<p>  </p> <p>Radian angle measure is central to learning trigonometry, with researchers providing evidence that a coherent understanding of radian contributes to a coherent understanding of trigonometric and inverse trigonometric functions. However, there are few opportunities for students to engage with curricular situations that involve radian angle measure. The purpose of this dissertation is to explore and provide insights into preservice mathematics teachers’ (PMTs’) conceptions of radian angle measure using three curricular situations. The first chapter reviews the relevant literature, which reported that PMTs’ conceptions of radian angle measure involve angles measured in terms of π, in relation to degrees, and in relation to the unit circle. In chapter two, I explored PMTs’ conceptions of radian angle measure using textbook representations. Seven PMTs participated in a think-aloud semi-structured interviews, where they defined radian angle measure from six textbook diagrams of radian, including a diagram of the unit circle. In chapter three, building on literature that reported that PMTs’ conceptions of radian angle measure involve relating radian to degrees, I explored how PMTs conceptualize this relationship. Five PMTs participated in semi-structured interviews, where they described radian angle measure given the angle measure in degrees. In chapter four, I explored the PMTs’ conceptions of radian angle measure given a novel context. Four PMTs participated in semi-structured virtual interviews, where they engaged with a digital activity that involves radian angle measure in the context of light reflection. Some of the dissertation’s findings align with previous research, where PMTs’ conceptualized radian angle measure in relation to the unit circle. However, this dissertation provides empirical evidence of why the PMTs refer to the unit circle. The PMTs acknowledged knowing the unit circle from memorization, but also explained that the purpose for using the unit circle is efficiency. At the same time, the PMTs acknowledged limitations in the unit circle and in their conceptions of it. Overall findings from the dissertation demonstrate the complexity of PMTs’ conceptions of radian angle measure. The PMTs’ conceptions were reported as concept definitions, ways of thinking, and spatial ways of thinking. The PMTs demonstrated flexibility with reasoning about radian angle measure using foundational conceptions in learning higher mathematics topics (e.g., proportional reasoning concepts, spatial ways of thinking). By positioning the PMTs as knowers and thinkers with valuable insights to provide, I was able to uncover and report a collection of conceptions that were demonstrated by PMTs when a curricular situation involved radian angle measure. The findings from this dissertation extend existing research that explored conceptions of angle measure and radian angle measure by reporting PMTs’ conceptions of radian angle measure given three different curricular situations. While there is still much that needs to be investigated about complexities in PMTs’ conceptions of radian angle measure, this dissertation represents one step toward providing insights about those complexities. </p>

abstract: There has been a growing emphasis on the education of future generations of engineers who will have to tackle complex, global issues that are sociotechnical in nature. The National Science Foundation invests millions of dollars in interdisciplinary engineering education research (EER) to create an innovative and inclusive culture aimed at radical change in the engineering education system. This exploratory research sought to better understand ways of thinking to address complex educational challenges, specifically, in the context of engineering-social sciences collaborations. The mixed methods inquiry drew on the ways of thinking perspectives from sustainability education to adapt futures, values, systems, and strategic thinking to the context of EER. Using the adapted framework, nine engineer-social scientist dyads were interviewed to empirically understand conceptualizations and applications of futures, values, systems, and strategic thinking. The qualitative results informed an original survey instrument, which was distributed to a sample of 310 researchers nationwide. Valid responses (n = 111) were analyzed to uncover the number and nature of factors underlying the scales of futures, values, systems, and strategic thinking. Findings illustrate the correlated, multidimensional nature of ways of thinking. Results from the qualitative and quantitative phases were also analyzed together to make recommendations for policy, teaching, research, and future collaborations. The current research suggested that ways of thinking, while perceived as a concept in theory, can and should be used in practice. Futures, values, systems, and strategic thinking, when used in conjunction could be an important tool for researchers to frame decisions regarding engineering education problem/solution constellations. / Dissertation/Thesis / Doctoral Dissertation Learning, Literacies and Technologies 2019

Food insecurity is on the rise worldwide and within Canada due to a myriad of factors such as climatic instability, rising food prices and unsustainable food production practices. In this context, educational systems (e.g. schools and universities) can contribute to developing knowledge and awareness of food insecurity as well as fostering new ways of thinking and engaging with food premised on just and sustainable food systems. This study is situated within the field of environmental education where there is a growing body of research at the intersections of food and the environment. Likewise, it was guided by the theoretical framing of EcoJustice Education, which offers a way of teaching and learning premised on the belief that our thoughts and actions can foster and enhance more social and ecologically equitable connections between food and the environment. By engaging in semi-structured interviews with teacher educators in select teacher education programs in Ontario and conducting document reviews, I investigated how the integration of the topic of food security is taking place—or not—in the initial training of future teachers in the province. The results showed that integration is not consistent across the different organizational levels of the programs investigated (i.e. whole-program level and classroom level). Moreover, the interviews with teacher educators revealed that any practices aiming at the integration of food security topics in BEd programs were primarily guided by a sustainable cultures perspective, which sees the world as having interconnected relationships amongst all living things. This view is supported by the data analysis of interviews with teacher educators, the Ontario Ministry of Education curriculum documents, and select course syllabi. On the other hand, the school curriculum documents contained conflicting views on the topic, including an understanding of the world as being based on hierarchized relationships. This research advances the field of environmental education by further adding to the limited scholarship on the topic of food security in the context of EE, as well as contributing to an account of food security education and EE with a focus on teacher education in Ontario.

Computer programming is a core area in computer science education that involves practical as well as conceptual learning goals. The literature in programming education reports however that novice students have great problems in their learning. These problems apply to concepts as well as to practise. The empirically based research presented in this thesis contributes to the body of knowledge on students' learning by investigating the relationship between conceptual and practical learning in novice student learning of programming. Previous research in programming education has focused either on students' practical or conceptual learning. The present research indicates however that students' problems with learning to program partly depend on a complex relationship and mutual dependence between the two. The most significant finding is that practise, in terms of activities at different levels of proficiency, and qualitatively different conceptual understandings, have dimensions of variation in common. An analytical model is suggested where the dimensions of variation relate both to concepts and activities. The implications of the model are several. With the dimensions of variation at the center of learning this implies that when students discern a dimension of variation, related conceptual understandings and the meaning embedded in related practises can be discerned. Activities as well as concepts can relate to more than one dimension. Activities at a higher level of proficiency, as well as qualitatively richer understandings of concepts, relate to more dimensions of variation. Concrete examples are given on how variation theory and patterns of variation can be applied in teaching programming. The results can be used by educators to help students discern dimensions of variation, and thus facilitate practical as well as conceptual learning.