No description available.

Visualization in mathematics can be discussed in many ways; it is a broad term that references physical visualization objects as well as the process in which we picture images and manipulate them in our minds. Research suggests that visualization can be a powerful tool in mathematics for intuitive understanding, providing and/or supporting proof and reasoning, and assisting in comprehension. The literature also reveals some difficulties related to the use of visualization, particularly how illustrations can mislead students if they are not comfortable seeing concepts represented in varied ways. However, despite the extensive research on the benefits and challenges of visualization there is little research into what types of figures students are exposed to through their textbooks. This study examines 14 high school geometry textbooks in total, comprised of eight physical textbooks from the top three major textbook publishers in the United States and six FlexBooks created by a non-profit organization developing free and customizable textbooks online. In each textbook the printed images from four topics were classified: Parallel Lines and Transversals, Classifying Triangles, Parallelograms, and Trapezoids. The ‘typical’ images in each of the four topics were defined and the percentages of images that were typical for each textbook in both the lesson and exercise portions were calculated. Results indicate that lesson portions of sections generally contain more typical images than exercise portions and that the total percentage of typical images in an average section varies from 51.9% typical images in the Parallel Lines and Transversals section to 75.2% typical images in the Trapezoid section. Based on these results we list possible avenues for further research in this area.

Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-14T12:39:43Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_PauloSergioDietrich.pdf: 3252469 bytes, checksum: 9b9c2edf1f6704ba4087125399e7d207 (MD5) / Made available in DSpace on 2018-08-14T12:39:43Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_PauloSergioDietrich.pdf: 3252469 bytes, checksum: 9b9c2edf1f6704ba4087125399e7d207 (MD5) Previous issue date: 2010-01-12 / This essay presents the results of the research accomplished with students of the Course of Degree in Mathematics registered in the discipline of Calculation II of the Regiona Integrated University of High Uruguay and Missions-URI, Campus of Frederico Wesphalen-RS. The main investigation focus was the analysis of the possibilities of acquisition of the basic concepts of integral defined, through the methodology of the Didactic Engineering, under the optics of the theory of concept image and concept definition proposed by Tall and Vinner (1981). The applied methodology allied to the pedagogic procedures of the application of a didactic sequence, built with base in the results of the previous analysis composed of the analysis of text books used by the teachers of the discipline of Calculation II and the application of a test diagnosis, made possible to the students who were involved in the present study ,a meaningful assimilation of the concepts and properties of integral defined. The results of the investigation demonstrated that the didactic sequence proposed contributed to the creation of conceptual images and it favored the understanding of the concepts and properties of the integral defined, in reason of the employed methodological process to have offered to the students opportunity of working with situations that allowed the construction of the integral defined concept starting from their own experiences. / Esta dissertação apresenta os resultados da pesquisa realizada com alunos do Curso de Licenciatura em Matemática matriculados na disciplina de Cálculo II da Universidade Regional Integrada do Alto Uruguai e das Missões – URI,Campus de Frederico Wesphalen-RS. O foco principal da investigação foi a análise das possibilidades de aquisição dos conceitos básicos de integral definida, por meio da metodologia da Engenharia Didática, sob a ótica da teoria de conceito imagem e conceito definição proposta por Tall e Vinner (1981). A metodologia utilizada, aliada aos procedimentos pedagógicos da aplicação de uma sequência didática, construída com base nos resultados da análise prévia, a qual foi composta da análise de livros didáticos utilizados pelos professores da disciplina de Cálculo II e a aplicação de um teste diagnóstico, possibilitou que os alunos envolvidos no presente estudo tivessem uma assimilação significativa dos conceitos e propriedades da integral definida. Os resultados da investigação demonstraram que a sequência didática proposta contribuiu para a criação de imagens conceituais e favoreceu a compreensão dos conceitos e propriedades da integral definida, em razão de o processo metodológico empregado ter oferecido aos alunos oportunidade de trabalhar com situações que permitiram a construção do conceito de integral definida a partir de suas próprias experiências.

The aim of the thesis is to describe the decision making process of students in the so-called simple categorization, i.e., decision whether a particular object is or is not an element of a category. This process is examined in the context of categories of mathematical objects. The theoretical part of the thesis presents arguments why the study of simple categorization of mathematical objects is important for mathematics education. These arguments are not only based on the available literature in mathematics education, but also partly draw on historical, mathematical and psychological literature. The practical chapters of the thesis describe the design and piloting of a research tool suitable for this research. The dominant elements of this tool are the measurement of the binary answers (yes / no) of the respondent and of his/her reaction time. This tool is then used in the Main study based on mixed, qualitative-quantitative methodology. It was found that with the help of the proposed tool, while adhering to appropriate methodological rules, it is possible to distinguish different approaches of respondents to categorization. In addition, the basic patterns in the decision-making process of the respondents were described. These are, for instance, differences in the categorization of examples and non-...

Made available in DSpace on 2016-04-27T16:57:30Z (GMT). No. of bitstreams: 1 Joelma Iamac Nomura.pdf: 7399337 bytes, checksum: 3b1b78708c15a38620c94201d8ab977e (MD5) Previous issue date: 2014-03-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The objective of this research harnesses to the results obtained in the Master's Dissertation defended in September 2008 in Postgraduate Studies Program in Mathematics Education at PUC - SP. In this same essay, issues related to teaching and learning of linear algebra sought to answer and find new ways of targeting and perspectives of students in a graduate in Electrical Engineering, asking Why and How should it be taught the discipline of linear algebra on a course with this profile? Among the results, we identified that the interdisciplinarity inherent to the topics of Linear Algebra and specific content of engineering or applied constituted an essential factor for the recognition of mathematical disciplines as theoretical and conceptual basis. Interdisciplinarity reflected in specific mathematical objects of linear algebra and practical situations of engineering materials for the formation of conceptual and general engineer seeking the theoretical foundation and basic justification for the technological improvement of its area. Based on a scenario and results envisioned in the dissertation we propose to investigate the cognitive structures involved in the construction of mathematical object eigenvalue and eigenvector in the initial and final student education phases in Engineering courses, showing the cognitive schemes in their mathematical minds. For this, the following issues are highlighted: ( 1 ) What conceptions (action - process -object- schema ) are evidenced in students after studying the mathematical object eigenvalue and eigenvector in the initial and final phases of their academic training courses in Engineering? and ( 2 ) these same phases, which concept image and concept definition are highlighted in the study of eigenvalue and eigenvector mathematical object? Substantiated by the theoretical contributions of Dubinsky (1991), on the APOS Theory and Vinner (1991), about the concept image and concept definition, we consider the cognitive processes involved in the construction of mathematical object, identifying the nature of their cognitive entities portrayed in mathematical mind. The discussion focuses on mathematical mind both the mathematical structure that is designed and shared by the community as the design in which each mental biological framework handles such ideas. To do so, we consider the relationship between the ideas which constitute the APOS theory, concepts image and definition and some aspects of Cognitive Neuroscience. Characterized as multiple case studies, data collection covered the speech of students in engineering courses in various training contexts, established by the institutions. The analysis of the specific mathematical concept called genetic decomposition led to this concept, which was proposed by System Dynamic Discrete problem, described by the difference equation K K x A.x 1 = + , (K = 0,1,2 , ... ) . Based on the ideas of Stewart (2008) and Trigueros et al. (2012) it was possible to us to identify some characteristics of showing the different conceptions of the students. Moreover, we consider some ideas that characterize the concept image and concept definition according Vinner (1991) and Domingos (2003). As a result of our investigation, we identified that the students of the first case study, at different stages of training, present the design process and the concept image on an instrumental level mathematical object eigenvalue and eigenvector. Have students in the second case, particularly, all of the first phase, and two of the second, showed signs of action and concept image incipient level. As a student of the second phase, have also highlighted the design process and the concept image on an instrumental level as the subject of the first case study. Therefore, we find no significant evolution between the inherent APOS Theory concepts and the concepts image of the object of study. We show that all students presented their speeches in relations between the Linear Algebra course and other courses in the program, such as Numerical Calculation, Electrical Circuits , Computer Graphics and Control Systems, with lesser or greater degree of depth and knowledge. We realize that students attach importance to mathematical disciplines in its formations and seek for a new approach to teaching that address the relationships between them and the disciplines of Engineering / O objetivo desta pesquisa atrela-se aos resultados obtidos na Dissertação de Mestrado defendida em setembro de 2008 no Programa de Estudos Pós-Graduados em Educação Matemática da PUC-SP. Nesta mesma dissertação, questões relacionadas ao ensino e aprendizagem de Álgebra Linear buscaram responder e encontrar novas formas de direcionamento e perspectivas de ensino em uma graduação em Engenharia Elétrica, indagando Por que e Como deve ser lecionada a disciplina de Álgebra Linear em um curso com este perfil? Dentre os resultados obtidos, identificou-se que a interdisciplinaridade inerente aos tópicos de Álgebra Linear e conteúdos específicos ou aplicados da Engenharia constituiu-se de fatores imprescindíveis para ao reconhecimento das disciplinas matemáticas, como base teórica e conceitual. A interdisciplinaridade refletida em objetos matemáticos específicos da Álgebra Linear e situações práticas da Engenharia prima pela formação do engenheiro conceitual e generalista que busca na fundamentação teórica e básica a justificativa para o aprimoramento tecnológico de sua área. Com base no cenário e resultados vislumbrados na defesa da dissertação, propusemonos investigar as estruturas cognitivas envolvidas na construção do objeto matemático autovalor e autovetor nas fases inicial e final de formação do aluno dos cursos de Engenharia, evidenciando os esquemas cognitivos e a mente matemática dos estudantes, sujeitos de nossa investigação. Para tanto, as seguintes questões são destacadas: (1) Quais concepções (ação-processo-objeto-esquema) são evidenciadas nos alunos, após o estudo do objeto matemático autovalor e autovetor nas fases inicial e final de sua formação acadêmica em cursos de Engenharia?; e (2) Nessas mesmas fases, quais conceitos imagem e definição são evidenciados no estudo do objeto matemático autovalor e autovetor? Fundamentados pelos aportes teóricos de Dubinsky (1991), sobre a Teoria APOS, e Vinner (1991) nos conceitos imagem e definição, foram considerados os processos cognitivos envolvidos na construção do objeto matemático, identificando a natureza de suas entidades cognitivas retratadas na mente matemática. A discussão sobre mente matemática foca-se tanto na estrutura matemática que é concebida e compartilhada pela comunidade como no delineamento em que cada estrutura biológica mental trata essas mesmas ideias. Para tanto, considerou-se a relação entre as ideias que constituem a Teoria APOS, os conceitos imagem e definição e alguns aspectos da Neurociência Cognitiva. A pesquisa caracterizada como estudos de caso múltiplos, identificou os dados a partir do discurso dos estudantes dos cursos de Engenharia em contextos diversos de formação, estabelecidos pelas instituições de ensino. A análise do conceito matemático específico levou à chamada decomposição genética desse conceito, que foi proposto pelo problema de Sistema Dinâmico Discreto, descrito pela equação de diferença K K x A.x 1 = + (K=0,1,2,...). Com base nas ideias de Stewart (2008) e Trigueros et al. (2012), foi possível identificar algumas características que evidenciassem as diferentes concepções dos estudantes. Além disso, foram consideradas algumas ideias que caracterizam o conceito imagem e definição de acordo com Vinner (1991) e Domingos (2003). Como resultado desta investigação, identificou-se que os alunos do primeiro estudo de caso, em fases distintas de formação, apresentam a concepção processo e o conceito imagem em nível instrumental do objeto matemático autovalor e autovetor. Já os alunos do segundo de caso, particularmente, todos os da primeira fase, e dois da segunda apresentaram indícios da concepção ação e conceito imagem em nível incipiente. Apenas um aluno da segunda fase também evidenciou ter a concepção processo e o conceito imagem em nível instrumental, como os sujeitos do primeiro estudo de caso. Portanto, constatou-se que não houve evolução significativa entre as concepções inerentes à Teoria APOS e os conceitos imagem do objeto de estudo. Evidenciou-se que todos os alunos apresentaram em seus discursos relações existentes entre a disciplina Álgebra Linear e demais disciplinas do curso, como Cálculo Numérico, Circuitos Elétricos, Computação Gráfica e Sistemas de Controle, com menor ou maior grau de profundidade e conhecimento. Percebe-se que os alunos atribuem relevância às disciplinas matemáticas em suas formações e buscam por um novo enfoque de ensino que contemple as relações entre as mesmas e as disciplinas da Engenharia

Made available in DSpace on 2016-04-27T16:58:08Z (GMT). No. of bitstreams: 1 dissertacao_cristina_meyer.pdf: 893209 bytes, checksum: cf688e40bd77ad09e1cf9cbe28191ac0 (MD5) Previous issue date: 2003-04-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This is a diagnostic research. The objective of this research is to investigate elements of the concept image and concept definition related to the concept of derivative while interpreted geometrically. It is based on the David Tall and Shlomo Vinner theory about concept image and concept definition. The investigated persons are students of the Formation Course for Mathematic Teachers in a private university of the State of São Paulo, who already had attended classes of Calculus I e II. Two kind of methodological instruments were used: questionnaire and interview. Among the conclusions we detach: the interpretation of the equation of the tangent straight line to the graph of function f as being the function derived from f; the interpretation of the derivative of function f in the point of absciss x = a, as being the y coordinate b of the point (a,b) in which the straight line is tangent to the graph of function f; the existence of persons that verbalize a concept definition related to the derivative concept whose elements are related of a coherent form with the geometric definition of this concept, which is ignored by them at the moment of the elaboration of the answers to the proposed questions / Esta é uma pesquisa de caráter diagnóstico. Objetiva investigar elementos da imagem conceitual e definição conceitual, relativas ao conceito de derivada, quando interpretado geometricamente. É referenciada na teoria de David Tall e Shlomo Vinner sobre imagem conceitual e definição conceitual. Os sujeitos investigados são estudantes do curso de licenciatura em Matemática de uma Universidade particular do Estado de São Paulo, que já cursaram as disciplinas de Cálculo I e II. Dois tipos de instrumentos metodológicos foram utilizados: questionário e entrevista. Entre as conclusões, destacamos: interpretação da equação da reta tangente ao gráfico da função f como sendo a função derivada de f; interpretação da derivada da função f no ponto de abscissa x = a como sendo a ordenada b do ponto (a,b) no qual a reta tangencia o gráfico da função f; existência de sujeitos que verbalizam uma definição conceitual, relativa ao conceito de derivada, cujos elementos estão coerentemente relacionados com a definição geométrica desse conceito, mas ignorada por eles no momento da elaboração das respostas às questões propostas

Made available in DSpace on 2018-06-27T19:13:02Z (GMT). No. of bitstreams: 2 Laura Moreira Bordin.pdf: 2775738 bytes, checksum: 12891239889c50b7d801757bc8d26043 (MD5) Laura Moreira Bordin.pdf.jpg: 3569 bytes, checksum: b503812f643369cda67fcc097918310b (MD5) Previous issue date: 2011-08-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The present work´s aim was to analyze the contributions which the Teaching-Learning-Evaluation methodology through Problem Solving provides to the Equation of Differences concept learning, founded in the Concept Image and Concept Definition theory of Tall and Vinner (1981).The participants of the research were undergraduated students of the third grade of a Mathematics Teaching License, from UNIPAR Universidade Paranaense, Cascavel, PR. The research was a qualitative type and the collected dada was based on the participant observation during the development of the activities developed by the students in groups, registered in the researcher and the students ´ field diary and audio recording. The activities developed in the classroom followed the steps of Problem Solving method, suggested by Onuchic e Allevato (2009): problem preparation, individual reading of the problem, group reading, problem solving, observation and encouraging, solving record on the blackboard, plenary, search of agreement and formalization of the content. The activities developed in the classroom were organized in three learning units, aiming the concept building of Difference Equations. In the first unit problem-situations were studied involving problem situations which allowed the students to build the concept of Linear Equations of Differences of homogeneous first order; in the second unit problem situations involving Linear Equations of Differences of non-homogeneous first order and in the last unit problem situations were proposed related to Linear Equation of Differences of homogeneous of second order. The results indicated that in front of an unknown math concept, the students tried to mean it by means of their conceptual images which already exist and the new ones built during the research. It was possible to identify an active involvement of the participants in the new concept building which allowed their learning. The results of the research indicate the importance of problem solving as a teaching strategy, which can provide the students the building of their own knowledge. / O presente trabalho teve por finalidade analisar as contribuições que a metodologia de Ensino-Aprendizagem-Avaliação através da Resolução de Problemas proporciona à aprendizagem de conceitos de Equações de Diferenças, alicerçada na teoria de imagem de conceito e definição de conceito de Tall e Vinner (1981). Os sujeitos da pesquisa foram alunos de uma turma da terceira série do curso de Licenciatura em Matemática, da Universidade Paranaense-UNIPAR de Cascavel-PR. Para tal, foi realizada uma pesquisa de natureza qualitativa que teve como instrumentos de coleta de dados a observação participante, durante o desenvolvimento das atividades realizadas pelos alunos reunidos em grupos e registradas no diário de campo da pesquisadora e dos alunos, e gravações em áudio. As atividades em sala de aula seguiram os passos da metodologia de resolução de problemas, sugeridos por Onuchic e Allevato (2009): preparação do problema; leitura individual do problema; leitura em conjunto; resolução do problema; observar e incentivar; registro das resoluções na lousa; plenária; busca do consenso e formalização do conteúdo. As atividades de sala de aula foram organizadas em três unidades de ensino, visando à construção dos conceitos de Equações de Diferenças. Na primeira unidade foram trabalhadas situações-problema que permitiram aos alunos a construção do conceito de Equações de Diferenças Lineares de Primeira Ordem Homogênea; na segunda unidade foram trabalhadas situações-problema envolvendo Equações de Diferenças Lineares de Primeira Ordem não Homogênea e na última unidade foram propostas situações-problema relacionadas com Equações de Diferenças Lineares de Segunda Ordem Homogêneas. Os resultados indicaram que frente a um conceito matemático desconhecido, os alunos buscaram significá-lo por meio de suas imagens conceituais já existentes e as novas construídas no decorrer da pesquisa. Verificou-se um envolvimento ativo dos participantes na construção dos novos conceitos, que permitiu a sua aprendizagem. Os resultados da pesquisa apontam para a importância da inserção da resolução de problemas como estratégia de ensino, a qual pode proporcionar aos alunos a construção do seu próprio conhecimento.

Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-20T13:32:06Z No. of bitstreams: 2 Dissertacao_LeandroFontoura.pdf: 7487127 bytes, checksum: a19a126dd987b5f8aeb25940074b89b1 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-08-20T13:32:06Z (GMT). No. of bitstreams: 2 Dissertacao_LeandroFontoura.pdf: 7487127 bytes, checksum: a19a126dd987b5f8aeb25940074b89b1 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-07-14 / This dissertation has the aim to investigate the contribution from the application of a didactic sequence elaborated in accordance with the assumptions of Didactic Engineering and the usage of a computer application from software Maple for teaching and learning of the double integrals subject for Mathematics Degree students. The theoretical background for analysing the results was based in the Tall and Vinner (1981) theory about the image concept and definition concept and it was used the Domingos (2003) categorization in order to classify the students in three different levels of image concept: incipient image concept, instrumental image concept and relational image concept. From the preliminary studies resulted from the didactic books analysis and from results of the application of a diagnostic test, it was elaborated one didactic sequence, compounded by four activities to build the Double Integral concept and used it in applications related to volume calculations of geometric solids. The data for analyses were obtained through participant observation from the researcher’s field logbook and from documents produced by students. Participated in this research Mathematics Degree students enrolled in the course Differential and Integral Calculus IV. The results shown that the proposed didactic sequence and the usage of a computer application programmed with the software Maple facilitated the formation of image concepts about the Double Integrals subject and helped students to calculate the volume of geometric solids. / Esta dissertação tem como objetivo geral investigar as contribuições da aplicação de uma sequência didática, elaborada de acordo com os pressupostos da Engenharia Didática, e da utilização de um aplicativo computacional do software Maple, para o ensino e aprendizagem do conteúdo de integrais duplas para alunos de um curso de Licenciatura em Matemática. O referencial teórico para análise dos resultados baseou-se na teoria de Tall e Vinner (1981) sobre conceito imagem e conceito definição, e foi utilizada a categorização de Domingos (2003) para classificar os alunos em três diferentes níveis de conceito imagem: conceito imagem incipiente, conceito imagem instrumental e conceito imagem relacional. A partir dos estudos preliminares resultantes da análise de livros didáticos e dos resultados da aplicação de um teste diagnóstico, elaborou-se uma sequência didática composta de quatro atividades, para construir o conceito de integral dupla e utilizá-lo nas aplicações referentes ao cálculo do volume de sólidos geométricos. Os dados para análise foram obtidos por meio da observação participante, do diário de campo do pesquisador e dos documentos produzidos pelos alunos. Participaram desta pesquisa alunos do curso de Licenciatura em Matemática matriculados na disciplina de Cálculo diferencial e integral IV. Os resultados demonstraram que a sequência didática proposta e o emprego de um aplicativo computacional, programado com o software Maple, facilitaram a formação de conceitos imagem sobre o conteúdo de integrais duplas e auxiliaram os alunos no cálculo do volume de sólidos geométricos.

Researchers argue that students need to understand the language and the key concepts of each discipline to understand the academic content. The purpose of this study is to develop, analyze and evaluate a development project in civics. Therefore, the study answer following questions:  How do subject-specific concepts take place during the lessons according to the students? In what way can concept definition maps encourage students to talk about subject -specific concepts? How do students experience the work with concept definition maps? Two qualitative methods are used to answer the questions, both interviews and classroom observations. The interviews focus on how the students describe their experience with subject-specific concepts during civics lessons but also how they experience the work with concept definition maps which was implemented during the intervention. The observations onthe other hand show in what way the concept definition maps encourage students to talk about subject-specific concepts. The theoretical framework of the study is based on the sociocultural perspective and other theories that stress the importance of encouraging students to generate their own meaning of terms but also how the language develops in interaction with others.The result shows that the students often rely on the definition they get from their teacher orthe textbook but there is no room for students to explore the concepts themselves. All the students felt that the concept definition maps had good impact on their understanding of the subject-specific concepts. The results also showed that the there was a positive interdependence during the intervention where the students learned from each other’s prior knowledge and paths of thinking.

<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>