Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-07-14T15:19:06Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Mestrado Gleidson Souza 2015.pdf: 4384688 bytes, checksum: 2b4f7a78258ef6b6b60955ac938994a0 (MD5) / Made available in DSpace on 2016-07-14T15:19:06Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Mestrado Gleidson Souza 2015.pdf: 4384688 bytes, checksum: 2b4f7a78258ef6b6b60955ac938994a0 (MD5) Previous issue date: 2015-05-15 / CAPEs / A probabilidade é um importante conteúdo e possui aplicações em outras áreas do conhecimento. Pesquisadores destacam a importância de conteúdos de probabilidade por possibilitar o desenvolvimento da criticidade e leitura de mundo pelos educandos, contudo realçam a escassez de estudos na área. Nesta pesquisa investigamos conhecimentos de estudantes do 9º ano do Ensino Fundamental sobre probabilidade com o uso da ferramenta Sampler do TinkerPlots 2.0. Buscou-se em específico, identificar conhecimentos prévios dos estudantes sobre probabilidade; descrever conhecimentos dos estudantes sobre probabilidade a partir de experimento aleatório não realizado e realizado no ambiente físico; e explorar a ferramenta Sampler do TinkerPlots para mobilizar conhecimentos dos estudantes sobre probabilidade. Quatro estudantes do 9º ano oriundos de uma escola pública do Recife participaram de quatro etapas de coleta de dados: 1) entrevista semi-estruturada com ênfase na investigação do perfil dos estudantes; 2) teste diagnóstico com questões sobre concepções e sobre experimento aleatório não realizado e realizado no ambiente físico. Nessas etapas iniciais os estudantes trabalharam individualmente. Em seguida, eles realizaram em duplas uma fase final do experimento realizado no ambiente físico; 3) na terceira etapa, subdividida em dois momentos, os estudantes em duplas, participaram de uma sessão de familiarização com o TinkerPlots para conhecer sua interface e ferramentas. Após esse contato inicial com o software eles participaram de uma sessão com o uso mais autônomo da ferramenta Sampler. Todo o processo ocorreu em dias alternados; 4) ao final da atividade de simulação com o Sampler, os estudantes realizaram novamente o teste diagnóstico administrado inicialmente, na segunda etapa, seguindo os mesmos procedimentos. Existem evidências de que as concepções iniciais dos estudantes estiveram mais voltadas para conhecimentos intuitivos sobre probabilidade. Ao longo das situações de pesquisa, mais particularmente no teste final, essas concepções foram ampliadas para incluir ideias de aleatoriedade, incerteza e chance. Seus conhecimentos sobre espaço amostral também foram se tornando mais explícitos a partir de suas reflexões sobre os experimentos aleatórios. O trabalho com a ferramenta Sampler do TinkerPlots permitiu que os estudantes realizassem simulações com tamanhos crescentes de amostras e verificassem as alterações nas representações gráficas produzidas. Esse processo foi realizado de forma dinâmica e mediado por diálogos e intervenções específicas do pesquisador. O TinkerPlots, por possibilitar a simulação de experimentos com tamanhos de amostras variadas pode ter contribuído para os estudantes refletirem melhor sobre a relação entre probabilidade teórica e probabilidade advinda da experimentação. Conclui-se que embora os estudantes não possuam conhecimentos formalizados sobre probabilidade, eles foram capazes de aprofundar suas ideias iniciais sobre aspectos cruciais desse conceito como é o caso de espaço amostral e da relação sobre probabilidade teórica e frequencial. Novas pesquisas devem ser feitas para investigar outras possibilidades do uso do software TinkerPlots para mobilizarem conhecimentos sobre probabilidades. / The probability is an important discipline and has application in many branch of human knowledge. Researchers emphasize the importance of probability related disciplines for enhancement students critical thinking about the real world problems; however they also highlight the shortage of studies in the area. In this research we investigate students knowledge in 9th grade about probability with the support of the software tool Sampler do TinkerPlots 2.0. Specifically, we sought to identify prior knowledge of students about probability; describe students knowledge about probability based on experiments realized and not realized in physical environment; and to explore the Sampler tool of TinkerPlots in order to students enhance and apply their knowledge about probabilit. Four 9th grade students from a public school in Recife participated in the four data collection steps: 1) semi-structured interview with an emphasis on investigating the student profile; 2) diagnostic test with questions about concepts and random experiments realized both in physical environment and not physical environment. In these initial steps the students worked individually. Then they performed, in pairs, the final experiment phase in a physical environment; 3) in the third step, divided into two different moments, the pairs were introduced to the TinkerPlots interface and tools. After this initial contact with the software, they attended a session with more autonomous use of the Sampler tool. The whole process was performed on alternate days; 4) at the end of the activity involving simulation with the use of the Sampler, the students were submited again to the diagnostic test administered initially in the second step, following the same procedures. There is evidence that the initial conceptions of the students were more focused on intuitive knowledge of probability. Over the situations found during the research, particularly in the final test, these conceptions were broadened in order to include ideas of randomness, uncertainty and chance. Students knowledge about sample space became more explicit after their reflections on experiments about randomness. The work with the Sampler tool of TinkerPlots allowed students to process simulations with an increasing number of sample and observe the changes in the graphs produced and its visual effect. This process was dynamically conducted and mediated by the researcher with dialogues and interventions. The TinkerPlots software, by allowing simulation experiments with different sample sizes, may have contributed to students reflection on the relationship between theoretical probability and probability arising from experimentation. We can conclude that although students may not have a formal knowledge of probability, they were able to deepen their initial ideas about crucial aspects of these concept such as sample space and the relationship between theoretical and frequency probability. New researches should be done in order to investigate other possibilities for the use of the software TinkerPlots improve students knowledge about probabilities.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O presente trabalho tem por finalidades: demonstrar o problema da agulha de Buffon, fazer uma pequena generalizaÃÃo do resultado obtido e apresentar aplicaÃÃes baseadas nos fundamentos do referido problema. O problema da agulha de Buffon està inserido no estudo da Teoria das Probabilidades, particularmente na subÃrea de probabilidade geomÃtrica. Para chegarmos à soluÃÃo desta questÃo, alÃm dos conceitos e propriedades atinentes à Teoria das probabilidades à necessÃrio o conhecimento de noÃÃes bÃsicas do cÃlculo integral. Nos capÃtulos 2, 3 e 4 à apresentado um estudo preliminar sobre probabilidade, com os conceitos bÃsicos, propriedades e a formulaÃÃo de alguns modelos probabilÃsticos. Durante o desenvolvimento do trabalho, sempre que possÃvel, os conceitos e definiÃÃes sÃo inseridos com o auxÃlio de um problema motivador e para fixaÃÃo dos mesmos sÃo mostrados exemplos resolvidos. O Ãltimo capÃtulo evidencia a importÃncia do problema de Buffon como mÃtodo para realizar estimativas e como fundamento para o processo de captaÃÃo de imagens pelos aparelhos de tomografia computadorizada, um grande avanÃo para a Medicina no que diz respeito ao diagnÃstico por imagens. / This paper has the objective of showing Buffon's needle problem, doing a minor generalization of the results obtained hereby, and also presenting some applications based upon the fundamentals of such problem. Buffon's needle problem has been inserted into the study of Theory of Probability, particularly in its sub-area of geometrical probability. In order to attain the solution to this question, in addition to the concepts and the properties concerning the theory of probabilities, it is necessary that one should have some basic knowledge about integral calculus. In chapters 2, 3, and 4 there is a preliminary study of probability, with the basic concepts, properties and formulation of some probabilistic models being presented. During the development of this paper, whenever it was possible, the concepts and definitions were inserted with the aid of a motivational problem and they were solved by means of fixing the same examples as shown. The final chapter presents the importance of Buffon's needle problem as a method of making estimates and as a foundation for the process of capturing images in CT (computerized tomography) scanning machines, such a great breakthrough in what concerns the diagnosis by means of imaging.

This paper presents an innovat ive cons t ruct ion of a probabilistic model for predicting chance situations. It describes the construction of a four phase model, derived from an intense qualitative analysis of the written responses of 94 mathematically talented middle school students to the probabilistic compound event problem: “How many doubles are expected when rolling two dice fifty times?” We found that the students’ comprehension process of compound event situations can be broken down into a four phase model: beliefs, subjective estimations, chance estimations and probabilistic calculations. The paper focuses on the development of the model over the course of the experiment, identifying the process the students underwent as they attempted to answer the question. We explain each phase as it was reflected in the students\' rationalizations. All phases, including their definitions and students’ citations, will be presented in the paper. While not every student necessarily goes through all four phases, an awareness and understanding of them all allows for efficient, effective intervention during the learning process. We found that guidance and learning intervention helped shorten the preliminary phases, leading to more relative time spent on probabilistic calculations.

Este trabalho tem como objetivo introduzir os conceitos bÃsicos da Teoria das Probabilidades e apresentar noÃÃes sobre alguns modelos probabilÃsticos para o estudante do Ensino MÃdio. Iniciaremos o trabalho apresentando no capÃtulo 1 as noÃÃes de experimento determinÃstico, experimento aleatÃrio, espaÃo amostral e eventos, seguidos de algumas definiÃÃes de Probabilidade, conceitos que constituem a base para essa teoria. No capÃtulo 2 abordaremos os conceitos de Probabilidade Condicional e IndependÃncia de Eventos, apresentando alguns teoremas importantes que decorrem desses conceitos, bem como algumas de suas aplicaÃoes. No capÃtulo 3 apresentaremos de maneira simples alguns modelos probabilÃsticos discretos bastante Ãteis por modelarem de forma eficaz um bom nÃmero de experimentos aleatÃrios contribuindo assim para o cÃlculo das probabilidades de seus resultados. Por fim, no capÃtulo 4 serà apresentado o modelo probabilÃstico conhecido como DistribuiÃÃo de Poisson, que nos permite calcular a probabilidade de um evento ocorrer em um dado intervalo de tempo ou numa dada regiÃo espacial. / This work has as objective introduce the basic concepts of the Theory of Probabilities and present notions on some probabilistic models for the student of the High School. We will begin the work presented in chapter I the notions of experiment deterministic, random experiment, sample space and events, followed by some definitions of Probability concepts that constitute the basis for this theory. In chapter II we will discuss the concepts of Conditional Probability and Independence of Events showcasing some important theorems that derive from these concepts, as well as some of its applications. In chapter III we will present in a simple way some probabilistic models discrete quite useful for shape effectively a good number of random experiments thus contributing to the calculation of the probabilities of its results. Finally, in chapter IV will be presented the probability model known as Poisson distribution, which allows us to calculate the probability that an event will occur in a given time interval or in a given spatial region.

This paper presents an innovat ive cons t ruct ion of a probabilistic model for predicting chance situations. It describes the construction of a four phase model, derived from an intense qualitative analysis of the written responses of 94 mathematically talented middle school students to the probabilistic compound event problem: “How many doubles are expected when rolling two dice fifty times?” We found that the students’ comprehension process of compound event situations can be broken down into a four phase model: beliefs, subjective estimations, chance estimations and probabilistic calculations. The paper focuses on the development of the model over the course of the experiment, identifying the process the students underwent as they attempted to answer the question. We explain each phase as it was reflected in the students\'' rationalizations. All phases, including their definitions and students’ citations, will be presented in the paper. While not every student necessarily goes through all four phases, an awareness and understanding of them all allows for efficient, effective intervention during the learning process. We found that guidance and learning intervention helped shorten the preliminary phases, leading to more relative time spent on probabilistic calculations.