Philosophiae Doctor - PhD / The purpose of this study was to explore errors that are displayed by students when learning derivatives of trigonometric functions in an extended curriculum programme. The first aim was to identify errors that are displayed by students in their solutions through the lens of the APOS theory. The second aim was to address students' errors by using the two principles of Vygotsky's socio-cultural theory of learning, namely the zone of proximal development and more knowledgeable others. The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The participants in this study comprised a group of students who registered for mathematics in the ECP at Cape Peninsula University of Technology, Cape Town, South Africa. The study was piloted in 2008 with a group of twenty students who registered for mathematics in the ECP for Chemical Engineering. In 2009 thirty students from the ECP registered for mathematics in Chemical Engineering were selected to participate in the main study. This study was conducted over a period of four and a half years. Data collection was done through students' written tasks; classroom audio and video recordings and indepth interviews. Data were analysed through categorising errors from students' written work, and finding common themes and patterns in audio and video recordings and from the in-depth interviews. The findings of this study revealed that students committed interpretation, arbitrary, procedural, linear extrapolation and conceptual errors. Interpretation errors arise when students fail to interpret the nature of the problem correctly owing to over-generalisation of certain mathematical rules. Arbitrary errors arise when students behave arbitrarily and fail to take account of the constraints laid down in what is given. Procedural errors occur when students fail to carry out manipulations or algorithms although they understand concepts in problem. Linear extrapolation errors happen through an overgeneralisation of the property f (a + b) = f (a) + f (b) , which applies only when f is a linear function Conceptual errors occur owing to failure to grasp the concepts involved in the problem or failure to appreciate the relationships involved in the problem. The findings were consistent with literature indicated that errors are based on students’ prior knowledge, as they over-generalise certain mathematical procedures, algorithms and rules of differentiation in their solutions. The use of learning activities in the form of written tasks; as well as classroom audio and video recordings assisted the lecturer to identify and address errors that were displayed by students when they learned derivatives of trigonometric functions. The students claimed in their interviews that they benefited from class discussions as they obtained immediate feedback from their fellow students and the lecturer. They also claimed that their performances improved as they continued to practice with the assistance of more knowledgeable students, as well as the lecturer. This study supports the view from the literature that identification of errors has immense potential to address students’ poor understanding of derivatives of trigonometric functions. This thesis recommends further research on errors in various sections of Differential Calculus, which is studied in an extended curriculum programme at Universities of Technology in South Africa.

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Doctor of Philosophy / Department of Mathematics / Andrew Bennett / Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.

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Made available in DSpace on 2016-04-27T16:57:04Z (GMT). No. of bitstreams: 1 Rogerio Osvaldo Chaparin.pdf: 2644934 bytes, checksum: c8a8cb59aa59e9acbc623f18e66cbb61 (MD5) Previous issue date: 2010-10-08 / Secretaria da Educação do Estado de São Paulo / This study aims to investigate the students' conceptions of a first year high school on the concept of divisibility of natural numbers. The relevance of this study is the importance that, according to Campbell and Zazkis (1996) and Resende (2007), has the divisibility concepts relevant in the development of mathematical thinking, in research activities at any level of education, identification and pattern recognition, in the formulation of conjectures and especially in solving problems. To achieve this I used as the theoretical APOS Theory to analyze the protocols, Sfard in formulating the idea of design and research Rina Zazkis building activities. To collect the data I have chosen a didactic sequence consists of four activities performed in pairs of first year students of high school I teach at school. These survey results show that students had great difficulty in handling the operation of the division, designing mostly divisibility through actions, algorithms, and procedures. They did not know deduce relations, information, ie, mainly not understand that the representation in prime factors is a very important way to relate the concepts of multiple and divisor. The students were unable to apply the concepts mentioned above in a situation contextualized in a situation of daily life. Thus concludes that it is necessary to give greater emphasis to basic issues of the Elementary Theory of Numbers in the teaching of mathematics / Este trabalho tem como objetivo investigar quais as concepções dos alunos de um primeiro ano do ensino médio sobre o conceito de divisibilidade dos números naturais. A relevância deste estudo está na importância que, segundo Campbell e Zazkis (1996) e Resende (2007), tem os conceitos pertinentes a divisibilidade no desenvolvimento do pensamento matemático, nas atividades investigativas em qualquer nível de ensino, na identificação e reconhecimento de padrões, na formulação de conjecturas e principalmente na resolução de problemas. Para alcançar tal objetivo usei como aporte teórico a Teoria APOS para análise dos protocolos, Sfard na formulação da idéia de concepção e as pesquisas de Rina Zazkis na elaboração de atividades. Para a coleta de dados optei por uma sequência didática composta por 4 atividades realizada em duplas de alunos do primeiro ano do ensino médio na escola que leciono. Os resultados dessa pesquisa revelam que os alunos tiveram grande dificuldade na manipulação da operação da divisão, concebem na sua maioria a divisibilidade por meio de ações, algoritmos, procedimentos. Não souberam deduzir relações, informações, ou seja, principalmente não compreenderam que a representação em fatores primos é uma forma muito importante para relacionar os conceitos de múltiplo e divisor. Os sujeitos não conseguiram aplicar os conceitos citados acima numa situação contextualizada em uma situação do cotidiano. Desta forma conclui que é necessário dar uma ênfase maior para os assuntos básicos da Teoria Elementar dos Números no ensino da matemática

Made available in DSpace on 2016-04-27T16:57:20Z (GMT). No. of bitstreams: 1 Fabio Rodrigues de Siqueira.pdf: 2570104 bytes, checksum: 932e47e8f7aba3ad1adce0aba39a8a34 (MD5) Previous issue date: 2012-10-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research has as objective to check if the proposal of an algorithm to be converted into a computer program can contribute to the learning of an object of mathematical study. For that, we have formulated the following research question: how can the elaboration of a converted algorithm into a computer program help high school students in the learning of the zeros of the 2nd degree polynomial function? The research was conducted in two stages. The first stage, with two initial activities, consisted of two parts: the first one with a 1st year high school student in order to verify if the activities were appropriate to the research and in the second part, after the application of the two activities, we have selected four participants for the second stage that had presented different results. After the analysis of the first stage development, the activities were improved to the second one, composed of three activities, among which the software Visualg 2.0 was used for the algorithm edition and its conversion to a computer program. The APOS theory by Ed Dubinsky, theoretical support of the research, presents the action levels, process, object and scheme, that allow the verification of the individual`s capacity to develop actions over an object and think about its properties. The research participants had improvements in their learning, because besides developing a computer program to determine the zeros of 2nd degree polynomial function, they have started to elaborate other functions previewing their possible solutions, presenting all the levels of the APOS theory. As research methodology we have adopted the Design Experiments. We have justified its use, for adjustments could be done during the work development. Analyzing the activities which were done we have concluded that the students have achieved a satisfactory learning level over the object of study / Esta pesquisa tem como objetivo verificar se a proposta de um algoritmo a ser convertido em um programa de computador pode contribuir na aprendizagem de um objeto de estudo matemático. Para tanto, formulamos a seguinte questão de pesquisa: como a elaboração de um algoritmo convertido em um programa pode auxiliar alunos do ensino médio na aprendizagem dos zeros da função polinomial do 2º grau? A pesquisa foi realizada em duas etapas. A primeira etapa, com duas atividades iniciais, foi composta por dois momentos: o primeiro com um aluno da 1ª série do ensino médio a fim de verificar se as atividades estavam adequadas à pesquisa e no segundo momento, após a aplicação das duas atividades, selecionamos quatro participantes para a segunda etapa que apresentaram resultados diferenciados. Após análise do desenvolvimento da primeira etapa, as atividades foram aprimoradas para a segunda, composta de três atividades, entre as quais foi utilizado o software Visualg 2.0 para edição do algoritmo e sua conversão para programa. A teoria APOS de Ed Dubinsky, aporte teórico da pesquisa, apresenta os níveis ação, processo, objeto e esquema, que permitem a verificação da capacidade do indivíduo em desenvolver ações sobre um objeto e raciocinar sobre suas propriedades. Os participantes da pesquisa tiveram melhorias em seu aprendizado, pois além de desenvolver um programa de computador para se determinar os zeros de funções polinomiais do 2º grau, passaram a elaborar outras funções já prevendo as possíveis soluções, apresentando todos os níveis da teoria APOS. A metodologia adotada nessa pesquisa foi o Design Experiments. Justificamos seu uso, pois adequações puderam ser realizadas durante o desenvolvimento do trabalho. Ao analisarmos as atividades realizadas concluímos que os alunos atingiram um nível de aprendizagem satisfatório acerca do objeto de estudo

Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-20T17:53:01Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_JaniceRachelli.pdf: 4818944 bytes, checksum: 0c02a81d2b4c04364b21e1ddddc2fe58 (MD5) / Made available in DSpace on 2018-08-20T17:53:01Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese_JaniceRachelli.pdf: 4818944 bytes, checksum: 0c02a81d2b4c04364b21e1ddddc2fe58 (MD5) Previous issue date: 2017-10-03 / The present study is on the field of Mathematics Education in higher education and is focused on the teaching and learning of Calculus concepts, specifically related to the concepts of classical derivative and weak derivative. The work, developed in the context of a qualitative research, aims to investigate how some students of the Master degree course in Teaching Mathematics of a community institution in Rio Grande do Sul comprehend the concepts of classical derivative and weak derivative. The APOS theory serves as a theoretical and methodological reference for the elaboration of the genetic decomposition in which the possible mental constructions used by the students were described in what the understanding of the concepts of classical derivative and weak derivative concern. We proposed some activities on the historical constructions of the classical concept, the derivative in the present times, and the passage from the classical derivative to the weak derivative. The teaching situations were developed in the classroom in the second semester of 2016, within the subject of Fundamentals of Differential and Integral Calculus. The basis for these activities was the ACE teaching cycle. The results obtained by analyzing the students' records in the proposed activities and the observations recorded in the field diary indicate that the students were able to coordinate actions and processes in order to obtain the derivative and verify if a function is differentiable. They could also coordinate the interpretations of the derivative, such as slope of the tangent line, instantaneous velocity and rate of variation, besides using mechanisms of generalization and reversibility in the analysis of the graphs of functions and their derivatives and encapsulating the processes necessary for a satisfactory understanding of the concept of the classical derivative. By means of the integral equation, the integration formula by parts, and the fundamental theorem of Calculus, the students were able to coordinate the function and intervals, the functions with compact support by means of internalizing actions and processes for the encapsulation of the mathematical object and weak derivative. Although there are some errors in these processes, there is evidence that the concepts of classical derivative and weak derivative have been understood by the students. These evidences developed mental mechanisms of reflective abstraction that allowed the construction of the mental structures of action, process, object and scheme present in the genetic decomposition that allowed them to understand the concepts. Moreover, the treatment with the historical context of the derivative and the collaborative work of the students were significant factors to obtain the results of the research. / O presente estudo se situa no campo da Educação Matemática no ensino superior e se insere na linha de investigação voltada ao ensino e aprendizagem de conceitos do Cálculo, especificamente ligados aos conceitos de derivada clássica e derivada fraca. O trabalho, desenvolvido no contexto de uma pesquisa qualitativa, teve como objetivo investigar como se dá a compreensão dos conceitos de derivada clássica e derivada fraca por estudantes de um curso de mestrado em Ensino de Matemática de uma instituição comunitária do Rio Grande do Sul. Tendo a teoria APOS como referencial teórico e metodológico, elaborou-se a decomposição genética, em que foram descritas as possíveis construções mentais utilizadas pelos estudantes para a compreensão dos conceitos de derivada clássica e derivada fraca. Foram organizadas situações de ensino compostas por atividades sobre as construções históricas do conceito clássico, a derivada nos tempos atuais e a passagem da derivada clássica para a derivada fraca. As situações de ensino foram desenvolvidas em sala de aula, no segundo semestre de 2016, na disciplina de Fundamentos de Cálculo Diferencial e Integral, tendo como base o ciclo de ensino ACE. Os resultados obtidos, por meio da análise dos registros dos alunos nas atividades propostas e das observações anotadas no diário de campo, indicam que os estudantes foram capazes de coordenar ações e processos para obter a derivada e verificar se uma função é diferenciável, coordenar as interpretações da derivada como inclinação da reta tangente, velocidade instantânea e taxa de variação, além de, utilizar mecanismos de generalização e reversibilidade na análise dos gráficos das funções e suas derivadas e de encapsular os processos necessários para a compreensão, de forma satisfatória, do conceito da derivada clássica. Por meio da equação integral, da fórmula de integração por partes e do teorema fundamental do Cálculo, os alunos coordenaram a função e os intervalos, funções com suporte compacto, interiorizando ações e processos para a encapsulação do objeto matemático, derivada fraca. Embora com alguns erros cometidos nesses processos, há evidências de que houve compreensão dos conceitos de derivada clássica e derivada fraca pelos estudantes. Estes evidenciaram desenvolver mecanismos mentais de abstração reflexionante que possibilitaram a construção das estruturas mentais de ação, processo, objeto e esquema presentes na decomposição genética que lhes permitiu compreender os conceitos. Além do mais, o trato com o contexto histórico da derivada e o trabalho colaborativo dos alunos foram fatores significativos para a obtenção dos resultados da pesquisa.

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

Doctor of Philosophy / Electrical and Computer Engineering / Steven Warren / Linear Systems (a.k.a., Signals and Systems) is an important class in an Electrical Engineering curriculum. A clear understanding of the topics in this course relies on a well-developed notion of lower-level mathematical constructs and procedures, including the roles these procedures play in system analysis. Students with an inadequate math foundation regularly struggle in this class, as they are typically able to perform sequences of the underlying calculations but cannot piece together the higher-level, conceptual relationships that drive these procedures. This dissertation describes an investigation to assess and improve students’ higher-level understanding of Linear Systems concepts. The focus is on the topics of (a) time-domain, linear time-invariant (LTI) system response visualization and (b) Fourier series conceptual understanding, including trigonometric Fourier series (TFS), compact trigonometric Fourier series (CTFS), and exponential Fourier series (EFS). Support data, including exam and online homework data, were collected since 2004 from students enrolled in ECE 512 - Linear Systems at Kansas State University. To assist with LTI response visualization, two online homework modules, Zero Input Response and Unit Impulse Response, were updated with enhanced plots of signal responses and placed in use starting with the Fall 2009 semester. To identify students’ conceptual weaknesses related to Fourier series and to help them achieve a better understanding of Fourier series concepts, teaching-learning interviews were applied between Spring 2010 and Fall 2012. A new concept-based online homework module was also introduced in Spring 2011. Selected final exam problems from 2007 to 2012 were analyzed, and these data were supplemented with detailed mid-term and final exam data from 77 students enrolled in the Spring 2010 and Spring 2011 semesters. In order to address these conceptual learning issues, two frameworks were applied: Bloom’s Taxonomy and APOS theory. The teaching-learning interviews and online module updates appeared to be effective treatments in terms of increasing students’ higher-level understanding. Scores on both conceptual exam questions and more traditional Fourier series exam questions were improved relative to scores received by students that did not receive those treatments.

D.L. A 171-2008