem instructional unit designed to introduce second-grade students to word problems in the semantic category of "compare with the word more, smaller unknown." The 10-day instructional unit incorporated three elements:…

Algebra I encompasses several topics that serve as a basis for students' subsequent mathematics courses as they progress in school. Some of the key topics that students struggle with is solving linear equations and algebraic word problems. There are several factors that may contribute to this ongoing struggle for students such as the structure of…

Background: With around 200,000 autistic students set to enter universities and the workforce in the next decade, inclusive programs are receiving increased attention to helping educational systems carefully integrate these students' needs. Moreover, educational programs recognize that problem-solving is critical for school success and optimal…

Online courses have become an important part of a student's academic career. The variety and number of online course offerings grow each year. Unfortunately, there are a few drawbacks to current online courses, including differences between success rates of general education courses and their face-to-face counterparts. Current research indicates…

Since most worldly phenomena can be expressed via language, language is a crucial medium for transferring information and integrating multiple information sources. For example, humans can describe what they see, hear and feel, and also explain how they move with words. Conversely, humans can imagine scenes, sounds, and feelings, and move their…

Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…

Spatial inscriptions of information--visualizations--can be a great asset to problem solving. Calendars and Cartesian graphs are two examples of visualizations that reorganize non-spatial information (temporal and quantitative) into a visual format. People readily rely on these visualizations to plan and see patterns, which intuitively attests to…

This dissertation targets young students' developing knowledge relevant to a fundamental practice in academic mathematics: the use of diagrams to represent idealized mathematical objects whose properties are established by definition (as in the use of drawn dots to represent zero-dimensional points and drawn lines to represent one-dimensional…

Despite the growing attention being paid to teaching mathematics for students with disabilities, the existing research tends to focus on mathematical skill acquisition, but not on skill maintenance. Maintenance of mathematical skills is especially important as mathematics has applications in daily life and is directly related to important life…

Mathematics classrooms have reflected little change over the past century, consisting of the learning of inert, disjoint procedures with few connections between mathematics concepts and students' everyday experiences. However, problem posing, that is, the reformatting of existing problems and the generation of new ones, has been promoted by the…

The study of student representation use and specifically the distinction between analytic and visual representations has fueled a long line of mathematics education literature that began more than 35 years ago. This literature can be partitioned into two bodies of work, one that is primarily cognitive and one that is primarily social. In spite of…

The work described here represents an effort to understand and influence visual attention while solving physics problems containing a diagram. Our visual system is guided by two types of processes--top-down and bottom-up. The top-down processes are internal and determined by ones prior knowledge and goals. The bottom-up processes are external and…

Science, technology, engineering, and mathematics (STEM) education is a national focus. Engineering education, as part of STEM education, needs to adapt to meet the needs of the nation in a rapidly changing world. Using computer-based visualization tools and corresponding 3D printed physical objects may help nontraditional students succeed in…

This thesis explores the role of multiple representations in introductory physics students' problem solving performance through several investigations. Representations can help students focus on the conceptual aspects of physics and play a major role in effective problem solving. Diagrammatic representations can play a particularly important role…