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Ensino De Geometrias Não-Euclidianas Usando Arte E Matemática UNIVERSIDADE TECNOLÓGICA FEDERAL DO PARANÁ PROGRAMA DE PÓS-GRADUAÇÃO EM ENSINO DE CIÊNCIA E TECNOLOGIA MESTRADO PROFISSIONAL EM ENSINO DE CIÊNCIA E TECNOLOGIA SIMONE SEMMER ENSINO DE GEOMETRIAS NÃO-EUCLIDIANAS USANDO ARTE E MATEMÁTICA DISSERTAÇÃO PONTA GROSSA 2013 SIMONE SEMMER ENSINO DE GEOMETRIAS NÃO-EUCLIDIANAS USANDO ARTE E MATEMÁTICA Dissertação apresentada como requisito parcial à obtenção do título de Mestre em Ensino de Ciência e Tecnologia, do Programa de Pós-Graduação em Ensino de Ciência e Tecnologia da Universidade Tecnológica Federal do Paraná, Campus Ponta Grossa. Área de Concentração: Ciência, Tecnologia e Ensino. Orientador: Profa. Dra. Sani de Carvalho Rutz da Silva. Coorientador: Prof. Dr. Marcos Cesar Danhoni Neves PONTA GROSSA 2013 Ficha catalográfica elaborada pelo Departamento de Biblioteca da Universidade Tecnológica Federal do Paraná, Campus Ponta Grossa n.028/13 S471 Semmer, Simone Ensino de geometrias não-euclidianas usando arte e matemática. / Simone Semmer. -- Ponta Grossa, 2013. 268 f. : il. ; 30 cm. Orientadora: Profa. Dra. Sani de Carvalho Rutz da Silva Co-orientador: Prof. Dr. Marcos Cesar Danhoni Neves Dissertação (Mestrado em Ensino de Ciência e Tecnologia) - Programa de Pós- Graduação em Ensino de Ciência e Tecnologia, Universidade Tecnológica Federal do Paraná. Ponta Grossa), 2013. 1. Geometria não-euclidiana. 2. Anamorfose. 2. Matemática na arte. I. Silva, Sani de Carvalho Rutz da. II. Neves, Marcos Cesar Danhoni. III. Universidade Tecnológica Federal do Paraná, Campus Ponta Grossa. IV. Título. CDD 507 Universidade Tecnológica Federal do Paraná Campus de Ponta Grossa PR Diretoria de Pesquisa e Pós-Graduação UNIVERSIDADE TECNOLÓGICA FEDERAL DO PARANÁ PROGRAMA DE PÓS-GRADUAÇÃO EM ENSINO DE CIÊNCIA E TECNOLOGIA FOLHA DE APROVAÇÃO Título de Dissertação No. 62/2013 ENSINO DE GEOMETRIAS NÃO-EUCLIDIANAS USANDO ARTE E MATEMÁTICA por Simone Semmer Esta dissertação foi apresentada às 14horas de 24 de maio de 2013 como requisito parcial para a obtenção do título de MESTRE EM ENSINO DE CIÊNCIA E TECNOLOGIA, com área de concentração em Ciência, Tecnologia e Ensino, do Programa de Pós-Graduação em Ensino de Ciência e Tecnologia. O candidato foi arguido pela Banca Examinadora composta pelos professores abaixo assinados. Após a deliberação, a Banca Examinadora considerou o trabalho aprovado. Prof. Dr. José Carlos Pinto Leivas Profa. Dra. Nilcéia Aparecida Maciel Pinheiro (UNIFRA) (UTFPR) Prof. Dr. Marcos Cesar Danhoni Neves Profa. Dra. Sani de Carvalho Rutz da Silva (UEM) - Coorientador (UTFPR) – Orientador Profa. Dra. Rosemari Monteiro Castilho Foggiatto Silveira (UTFPR) Visto do Coordenador: Profa Dra Sani de Carvalho Rutz da Silva Coordenadora do PPGECT A FOLHA DE APROVAÇÃO ASSINADA ENCONTRA-SE NO DEPARTAMENTO DE REGISTROS ACADÊMICOS DA UTFPR – CÂMPUS PONTA GROSSA Dedico este trabalho a minha família e aos meus amigos, por me fazerem acreditar que os momentos de ausência, paciência e de desespero seriam passageiros, e que no final restaria somente sensação de dever cumprido. Ao meu filho Carlos, para que não desista de seus sonhos, que podem ser realizáveis. AGRADECIMENTOS Certamente estes parágrafos não irão atender a todas as pessoas que fizeram parte dessa importante fase de minha vida. Portanto, desde já peço desculpas àquelas que não estão presentes entre essas palavras, mas elas podem estar certas de que fazem parte do meu pensamento e de minha gratidão. Agradeço às forças supremas do Universo, aos Grandes Mestres que acompanham a vida e o sucesso. A minha orientadora Prof.a Dr.a Sani de Carvalho Rutz da Silva, pela competência com que me guiou nesta trajetória; A meu coorientador Prof. Dr. Marcos Cesar Danhoni Neves, pelas colocações certas, nos momentos adequados; Ao Prof. Dr. José Carlos Pinto Leivas, a oportunidade de ter visto meu trabalho já na qualificação e, assim, contribuído com a sua linguagem e considerações; À Prof.a Dr.a Nilcéia Aparecida Maciel Pinheiro, pelas sugestões de encaminhamento da metodologia da pesquisa que nortearam o trabalho; À Prof.a Dr.a Rosemari Monteiro Castilho Foggiatto Silveira, pelas informações necessárias, principalmente a indicação bibliográfica que abriu o pensamento e norteou a análise dos dados desta pesquisa; À Tânia Mara Niezer, por ter indicado a UTFPR e pela ajuda dedicada, por ser amiga e confidente, auxiliando, lembrando e orientando, não tenho adjetivo suficiente para descrever minha gratidão; A Simone Perpétuo Nishi, que em tantos momentos soube ouvir e compartilhar momentos de agonia e êxtase. Sem dúvida, uma grande amiga cuja amizade iniciou em um padrão de coincidências e perdurará num referencial de boas lembranças; A todos os meus colegas de sala do PPGECT, pela companhia, força e determinação nos trabalhos iniciais e, mais tarde, na atenção e incentivos, uma grande amizade sem palavras para descrever e agradecer; Aos professores Zamir K. Nogueira e Mara Ribeiro Bartinik, da Universidade do Contestado - UnC, que, de uma forma ou outra, fizeram parte dessa pesquisa, seja por meio de prorrogação de prazos de trabalho e incentivos com palavras de conforto seja com sugestões de encaminhamento; A Simone Stange, amiga de infância, colega de instituição, que conseguiu me tranquilizar num momento de desespero, indicando os caminhos a serem trilhados sem preocupação; A Mauricio Linécia, Vilson José Kotviski, Fabine Zadorosny e Fabiene Zadorosny, que gentilmente colaboraram para ampliação do conhecimento das pêssankas com suas práticas enriquecedoras. A Secretaria do PPGECT, pela cooperação. A todos os alunos da 2ª série do Ensino Médio do Colégio Estadual Dr. Ovande do Amaral, do ano letivo de 2012. A todos os professores do Colégio Estadual Dr. Ovande do Amaral, que auxiliaram na troca de horários, no empréstimo de livros, na compreensão pela ausência e cooperação, durante o último ano letivo e, principalmente, nas palavras de incentivo e de retomada da razão. Gostaria de deixar registrado, também, meu reconhecimento à minha família. Agradeço aos meus pais José Adalberto e Waldette, pela educação recebida, destacando sempre que o conhecimento e a cultura é o que move a vida. Agradeço à tia Waldine, pela harmonia com que me mostrou que com estética e proporção se faz Arte. A todos os familiares que compreenderam a minha ausência em reuniões de família. E agradeço, principalmente, ao meu filho Carlos, que foi enfermeiro, fisioterapeuta, carregador, motorista, revisor, confidente e soube conduzir suas palavras e comportamentos em função desse trabalho extenso e difícil. Sem o apoio deles seria muito difícil vencer esse desafio. Enfim, a todos os que por algum motivo contribuíram para a realização desta pesquisa. Depois que você enxergar a matemática como a ciência de padrões e compreender que fazer matemática é raciocinar sobre padrões, achará muito menos surpreendente a descoberta de que muitas criaturas vivas fazem alguma matemática. Uma vez que você se distancia da visão de lápis-e-papel da matemática que todos nós obtivemos quando estudantes e pensa na atividade mais fundamental para a qual esses métodos acadêmicos fornecem apenas um modo de execução, você descobre que a matemática está em toda parte. (Keith Devlin) RESUMO SEMMER, Simone. Ensino de geometrias não-euclidianas usando Arte e Matemática. 2013. 265 f. Dissertação (Mestrado em Ensino de Ciência e Tecnologia) - Universidade Tecnológica Federal do Paraná. Ponta Grossa, 2013. A presente dissertação teve como objetivo da introduzir conceitos básicos de geometrias não-euclidianas em aulas de Matemática do Ensino Médio, usando Arte e Matemática. Para tanto, utilizou-se de abordagem triangular, fundamentada por Barbosa e de registros de representações semióticas, baseados nos estudos de Duval. O estudo envolveu estudantes de 2as séries do Ensino Médio de um colégio público estadual do município de Rio Negro (PR). A pesquisa aplicada constou de duas etapas. Num primeiro momento, analisou-se pêssankas à procura de conceitos matemáticos empregados em sua composição e verificou-se a utilização, instintivamente, pelos artesãos, de conceitos como simetria, proporção, polígonos, elipses, biláteros, retas e pontos. Na segunda etapa abordou-se o ensino de geometrias não-euclidianas no Ensino Médio, usando Arte e Matemática. Do ponto de vista metodológico a abordagem foi qualitativa, de natureza interpretativa, com observação participante. Os dados foram recolhidos a partir da aplicação de sequências de atividades envolvendo anamorfose, geometria espacial e projetiva e, da aplicação de oficina investigativa, envolvendo geometrias plana, espacial, elíptica e projetiva. As atividades desenvolvidas com os alunos envolveram materiais manipuláveis, recursos tecnológicos, análise de imagem, contexto histórico e fazer artístico. Os resultados mostram a validade do trabalho docente com metodologia interdisciplinar, tornando as aulas de Matemática motivadoras e desafiantes. Como produto final, apresenta-se um manual pedagógico que tem por finalidade fornecer aos professores de Matemática e de Arte, interessados no assunto, informações sobre conexões entre Arte e Matemática que se fazem presentes no ensino de noções de geometrias não-euclidianas. Palavras-chave: Geometria não-euclidiana. Anamorfose. Ensino de geometria. Arte e Matemática. ABSTRACT SEMMER, Simone. Teaching of no-euclidean geometry using Art and Mathematics. 2013. 265 f. Dissertation (Masters in Teaching of Science and Technology) - Federal Technological University of Paraná. Ponta Grossa, 2013. The present dissertation had as objective to
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