Doctor of Philosophy

dissertationWith the ever-increasing amount of available computing resources and sensing devices, a wide variety of high-dimensional datasets are being produced in numerous fields. The complexity and increasing popularity of these data have led to new challenges and opportunities in visualization. Since most display devices are limited to communication through two-dimensional (2D) images, many visualization methods rely on 2D projections to express high-dimensional information. Such a reduction of dimension leads to an explosion in the number of 2D representations required to visualize high-dimensional spaces, each giving a glimpse of the high-dimensional information. As a result, one of the most important challenges in visualizing high-dimensional datasets is the automatic filtration and summarization of the large exploration space consisting of all 2D projections. In this dissertation, a new type of algorithm is introduced to reduce the exploration space that identifies a small set of projections that capture the intrinsic structure of high-dimensional data. In addition, a general framework for summarizing the structure of quality measures in the space of all linear 2D projections is presented. However, identifying the representative or informative projections is only part of the challenge. Due to the high-dimensional nature of these datasets, obtaining insights and arriving at conclusions based solely on 2D representations are limited and prone to error. How to interpret the inaccuracies and resolve the ambiguity in the 2D projections is the other half of the puzzle. This dissertation introduces projection distortion error measures and interactive manipulation schemes that allow the understanding of high-dimensional structures via data manipulation in 2D projections

Proceedings of the 15th International Conference on Technology in Mathematics Teaching: Making and Strengthening "Connections and Connectivity" for Teaching Mathematics with Technology

The 15th International Conference on Technology in Mathematics Teaching (ICTMT 15) took place on September 13–16, 2022, in the Danish School of Education, Aarhus University, located on campus Emdrup, in the Northwestern district of Copenhagen, Denmark. There were a total of 66 participants from 15 different countries. ICTMT 15 certainly focused on the impacts that the coronavirus pandemic has had on global mathematics education. However, it looked at the impacts of digital technology from a much wider perspective. In particular, the conference aimed to highlight how technology facilitates the multiple “Connections and Connectivity” between us all to achieve the goals of purposeful mathematics education in the early 21st century. By “Connections” we mean the interrelationships between researchers, teachers, students, parents, policymakers, and industry (big and small). “Connectivity” includes oral, aural, textual and gestural communications as mediated by the internet, learning environments and classroom activities. Together, “Connections and Connectivity” describes the relationships between people, between different ideas and strategies to teach, and between people and environments. It offers a frame through which to interpret assessment in mathematics education as a more formative process from the point of view of both teachers and students