Analyzing the structure of a dataset is a challenging problem in high-dimensions as the volume of the space increases at an exponential rate and typically, data becomes sparse in this high-dimensional space. This poses a significant challenge to machine learning methods which rely on exploiting structures underlying data to make meaningful inferences. This dissertation proposes the <i>n</i>-TARP method as a building block for high-dimensional data analysis, in both supervised and unsupervised scenarios.<div><br></div><div>The basic element, <i>n</i>-TARP, consists of a random projection framework to transform high-dimensional data to one-dimensional data in a manner that yields point separations in the projected space. The point separation can be tuned to reflect classes in supervised scenarios and clusters in unsupervised scenarios. The <i>n</i>-TARP method finds linear separations in high-dimensional data. This basic unit can be used repeatedly to find a variety of structures. It can be arranged in a hierarchical structure like a tree, which increases the model complexity, flexibility and discriminating power. Feature space extensions combined with <i>n</i>-TARP can also be used to investigate non-linear separations in high-dimensional data.<br></div><div><br></div><div>The application of <i>n</i>-TARP to both supervised and unsupervised problems is investigated in this dissertation. In the supervised scenario, a sequence of <i>n</i>-TARP based classifiers with increasing complexity is considered. The point separations are measured by classification metrics like accuracy, Gini impurity or entropy. The performance of these classifiers on image classification tasks is studied. This study provides an interesting insight into the working of classification methods. The sequence of <i>n</i>-TARP classifiers yields benchmark curves that put in context the accuracy and complexity of other classification methods for a given dataset. The benchmark curves are parameterized by classification error and computational cost to define a benchmarking plane. This framework splits this plane into regions of "positive-gain" and "negative-gain" which provide context for the performance and effectiveness of other classification methods. The asymptotes of benchmark curves are shown to be optimal (i.e. at Bayes Error) in some cases (Theorem 2.5.2).<br></div><div><br></div><div>In the unsupervised scenario, the <i>n</i>-TARP method highlights the existence of many different clustering structures in a dataset. However, not all structures present are statistically meaningful. This issue is amplified when the dataset is small, as random events may yield sample sets that exhibit separations that are not present in the distribution of the data. Thus, statistical validation is an important step in data analysis, especially in high-dimensions. However, in order to statistically validate results, often an exponentially increasing number of data samples are required as the dimensions increase. The proposed <i>n</i>-TARP method circumvents this challenge by evaluating statistical significance in the one-dimensional space of data projections. The <i>n</i>-TARP framework also results in several different statistically valid instances of point separation into clusters, as opposed to a unique "best" separation, which leads to a distribution of clusters induced by the random projection process.<br></div><div><br></div><div>The distributions of clusters resulting from <i>n</i>-TARP are studied. This dissertation focuses on small sample high-dimensional problems. A large number of distinct clusters are found, which are statistically validated. The distribution of clusters is studied as the dimensionality of the problem evolves through the extension of the feature space using monomial terms of increasing degree in the original features, which corresponds to investigating non-linear point separations in the projection space.<br></div><div><br></div><div>A statistical framework is introduced to detect patterns of dependence between the clusters formed with the features (predictors) and a chosen outcome (response) in the data that is not used by the clustering method. This framework is designed to detect the existence of a relationship between the predictors and response. This framework can also serve as an alternative cluster validation tool.<br></div><div><br></div><div>The concepts and methods developed in this dissertation are applied to a real world data analysis problem in Engineering Education. Specifically, engineering students' Habits of Mind are analyzed. The data at hand is qualitative, in the form of text, equations and figures. To use the <i>n</i>-TARP based analysis method, the source data must be transformed into quantitative data (vectors). This is done by modeling it as a random process based on the theoretical framework defined by a rubric. Since the number of students is small, this problem falls into the small sample high-dimensions scenario. The <i>n</i>-TARP clustering method is used to find groups within this data in a statistically valid manner. The resulting clusters are analyzed in the context of education to determine what is represented by the identified clusters. The dependence of student performance indicators like the course grade on the clusters formed with <i>n</i>-TARP are studied in the pattern dependence framework, and the observed effect is statistically validated. The data obtained suggests the presence of a large variety of different patterns of Habits of Mind among students, many of which are associated with significant grade differences. In particular, the course grade is found to be dependent on at least two Habits of Mind: "computation and estimation" and "values and attitudes."<br></div>

Learn how to realize a measurable increase in student engagement and achievement in mathematics through a guided, collaborative, and active process grounded in mathematics. Students and teachers collaboratively devise a data-driven plan of action that moves learning forward for all students and effectively supports at-risk secondary students in urban environments. Learn how teachers in the LMU Math and Science Teaching Program effectively implement assessments as motivations for student achievement and develop opportunities for students to demonstrate comprehension and retention of essential content over time. Students become active participants in the assessment process in an environment where learning is an individual progression and risk-taking is valued and encouraged. Find out how students, guided by teacher-provided descriptive feedback, make decisions in a process of self-reflection in which they critically analyze and compare their learning outcomes to expectations of content mastery. By comparing mastery to current performance, students utilize failure and engage in error analysis to deconstruct prior shortcomings and devise a plan of action that will move learning forward thereby overcoming failure.

<p> </p> <p>Traditional kinship systems involve the organization of individuals who are biologically connected. However, such systems have evolved beyond bloodlines to incorporate individuals that are biologically unassociated but operate in familial-like roles due to shared spaces and/or experiences. Historically, kinship systems or cultural networks have functioned as the cornerstone of survival for those of the Black lived experience. From the days of legalized human chattel slavery to present-day movements seeking justice for the minoritized, the foundation of kinship was typically built through the local church, the assumed maternal positions by Black women, Black secret societies and more. They each served, and continue to serve, as a means for survival and success against a systemically oppressive society. This study explores the notion and existence of kinship systems at historically Black colleges and universities (HBCUs). It specifically examines how fictive kinships through the lens of faculty-student dynamics, religion, and social activities, potentially influences the academic experience of Black students at HBCUs that currently have an above average retention rate. As America’s educational institution has lacked diversity, inclusion, justice, and equity for Black people for countless years, the primary mission of this study was to amplify Black student voices which have traditionally been suppressed. A supplemental goal of this study was to offer Black students tools for introspection that will aid them in navigating possible barriers to (post) educational success. In turn, this study gives insight to predominantly white institutions of higher learning on how to positively enhance the experience and retention of Black students, and the overall structure of diversity and inclusion on campus.</p>

<p>In this dissertation, I investigate the authentic learning experiences of students participating in the Engineering Projects in Community Service (EPICS) program at Purdue University within the framework of authentic education. Utilizing a quantitative approach, the study assesses the performance of new and returning students across five key outcomes that measure authentic learning during a single semester. The analysis of variance (ANOVA) revealed significant main effects for time of assessment and type of student on performance, with an overall improvement in all outcomes observed from mid-term to final evaluations and returning students typically outperforming new students. Interaction effects between time and type were also examined, revealing subtle yet complex dynamics in students’ learning trajectories. The findings hold implications for enhancing authentic learning, especially in engineering design contexts, and offer insights to guide future implementation of and improvements to authentic education initiatives, particularly the EPICS program. Despite certain limitations, the research opens avenues for future investigations into diverse aspects of authentic education in STEM and beyond.  </p>

<p dir="ltr">Institutions of higher education within the United States have not had a reputation of inclusivity. The discrimination and oppression of people with disabilities is an important topic of conversation within these educational spaces, not only to change the way that society thinks of disability on a whole, but to incite discussions surrounding the best ways to support students with disabilities and their educational goals. This paper will provide a deconstruction of what ableism is, how it impacts mental health and wellness and how it shows up within institutions of higher education. This paper will also provide details on a training course for higher education faculty members that provides practical applications of the ethical ways of creating a supportive learning environment for students with disabilities. This paper will explore how critical disability theory, the social model and some aspects of the medical model can be utilized to provide faculty and staff with the competency to understand and interact with students with disabilities in ways that not only support their learning but contribute to positive social change and the deconstruction of ableist actions.</p>

<p dir="ltr">Engineering Technology (ET) programs are pivotal in preparing graduates for the demands of the modern workforce, particularly in quality management systems (QMS). This study examines the alignment between QMS knowledge and experience gained by graduates in ABET-accredited ET programs and the expectations of industry. Through a dual-survey approach, targeting both industry leaders and academic educators, the research elucidates current QMS practices in industry, the scope of QMS education, and the resulting preparedness of graduates for professional roles. Findings indicate a discernible gap between industry expectations and current academic offerings in QMS education. While industry professionals rely on established QMS frameworks such as ISO 9001 and Lean Six Sigma, academic programs often limit their coverage to theoretical underpinnings rather than hands-on, practical applications. This discrepancy highlights the need for a more robust, application-oriented curriculum that bridges theoretical knowledge with real-world practice. Recommendations include a call for greater integration of practical QMS training within academic programs and stronger partnerships between academia and industry to foster educational content that aligns with professional QMS applications.</p>

The facilitator model for dual credit offers a way for student to earn directly transcripted credit to colleges and universities, overcoming many barriers faced by other dual credit models. Successful implementation of this model requires high degree of involvement from the cooperating institution. This IRB approved qualitative case study explored the needs of five teacher facilitators in both summer professional development and on-going support throughout the school year when implementing a facilitator model for dual credit with open-ended design coursework. Code-recode and axial coding techniques were applied to over 90 hours of transcribed data, artifacts, and observations from a seven month period to find emerging themes and offer recommendations for implementation.<p></p>

<p dir="ltr">This dissertation explores the intricate dynamics between the self-perceptions of undergraduate mathematics (UM) educators and their conceptions of excellent teaching practices conducive to student learning. Employing a sequential mixed methods approach, the study addresses two primary research questions. First, it investigates educators' self-perceptions within the realm of UM teaching, examining potential variances based on educators' Professional Status and Educational Institution (PSEI) affiliations and experience levels. Second, it delves into educators' perspectives on aspects of excellent UM teaching, scrutinizing potential disparities rooted in PSEI affiliations and experience levels, while also exploring the manifestations of Mathematics Teachers' Specialized Knowledge (MTSK) and teaching self-concept within these descriptors.</p><p dir="ltr">Drawing upon Shavelson's self-concept (1976) framework and Carrillo and colleagues' (2018) MTSK framework, data collection involved a Likert-style questionnaire augmented by open-ended inquiries, followed by qualitative case studies featuring eight participants from diverse Carnegie classifications. Findings demonstrate educators' overall confidence in their teaching abilities, with notable discrepancies observed among educators from associate's colleges and doctoral universities. Through thematic analysis, key dimensions of excellent teaching emerged, including active learning, student engagement, problem-solving, and positive learning environments.</p><p dir="ltr">This study yields implications for educational practice and institutional policy. Educators can leverage identified themes to inform professional development initiatives tailored to enhance UM teaching effectiveness. Furthermore, the validated instrument offers institutions a means to assess educators' confidence levels, facilitating targeted support within mathematics departments.</p><p dir="ltr">In conclusion, this dissertation contributes valuable insights into the multifaceted interplay between educators' self-perceptions, teaching practices, and student learning outcomes within the context of UM instruction.</p>

<p>For the projects titled Undergraduate Students’ Interpretation of Fischer and Haworth Carbohydrate Projections and Undergraduate Students' Interpretation of Glycosidic Bonds – there is a prevalent issue in biochemistry education in which students display fragmented knowledge of the biochemical concepts learned when asked to illustrate their understandings (via drawings, descriptions, analysis, etc.). In science education, educators have traditionally used illustrations to support students’ development of conceptual understanding. However, interpreting a representation is dependent on prior knowledge, ability to decode visual information, and the nature of the representation itself. With a prevalence of studies conducted on visualizations, there is little research with a focus on the students’ interpretation and understanding of carbohydrates and/or glycosidic bonds. The aim of these projects focuses on how students interpret representations of carbohydrates and glycosidic bonds. This study offers a description of undergraduate students’ understanding and interpretation using semi-structured interviews through Phenomenography, Grounded Theory and the Resources Frameworks. The data suggests that students have different combinations of (low or high) accuracy and productivity for interpreting and illustrating carbohydrates and glycosidic bonds, among other findings to be highlighted in their respective chapters. More effective teaching strategies can be designed to assist students in developing expertise in proper illustrations and guide their thought process in composing proper explanations in relation to and/or presence of illustrations.</p> <p><br></p> <p>For the project titled Impact of the Pandemic on Student Readiness: Laboratories, Preparedness, and Support – it was based upon research by Meaders et. al (2021) published in the International Journal of STEM Education. Messaging during the first day of class is highly important in establishing positive student learning environments.  Further, this research suggests that students are detecting the messages that are communicated.  Thus, attention should be given to prioritizing what information and messages are most important for faculty to voice. There is little doubt that the pandemic has had a significant impact on students across the K-16 spectrum.  In particular, for undergraduate chemistry instructors’, data on the number of laboratories students completed in high school and in what mode would be important information in considering what modifications could be implemented in the laboratory curriculum and in messaging about the laboratory activities – additionally on how prepared students feel to succeed at college work, how the pandemic has impacted their preparedness for learning, and what we can do to support student learning in chemistry can shape messaging on the first day and for subsequent activities in the course.  An initial course survey that sought to highlight these student experiences and perspectives will be discussed along with the impact on course messaging and structure.    </p> <p><br></p>