Acompanha: O lúdico associado à resolução de problemas e jogos no ensino e aprendizagem de funções: uma abordagem diferenciada / Este trabalho teve como objetivo investigar os resultados obtidos no ensino e aprendizagem de funções na 1ª série do ensino médio ao se utilizar o lúdico associado a outras estratégias de ensino. A pesquisa foi realizada com alunos de uma escola da Rede Estadual de Educação da cidade de Guarapuava, Paraná. Para a elaboração das estratégias que atendessem às necessidades de aprendizagem dos estudantes, optou-se pela aplicação do jogo “Torre de Hanói” e a resolução de problemas abordados de forma lúdica por meio de personagens e paródias para desenvolver o conteúdo de funções nas aulas de matemática. O referencial teórico que balizou a pesquisa está organizado em três capítulos. Os dois primeiros capítulos abordam o Ensino Médio de acordo com os Parâmetros Curriculares Nacionais e a Matemática no contexto da Base Nacional Comum, bem como fazem referências aos autores Machado (1993), Kuenzer (2010), Sacristán (2000) e Pais (2009). No terceiro capítulo, as referências contemplam o Lúdico na Escola, com a abordagem dos autores Grando (2008), Macedo (2005), Haydt (2006) e Macedo, Petty e Passos (2005). Faz-se referência do lúdico por meio de jogos, com base nos autores Kishimoto (2005), Smole e Diniz (2007) e Machado (2012). A Resolução de Problemas numa abordagem lúdica traz como referência os autores Pólya (2006), Pais (2009), Walle (2009) e Onuchic (1999). Finaliza-se o capítulo do lúdico abordando-se as paródias segundo os autores Hutcheon (1991), Fávero (1983) e Marchuschi (2008). O direcionamento, coleta e análise dos dados se deram por meio da pesquisa interpretativa, de natureza qualitativa com finalidade aplicada. Durante o desenvolvimento das atividades, percebeu-se maior envolvimento e interesse dos alunos com os conteúdos se comparados às estratégias de trabalhos anteriores, bem como acentuada evolução na aprendizagem, demonstrado por meio de avaliação diagnóstica, processual e formativa, usada de forma diversificada para atender à demanda dos estudantes. Com base no trabalho desenvolvido, originou-se um manual didático interativo em DVD, cujo objetivo é oferecer aos professores do ensino médio um roteiro de trabalho para desenvolver os conteúdos de função de maneira contextualizada, utilizando o jogo e a resolução de problemas em um cenário lúdico. / The present paper had as its aim to analyze the results obtained in teaching and learning functions in the first year of high school when using amusement applied to different pedagogical strategies. The research was carried out with high school first grade students in a state school in Guarapuava, Paraná. To organize some strategies that could fill the students learning necessities, the game “Hanoi´s Tower” was chosen, as well as some problems solving, which were undertaken through characters and parodies in order to develop the functions contents during the mathematics classes. The theoretical references that delimited the research are organized in three chapters. The first and the second ones deal with High School in conformation to the National Curriculum Patterns and the Mathematics within the Common National Basis, also referring to authors like Machado (1993), Kuenzer (2010), Sacristán (2000) and Pais (2009). In the third chapter, the references observe Amusement at School, bringing studies from Grando (2008). Macedo (2005), Haydt (2006) and Macedo, Peety and Passos (2005). The amusement allusion is made through games, with basis on Kishimoto (2005), Smole amd Diniz (2007) and Machado (2012). Problem Solving, in an amusing approach brings as references authors as Pólya (2006), Pais (2009), Walle (2009) and Onuchic (1999). The chapter about amusement finishes broaching the parodies according to Hutcheon (1991), Fávero (1983) and Marchuschi (2008). The orientation, gathering and analysis of the data happened by the interpretative research, of qualitative nature with applied purpose. During the activities development, it was possible to observe a higher students involvement and interest for the contents if compared to earlier work strategies, as well as a strong evolution in learning, what is demonstrated by a diagnostic, processual and formative evaluation, which was used in different ways to attend students‟ demands. Based on the developed work, an interactive didactic manual in DVD was produced. Its objective is to offer high school teachers an itinerary to develop functions contents in a contextualized manner, using games and problems solving in a playful scenery.

CAPES / Este trabalho teve como principal objetivo apresentar uma Proposta Metodológica fundamentada na teoria de Ponte (2005, 2006, 2014) sobre Diversificação de Tarefas, afim de que esta coloque os alunos em Atividade e possibilite o aprendizado significativo dos conceitos formais de Números Inteiros, do contexto histórico e das quatro operações. Para completar nossa Proposta, buscamos referências nas pesquisas que versam sobre Registros de Representações Semióticas, de Duval (1993, 2003) e nas teorias sobre Jogos em Matemática. Deste modo, nossa Proposta foi planejada e construída com base no movimento de ação reflexão-ação, sendo aplicada em uma turma regular do 7º ano (6ª série) do Ensino Fundamental 2, na qual o pesquisador é o professor de Matemática. Esta pesquisa-ação é majoritariamente qualitativa, visando avaliar se as mudanças realizadas na metodologia provocarão melhorias na prática do professor. Posteriormente, com base nos dados coletados por meio das Tarefas aplicadas, tecemos considerações estruturadas em Categorias de Análise, em acordo com a teoria de Bardin (1977), sobre Análise de Conteúdos. Os resultados apontam para uma real necessidade da diversificação de Tarefas no planejamento metodológico de qualquer professor que busque a apropriação do conhecimento por seus alunos. / This piece of work has as its main goal to present a Methodological Proposal based on the theory of Ponte (2005, 2006, and 2014) about Diversification of Tasks, in order that this put the students in Activity and make the process of learning possible and meaningful of the formal concepts of Integer, on the historical concept and the four operations. In order to accomplish our Proposal, we sought references on the researches that deal with Records of Semiotic Representations, of Duval (1993, 2003) and on the theories about Games in Math. This way, our Proposal was planned and built based on the movement of action-reflection-action, being applied in a regular classroom of the 7th year (6th grade) of the Middle School, in which the researcher is the Math teacher. This research-action is mostly qualitative, seeking to evaluate if the changes accomplished on the methodology will cause positive impact on the teacher’s role. Later on, based on the collected data through the applied Tasks, according to the theory of Bardin (1977), about Content Analysis. The results point to a real need of the diversification of Tasks on the methodological planning of any teacher that seeks the appropriation of knowledge for their students.

CAPES / Este trabalho traz uma breve apresentação histórica do surgimento dos Fractais, bem como do criador do termo Fractal, Benoit Mandelbrot. Aqui são apresentados alguns Fractais clássicos e destacam-se suas principais propriedades Matemáticas. Sugere-se um software livre, e de acesso online, para a criação dessas figuras. Posteriormente, aplica-se uma proposta didática baseada em um material paradidático que propõe a investigação de Fractais nos anos finais do ensino fundamental ou no ensino médio, relatando-se os resultados dessa aplicação. / This dissertation presents a brief historical overview of the origins of Fractals, as well as of the creator of the Fractal term, Benoit Mandelbrot. Some classical Fractals are presented and its main mathematical properties are highlighted. It suggests a free software, with on-line access, to the creation of these figures. Later, the didactic proposal is applied based on a paradidactic material, which proposes investigation of Fractals in the last years of primary school or in the secondary school and the results of this application are reported.

CAPES / Este trabalho teve como principal objetivo apresentar uma Proposta Metodológica fundamentada na teoria de Ponte (2005, 2006, 2014) sobre Diversificação de Tarefas, afim de que esta coloque os alunos em Atividade e possibilite o aprendizado significativo dos conceitos formais de Números Inteiros, do contexto histórico e das quatro operações. Para completar nossa Proposta, buscamos referências nas pesquisas que versam sobre Registros de Representações Semióticas, de Duval (1993, 2003) e nas teorias sobre Jogos em Matemática. Deste modo, nossa Proposta foi planejada e construída com base no movimento de ação reflexão-ação, sendo aplicada em uma turma regular do 7º ano (6ª série) do Ensino Fundamental 2, na qual o pesquisador é o professor de Matemática. Esta pesquisa-ação é majoritariamente qualitativa, visando avaliar se as mudanças realizadas na metodologia provocarão melhorias na prática do professor. Posteriormente, com base nos dados coletados por meio das Tarefas aplicadas, tecemos considerações estruturadas em Categorias de Análise, em acordo com a teoria de Bardin (1977), sobre Análise de Conteúdos. Os resultados apontam para uma real necessidade da diversificação de Tarefas no planejamento metodológico de qualquer professor que busque a apropriação do conhecimento por seus alunos. / This piece of work has as its main goal to present a Methodological Proposal based on the theory of Ponte (2005, 2006, and 2014) about Diversification of Tasks, in order that this put the students in Activity and make the process of learning possible and meaningful of the formal concepts of Integer, on the historical concept and the four operations. In order to accomplish our Proposal, we sought references on the researches that deal with Records of Semiotic Representations, of Duval (1993, 2003) and on the theories about Games in Math. This way, our Proposal was planned and built based on the movement of action-reflection-action, being applied in a regular classroom of the 7th year (6th grade) of the Middle School, in which the researcher is the Math teacher. This research-action is mostly qualitative, seeking to evaluate if the changes accomplished on the methodology will cause positive impact on the teacher’s role. Later on, based on the collected data through the applied Tasks, according to the theory of Bardin (1977), about Content Analysis. The results point to a real need of the diversification of Tasks on the methodological planning of any teacher that seeks the appropriation of knowledge for their students.

CAPES / Este trabalho traz uma breve apresentação histórica do surgimento dos Fractais, bem como do criador do termo Fractal, Benoit Mandelbrot. Aqui são apresentados alguns Fractais clássicos e destacam-se suas principais propriedades Matemáticas. Sugere-se um software livre, e de acesso online, para a criação dessas figuras. Posteriormente, aplica-se uma proposta didática baseada em um material paradidático que propõe a investigação de Fractais nos anos finais do ensino fundamental ou no ensino médio, relatando-se os resultados dessa aplicação. / This dissertation presents a brief historical overview of the origins of Fractals, as well as of the creator of the Fractal term, Benoit Mandelbrot. Some classical Fractals are presented and its main mathematical properties are highlighted. It suggests a free software, with on-line access, to the creation of these figures. Later, the didactic proposal is applied based on a paradidactic material, which proposes investigation of Fractals in the last years of primary school or in the secondary school and the results of this application are reported.

Text in Afrikaans / Die doel van hierdie navorsing was om vas te stel of leerders in Graad R die nodige wiskundige vaardighede deur gestruktureerde spel bemeester het wat as grondslag vir verdere wiskunde-onderrig kan dien. In die literatuurstudie is die verskillende wiskunde konsepte in diepte ondersoek. Daar is ook gefokus op die oordra van wiskunde konsepte aan jong leerders. Verder is daar in die literatuurstudie ondersoek ingestel na die doel, waardes, tipes, ontwikkeling van spel asook spel in die moderne era. Vir die empiriese ondersoek is 103 Graad R-leerders van ses verskillende skole betrek. Daar is gebruik gemaak van ʼn ewekansige steekproef by die skole wat bereid was om aan die studie deel te neem. Die ondersoek het plaasgevind deur die gebruik van die positivistiese kwantitatiewe navorsingsmetode. Daar is gebruik gemaak van vraelyste om te bepaal wat die bemeestering van wiskundige vaardighede by Graad R-leerders is. Gestruktureerde spel is as hulpmiddel gebruik vir die insameling van die navorsingsmateriaal. Die resultate is bekom deur die vraelyste statisties te verwerk. Daar is ook vraelyste aan die onderwysers gegee om meer agtergrond oor hulle onderrigmetodes asook die onderwysers se eie gevoel teenoor en kennis van wiskunde te bekom. Die navorsingsresultate bevestig dat die meerderheid van die Graad R-leerders oor ʼn goeie wiskundige grondslag beskik. Dus is daar deur die studie gevind dat dit belangrik is dat jong kinders baie moet speel om nuwe vaardighede aan te leer, maar ook die geleentheid moet hê om die vaardighede te beoefen. / The purpose of this research was to determine whether learners in Grade R mastered the necessary math skills through structured play that could serve as a basis for further mathematics training. In the literature various math concepts were studied in depth. The main focus was on the transfer of math concepts to young learners. The literature study further focussed on the purpose, values, and types of play, development of play and play in the modern era. An empirical study using random sampling was done using 103 Grade R learners from six different schools who were willing to participate in the study. The study was done using the positivism quantitative research methodology. Questionnaires were used to determine the mastery of mathematical skills by Grade R learners. Structured play was used as a tool to collect the research material. The results were obtained by processing the questionnaires statistically. Questionnaires were also completed by the teachers to confirm their background knowledge, teaching methods as well as to determine the teachers‟ own feelings towards and their knowledge of mathematics. The research results confirm that the majority of Grade R learners have obtained a good math foundation. The study also confirmed that it is important that young children need to play to learn new skills and must have the opportunity to practise the new skills. / Psychology of Education / D. Ed. (Sielkundige Opvoedkunde)

Text in Afrikaans / Die doel van hierdie navorsing was om vas te stel of leerders in Graad R die nodige wiskundige vaardighede deur gestruktureerde spel bemeester het wat as grondslag vir verdere wiskunde-onderrig kan dien. In die literatuurstudie is die verskillende wiskunde konsepte in diepte ondersoek. Daar is ook gefokus op die oordra van wiskunde konsepte aan jong leerders. Verder is daar in die literatuurstudie ondersoek ingestel na die doel, waardes, tipes, ontwikkeling van spel asook spel in die moderne era. Vir die empiriese ondersoek is 103 Graad R-leerders van ses verskillende skole betrek. Daar is gebruik gemaak van ʼn ewekansige steekproef by die skole wat bereid was om aan die studie deel te neem. Die ondersoek het plaasgevind deur die gebruik van die positivistiese kwantitatiewe navorsingsmetode. Daar is gebruik gemaak van vraelyste om te bepaal wat die bemeestering van wiskundige vaardighede by Graad R-leerders is. Gestruktureerde spel is as hulpmiddel gebruik vir die insameling van die navorsingsmateriaal. Die resultate is bekom deur die vraelyste statisties te verwerk. Daar is ook vraelyste aan die onderwysers gegee om meer agtergrond oor hulle onderrigmetodes asook die onderwysers se eie gevoel teenoor en kennis van wiskunde te bekom. Die navorsingsresultate bevestig dat die meerderheid van die Graad R-leerders oor ʼn goeie wiskundige grondslag beskik. Dus is daar deur die studie gevind dat dit belangrik is dat jong kinders baie moet speel om nuwe vaardighede aan te leer, maar ook die geleentheid moet hê om die vaardighede te beoefen. / The purpose of this research was to determine whether learners in Grade R mastered the necessary math skills through structured play that could serve as a basis for further mathematics training. In the literature various math concepts were studied in depth. The main focus was on the transfer of math concepts to young learners. The literature study further focussed on the purpose, values, and types of play, development of play and play in the modern era. An empirical study using random sampling was done using 103 Grade R learners from six different schools who were willing to participate in the study. The study was done using the positivism quantitative research methodology. Questionnaires were used to determine the mastery of mathematical skills by Grade R learners. Structured play was used as a tool to collect the research material. The results were obtained by processing the questionnaires statistically. Questionnaires were also completed by the teachers to confirm their background knowledge, teaching methods as well as to determine the teachers‟ own feelings towards and their knowledge of mathematics. The research results confirm that the majority of Grade R learners have obtained a good math foundation. The study also confirmed that it is important that young children need to play to learn new skills and must have the opportunity to practise the new skills. / Psychology of Education / D. Ed. (Sielkundige Opvoedkunde)