In the realm of academic engagement research, students are valuable sources of information to learn how and why students often engage unproductively in mathematics group work. However, although secondary mathematics students are often expected to engage in meaningful mathematical discourse in a small group setting, little research has been conducted to better understand student engagement in this setting from the perspective of the students themselves. This thesis attempts to understand how one junior high student described his own engagement in mathematics small group work as well as what factors influenced this engagement. By conducting several cycles of observations and interviews followed by qualitative analysis, we learned how this student engaged in a variety of ways in group work; on different occasions (and sometimes within the same class period), he talked with his peers about mathematics, remained silent, played on his phone, connected with peers across the room, and pursued off-topic conversation with his group mates. We also discovered that the student participant as well as his peers often ceased to engage productively when they encountered mathematics that they deemed too difficult. Several other factors impacted his engagement in complex ways, including his familiarity with group mates, fear of being singled out, and access to adequate help from a teacher. These results may inform researchers of new data collection and analysis methods to gain insights into student engagement and teachers of ways in which they may adapt instruction to better encourage students to engage productively.

Made available in DSpace on 2016-12-23T14:01:46Z (GMT). No. of bitstreams: 1 Thais Leal da Cruz Silva.pdf: 3736864 bytes, checksum: 897ad6d3bad96489ee68a426cb55f256 (MD5) Previous issue date: 2013-04-08 / This masters thesis, focusing on mathematics education, is linked to the Graduate Education Program, of the Education Center at Federal University of Espírito Santo - Brasil (UFES). The study investigated conceptions, memories and career choice of undergraduate pre-service secondary mathematics teachers from the Federal Institute of Education, Science and Technology of Espírito Santo (Ifes) - Brazil in Cachoeiro de Itapemirim campus. The studies of Ernest, Gómez Chacón, Lorenzato, Bridge, Santos, Santos-Wagner, and Thompson, among others offered theoretical support for this work. These studies show how much relevant is to value the conceptions of teachers and students relative to mathematics, and how it can interfere with the training, learning and performance of these individuals, in the environment in which they live. We followed a class of the undergraduate pre-service secondary mathematics teacher education course for a year, between the second and fourth period of the course. The study methodology was qualitative and adopted features of ethnographic research. The gathering and the production of data were performed by classroom observations, conversations, interviews, questionnaires and tasks proposed to undergraduates. Furthermore, we gave constantly feedback to the students to confirm data recorded and interpretations with them. The study showed that participants exhibited more than one conception about mathematics and furnished cues that differentiated their thoughts about mathematics and mathematics teaching. We noted that experiences with mathematics influence the undergraduates conceptions about this discipline and pedagogical aspects, as well as the professional choice. The teachers, the kinds of classes that students experienced, the social and family environment were important elements in the individual formation and conceptions about mathematics and mathematic teaching. The participants showed mathematics conceptions and study habits different in relation to the mathematics studied in basic education and college. The research process and its constant feedback contributed to improve the undergraduates knowledge about themselves, and for some students to become self-conscious about how they have been dedicating to the undergraduate secondary mathematics teacher course / Este trabalho de mestrado, com foco na educação matemática, vincula-se ao Programa de Pós-Graduação em Educação do Centro de Educação da Universidade Federal do Espírito Santo. O estudo investigou concepções, memórias e escolha profissional de licenciandos em matemática, do Instituto Federal de Educação, Ciência e Tecnologia do Espírito Santo (Ifes), no campus de Cachoeiro de Itapemirim. Os estudos de Ernest, Gómez Chacón, Lorenzato, Ponte, Santos, Santos-Wagner, e Thompson, dentre outros, ofereceram aportes teóricos para este trabalho. Estes estudos mostram o quanto é relevante valorizar concepções de professores e alunos, em relação à matemática e o quanto isso pode interferir na formação, aprendizagem e atuação desses indivíduos no ambiente em que estudam, aprendem e vivem. Acompanhamos uma turma do curso de licenciatura em matemática durante um ano, entre o segundo e o quarto período do curso. A metodologia do estudo teve natureza qualitativa e adotou recursos da pesquisa do tipo etnográfica. A coleta e a produção de dados foram realizadas por meio de observações de aulas, conversas, entrevistas, questionários e tarefas propostas aos licenciandos. Além disso, realizamos momentos de retorno da pesquisa para confirmar dados e interpretações com os participantes. Em nosso estudo, percebemos que os participantes exibiam mais de uma concepção sobre a matemática e forneciam pistas que diferenciavam seus pensamentos, a respeito da matemática e de seu ensino. Notamos que experiências com a matemática influenciam concepções dos licenciandos sobre a disciplina e seus aspectos pedagógicos, assim como a escolha profissional. Os professores, os tipos de aulas que os alunos vivenciaram, o ambiente social e familiar foram elementos importantes na formação do indivíduo e nas suas concepções, a respeito de matemática e seu ensino. Os participantes apresentaram concepções e hábitos de estudos distintos, em relação à matemática estudada na educação básica e na faculdade. O processo de pesquisa e o constante retorno aos participantes contribuíram para que os licenciandos se conhecessem melhor e para que alguns alunos se conscientizassem a respeito do quanto estão se dedicando para o curso de licenciatura

This project explored the Concrete to Representational to Abstract instruction (CRA instruction) as a strategy to teach abstract math concepts for secondary students with learning disabilities. Through the review of literature, multiple researchers suggested that students with learning disabilities need to be exposed to a variety of instructional strategies to develop problem solving skills in algebra concepts.

This research seeks to investigate the influence of lesson study on preservice secondary mathematics teachers' view of self as mathematics expert. The study acknowledges the commonly held belief that prospective mathematics teachers have that they know and understand secondary mathematics. The purpose in engaging the preservice teachers in lesson study is to dislodge this belief. In particular, this research report focuses on one preservice teacher and her experiences during lesson study. Using the data collected, the researcher reports on the baseline beliefs that the preservice teacher held toward her knowledge of secondary mathematics, her mathematical experiences during the actual lesson study phase of the research and the final status of her beliefs in relation to her secondary mathematics understanding. After assessing the preservice teacher's beliefs, the report focuses on the moves the preservice teacher makes to protect her identity as a knower of mathematics. The report details how the researcher probed the subject's views through a follow-up interview. The researcher discovered during the follow-up interview that the subject was finally able to admit her lack of mathematical knowledge and her desire to not be seen as 'dumb' in front of the interviewer. The implications of the study suggest that teacher educators should be sensitive to preservice secondary teachers' perceptions of their mathematical knowledge and teacher educators should watch for the moves preservice teachers make to shift conversation away from mathematics topics.

Submitted by ROSEMEIRE DE FATIMA BATISTELA null (rosebatistela@hotmail.com) on 2017-02-11T02:22:43Z No. of bitstreams: 1 tese finalizada 10 fevereiro 2017 com a capa.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2017-02-15T16:56:58Z (GMT) No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) / Made available in DSpace on 2017-02-15T16:56:58Z (GMT). No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) Previous issue date: 2017-02-02 / Apresentamos nesta tese uma proposta de inserção do tema teorema da incompletude de Gödel em cursos de Licenciatura em Matemática. A interrogação norteadora foi: como sentidos e significados do teorema da incompletude de Gödel podem ser atualizados em cursos de Licenciatura em Matemática? Na busca de elaborarmos uma resposta para essa questão, apresentamos o cenário matemático presente à época do surgimento deste teorema, expondo-o como a resposta negativa para o projeto do Formalismo que objetivava formalizar toda a Matemática a partir da aritmética de Peano. Além disso, trazemos no contexto, as outras duas correntes filosóficas, Logicismo e Intuicionismo, e os motivos que impossibilitaram o completamento de seus projetos, que semelhantemente ao Formalismo buscaram fundamentar a Matemática sob outras bases, a saber, a Lógica e os constructos finitistas, respectivamente. Assim, explicitamos que teorema da incompletude de Gödel aparece oferecendo resposta negativa à questão da consistência da aritmética, que era um problema para a Matemática na época, estabelecendo uma barreira intransponível para a demonstração dessa consistência, da qual dependia o sucesso do Formalismo e, consequentemente, a fundamentação completa da Matemática no ideal dos formalistas. Num segundo momento, focamos na demonstração deste teorema expondo-a em duas versões distintas, que para nós se nos mostraram apropriadas para serem trabalhadas em cursos de Licenciatura em Matemática. Uma, como possibilidade de conduzir o leitor pelos meandros da prova desenvolvida por Gödel em 1931, ilustrando-a, bem como, as ideias utilizadas nela, aclarando a sua compreensão. Outra, como opção que valida o teorema da incompletude apresentando-o de maneira formal, portanto, com endereçamentos e objetivos distintos, por um lado, a experiência com a numeração de Gödel e a construção da sentença indecidível, por outro, com a construção formal do conceito de método de decisão de uma teoria. Na sequência, apresentamos uma discussão focada na proposta de Bourbaki para a Matemática, por compreendermos que a atitude desse grupo revela a forma como o teorema da incompletude de Gödel foi acolhido nessa ciência e como ela continuou após este resultado. Nessa exposição aparece que o grupo Bourbaki assume que o teorema da incompletude não impossibilita que a Matemática prossiga em sua atividade, ele apenas sinaliza que o aparecimento de proposições indecidíveis, até mesmo na teoria dos números naturais, é inevitável. Finalmente, trazemos a proposta de como atualizar sentidos e significados do teorema da incompletude de Gödel em cursos de Licenciatura em Matemática, aproximando o tema de conteúdos agendados nas ementas, propondo discussão de aspectos desse teorema em diversos momentos, em disciplinas que julgamos apropriadas, culminando no trabalho com as duas demonstrações em disciplinas do último semestre do curso. A apresentação é feita tomando como exemplar um curso de Licenciatura em Matemática. Consideramos por fim, a importância do trabalho com um resultado tão significativo da Lógica Matemática que requer atenção da comunidade da Educação Matemática, dado que as consequências deste teorema se relacionam com a concepção de Matemática ensinada em todos os níveis escolares, que, muito embora não tenham relação com conteúdos específicos, expõem o alcance do método de produção da Matemática. / In this thesis we present a proposal to insert Gödel's incompleteness theorem in Mathematics Education undergraduate courses. The main research question guiding this investigation is: How can the senses and meanings of Gödel's incompleteness theorem be updated in Mathematics Education undergraduate courses? In answering the research question, we start by presenting the mathematical scenario from the time when the theorem emerged; this scenario proposed a negative response to the project of Formalism, which aimed to formalize all Mathematics based upon Peano’s arithmetic. We also describe Logicism and Intuitionism, focusing on reasons that prevented the completion of these two projects which, in similarly to Formalism, were sought to support mathematics under other bases of Logic and finitists constructs. Gödel's incompleteness theorem, which offers a negative answer to the issue of arithmetic consistency, was a problem for Mathematics at that time, as the Mathematical field was passing though the challenge of demonstrating its consistency by depending upon the success of Formalism and upon the Mathematics’ rationale grounded in formalists’ ideal. We present the proof of Gödel's theorem by focusing on its two different versions, both being accessible and appropriate to be explored in Mathematics Education undergraduate courses. In the first one, the reader will have a chance to follow the details of the proof as developed by Gödel in 1931. The intention here is to expose Gödel’ ideas used at the time, as well as to clarify understanding of the proof. In the second one, the reader will be familiarized with another proof that validates the incompleteness theorem, presenting it in its formal version. The intention here is to highlight Gödel’s numbering experience and the construction of undecidable sentence, and to present the formal construction of the decision method concept from a theory. We also present a brief discussion of Bourbaki’s proposal for Mathematics, highlighting Bourbaki’s group perspective which reveals how Gödel’s incompleteness theorem was important and welcome in science, and how the field has developed since its result. It seems to us that Bourbaki’s group assumes that the incompleteness theorem does not preclude Mathematics from continuing its activity. Thus, from Bourbaki’s perspective, Gödel’s incompleteness theorem only indicates the arising of undecidable propositions, which are inevitable, occurring even in the theory of natural numbers. We suggest updating the senses and the meanings of Gödel's incompleteness theorem in Mathematics Education undergraduate courses by aligning Gödel's theorem with secondary mathematics school curriculum. We also suggest including discussion of this theorem in different moments of the secondary mathematics school curriculum, in which students will have elements to build understanding of the two proofs as a final comprehensive project. This study contributes to the literature by setting light on the importance of working with results of Mathematical Logic such as Gödel's incompleteness theorem in secondary mathematics courses and teaching preparation. It calls the attention of the Mathematical Education community, since its consequences are directly related to the design of mathematics and how it is being taught at all grade levels. Although some of these mathematics contents may not be related specifically to the theorem, the understanding of the theorem shows the broad relevance of the method in making sense of Mathematics.

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

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There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)