▼ Complex APIs in new frameworks (Spark, R, TensorFlow, etc) have imposed steep learning curves on everyone, especially for people with limited programming backgrounds. For instance, due to the messy nature of data in different application domains, data scientists spend close to 80% of their time in data wrangling tasks, which are considered to be the "janitor work" of data science. Similarly, software engineers spend hours or even days learning how to use APIs through official documentation or examples from online forums. Program synthesis has the potential to automate complex tasks that involve API usage by providing powerful search algorithms to look for executable programs that satisfy a given specification (input-output examples, partial programs, formal specs, etc). However, the biggest barrier to a practical synthesizer is the size of search space, which increases strikingly fast with the complexity of the programs and the size of the targeted APIs. To address the above issue, this dissertation focuses on developing algorithms that push the frontiers of program synthesis. First, we propose a type-directed graph reachability algorithm in SyPet, a synthesizer for assembling programs from complex APIs. Second, we show how to combine enumerative search with lightweight constraint-based deduction in Morpheus, a synthesizer for automating real-world data wrangling tasks from input-output examples. Finally, we generalize the previous approaches to develop a novel conflict-driven synthesis algorithm that can learn from past mistakes. Advisors/Committee Members: Dillig, Isil (advisor), Mooney, Raymond (committee member), Kr ahenb uhl, Philipp (committee member), Aiken, Alex (committee member).

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Esta pesquisa se dedicou à introdução dos jogos lógicos na sala de aula, em específico, ela busca a investigação das dificuldades encontradas pelos alunos com os diferentes estilos de jogos apresentados e quais são os raciocínios lógicos utilizados para a resolução dos problemas propostos. Buscamos os benefícios que estes jogos podem trazer para o ensino e a aprendizagem na sala de aula. Aliado a isso também trabalhamos a parte geométrica dos tabuleiros, e com isso os alunos utilizaram como ferramentas a régua e o compasso para a construção dos tabuleiros apresentados.. Para tanto, a metodologia de pesquisa escolhida foi o Estudo de Caso, de acordo com Fiorentini e Lorenzato (2006), Ventura (2007) e Gil (2002). O referencial teórico é baseado nos trabalhos de Macedo (2007), Grando (2011), Skovsmose (2000), Huizinga (2000), Kishimoto (2006), Zuin (2001), bem como os PCNs e outros artigos/livros relacionados aos jogos lógicos e as construções geométricas com a régua e o compasso. As atividades foram desenvolvidas com uma turma do 9º ano do Ensino Fundamental de em uma Escola Municipal de Porto Alegre, no ano de 2015. Em especial sugerimos que é possível a inserção desses materiais a fim de serem usados como ferramentas de auxílio no ensino aprendizagem de matemática contribuindo positivamente para a formação dos alunos. Os registros coletados no estudo de caso possibilitaram a validação da proposta.

The present research has focused on the introduction of the logical games in the class. It aims to pin point difficulties presented by the students on the different styles of games proposed and a logical ratiocination is required to solve the tasks proposed. The benefits of the games along with the teaching and learning in the classroom was the goal. Additionally, the geometric segment of board games have been investigated and students used tools as ruler and compass to the construction of the boards presented. Therefore, the chosen researching methodology was the Case Study, according to Fiorentini e Lorenzato (2006), Ventura (2007) e Gil (2002). The theoretical referential is based on Works from Macedo (2007), Grando (2011), Skovsmose (2000), Huizinga (2000), Kishimoto (2006), Zuin (2001), along with the PCNs and others articles/books related to logical games and geometric constructions with a ruler and a compass. The activities were performed by a 9th grade group of the Elementary School in a Municipal School in Porto Alegre, in 2015. We showed in particular that is possible the insertion of these materials in order to be used as a sustenance tool on Mathematics Learning contributing positively to students formation. The data collected in the case study enabled the authentication of the proposal.

Advisors/Committee Members: Sant'Ana, Marilaine de Fraga.

▼ We propose and investigate several new principles of rational reasoning within the framework of Pure Inductive Logic, PIL, where probability functions defined on the sentences of a first-order language are used to model an agent's beliefs. The Elephant Principle is concerned with how learning, modelled by conditioning, may be uniquely `remembered'. The Perspective Principle requires that, from a given prior, conditioning on statistically similar experiences should result in similar assignments, and is found to be a necessary condition for Reichenbach's Axiom to hold. The Abductive Inference Principle and some variations are proposed as possible formulations of a restriction of C.S. Peirce's notion of hypothesis in the context of PIL, though characterization results obtained for these principles suggest that they may be too strong. The Finite Values Property holds when a probability function takes only finitely many values when restricted to sentences containing only constant symbols from some fixed finite set. This is shown to entail a certain systematic method of assigning probabilities in terms of possible worlds, and it is considered in this light as a possible principle of inductive reasoning. Classification results are given, stating which members of certain established families of probability functions satisfy each of these new principles. Additionally, we define the theory of a principle P of PIL to be the set of those sentences which are assigned probability 1 by every probability function which satisfies P. We investigate the theory of the established principle of Spectrum Exchangeability by finding separately the theories of heterogeneous and homogeneous functions. The theory of Spectrum Exchangeability is found to be equal to the theory of finite structures. The theory of Johnson's Sufficientness Postulate is also found. Consequently, we find that Spectrum Exchangeability, Johnson's Sufficientness Postulate and the Finite Values Property are all inconsistent with the principle of Super-Regularity: that any consistent sentence should be assigned non-zero probability.

▼ We propose and investigate several new principles of rational reasoning within the framework of Pure Inductive Logic, PIL, where probability functions defined on the sentences of a first-order language are used to model an agent's beliefs. The Elephant Principle is concerned with how learning, modelled by conditioning, may be uniquely `remembered'. The Perspective Principle requires that, from a given prior, conditioning on statistically similar experiences should result in similar assignments, and is found to be a necessary condition for Reichenbach's Axiom to hold. The Abductive Inference Principle and some variations are proposed as possible formulations of a restriction of C.S. Peirce's notion of hypothesis in the context of PIL, though characterization results obtained for these principles suggest that they may be too strong. The Finite Values Property holds when a probability function takes only finitely many values when restricted to sentences containing only constant symbols from some fixed finite set. This is shown to entail a certain systematic method of assigning probabilities in terms of possible worlds, and it is considered in this light as a possible principle of inductive reasoning. Classification results are given, stating which members of certain established families of probability functions satisfy each of these new principles. Additionally, we define the theory of a principle P of PIL to be the set of those sentences which are assigned probability 1 by every probability function which satisfies P. We investigate the theory of the established principle of Spectrum Exchangeability by finding separately the theories of heterogeneous and homogeneous functions. The theory of Spectrum Exchangeability is found to be equal to the theory of finite structures. The theory of Johnson's Sufficientness Postulate is also found. Consequently, we find that Spectrum Exchangeability, Johnson's Sufficientness Postulate and the Finite Values Property are all inconsistent with the principle of Super-Regularity: that any consistent sentence should be assigned non-zero probability. Advisors/Committee Members: VENCOVSKA, ALENA A, Paris, Jeffrey, Vencovska, Alena.

▼ The present research aims to evaluate the extent of deaf children’s delay in mathematics, identifying the moderators of this delay and determine the longitudinal predictors of their mathematical achievement. For five decades, studies have reported that deaf children lag behind their hearing peers in mathematics (Gottardis, Nunes and Lunt, 2011). Background factors such as age, degree of hearing loss, presence of cochlear implant and types of educational provision were previously hypothesised to be moderators of the extent of this delay but, up to now, they have not been tested. Pagliaro (2010) argued that number knowledge, working memory and degree of hearing loss could be possible causes of deaf children’s difficulties in mathematics but no clear conclusions were reached. The present investigation aims to provide insight into the causes of deaf children’s delay in mathematics. The survey study addressed the first aim of the present study. The maths test of the Performance Indicators for Primary School (PIPS) was used as outcome measure. Factors related to deaf children (degree of hearing loss, age, years in education, presence of cochlear implant, gender, causes of deafness) and background factors (highest maternal education, language used at home, type of educational provision) were assessed as possible predictors and moderators of the extent of deaf children’s delay in mathematics. The overall extent of deaf children’s delay in mathematics was of -1.76 SDs. The older the children get and the more years they spend in special schools for the deaf or in units for hearing impaired, the wider is their gap in mathematics achievement compared with their hearing peers. It is, therefore, necessary to intervene in their mathematical learning in the early years of schooling in order to create pathways for improvement. The second aim of the present study was addressed through a longitudinal design. Logical-mathematical reasoning, working memory and counting ability were chosen as predictors of deaf children’s mathematical attainment on the basis of theoretical framework, evidence from longitudinal studies and from the analysis of the difficulties that deaf children have in these factors compared with hearing peers. Hierarchical regression analyses were used to assess the independence of the contributions of logical-mathematical reasoning, working memory and counting ability to the prediction of deaf children’s mathematical achievement measured through the PIPS. Age, years in education, type of educational provision and non-verbal intelligence were used as controls. Counting ability and working memory did make independent contributions to the prediction of deaf children’s mathematical success but logical mathematical reasoning was by far the strongest predictor. When the predictors were entered in the model, none of the control variables predicted significantly deaf children’s mathematical achievement. This study makes several empirical contributions. First, it established age, years in education and types of educational…

▼ Reasoning is an essential element of intelligence. Automated reasoning in formal and symbolic systems is a major topic in computer science and artificial intelligence (AI). Programs for computer-assisted and automated theorem proving are being constructed and used by mathematicians and logicians. Recently, there has been a surge in research in inductive reasoning systems. Inductive programming is an example of applying induction to symbolic systems. Most of the reasoning systems in AI are narrow and specialized to particular tasks and domains. Research work in artificial general intelligence (AGI) aims at producing models of AI that are fully or partially independent of task types. AGI systems are ideally able to learn new knowledge and rules that were not intended during their construction. Automatic reasoning systems are traditionally developed by using heuristics to limit the computational resources. This thesis aims to produce models of easoning that use bounded cognitive resources. Since reasoning is a cognitive function, and human cognitive resources (such as working memory) are severely restricted, one possible method of reducing the computational complexity of reasoning systems is to introduce cognitive resources in the system and put limits on them similar to human cognitive limits. Another important aim of this thesis is to unite the deductive and inductive reasoning in symbolic systems, by using Occam's razor as a guiding principle for induction. The thesis uses an exploratory approach to search for a unified model of reasoning in arbitrary domains. This thesis is a collection of published scientific papers, each contributing iteratively to the construction of a larger and domain-independent model of reasoning in symbolic domains. The first two papers present proof formalisms for first-order logic and description logic that produce comprehensible proofs. The second part comprises of five papers, each extending a model of inductive reasoning that can learn axioms of any arbitrary symbolic domain from random examples. Some of the models in the thesis were able to outstrip human performance in arithmetic, logic and number series problems. This is an interdisciplinary thesis that contributes to a number of scientific areas, mainly formal logic and AGI. Further research in this area can potentially lead to a universal reasoner that is able to learn and reason in more complex symbolic systems such as higher order logics and computer programming.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Iremos apresentar a um grupo de alunos do Ensino Médio da rede pública de Ensino do Estado de São Paulo, o mundialmente conhecido quebra cabeças Sudoku, e realizar com eles várias atividades buscando apresentá-lo como subsídio didático na aprendizagem de conceitos matemáticos importantes, além de proporcionar oportunidades de aprimorar a concentração e o raciocínio lógico. Iremos explorar conceitos matemáticos ocultos por trás de suas linhas, colunas e blocos, partindo de uma das primeiras perguntas que podem ser feitas: Qual é a quantidade total de jogos válidos existentes? Para responde-la, será proposto a realização de diversas atividades, primeiramente com um Shidoku (matriz 4 × 4), em seguida iremos calcular o total desses jogos. O tamanho reduzido dessa grade, facilita os cálculos manuais, permitindo visualizar e compreender o processo utilizado, aproveitando para introduzir o princípio fundamental da contagem. A discussão principal desse trabalho, concentra-se na exploração de um método para se determinar a quantidade de jogos válidos existentes para um Sudoku, e para isso, utilizaremos as demonstrações de Bertrand Felgenhauer e Frazer Jarvis. Também apresentaremos um método capaz de gerar uma grade completa de Sudoku, partindo de uma matriz quadrada de ordem 3, que em seguida, será utilizada para gerar uma solução de Sudoku ortogonal. Finalizando, iremos apresentar e explorar algumas formas diferenciadas para os quebra cabeças Sudoku, mostrando variações no formato dos blocos, no tamanho das grades e uma variação que utiliza formas geométricas em suas pistas (Shapedoku). Como desafio de leitura, pesquisa e aprofundamento, será proposto o problema ainda em aberto do número mínimo de dados iniciais para se ter um jogo válido. Podemos afirmar que um dos objetivos esperados, é que tal atividade venha interferir na concentração e raciocínio, auxiliando nas atividades propostas nesse trabalho e que possam ser utilizadas em outros problemas do cotidiano.

We will present to a group of high school students of the public Education of Sao Paulo state, the world-known puzzle Sudoku, and perform with them several activities seeking to present it as a didactic subsidy in the learning important mathematical concepts, besides opportunities to enhance concentration and logical reasoning. We will explore hidden mathematical concepts…

Advisors/Committee Members: Universidade Estadual Paulista (UNESP), Vasconcellos, Luis Antonio da Silva [UNESP].

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Mestrado em Educação Especial

O desenvolvimento de competências matemáticas é essencial para o desenvolvimento da nossa autonomia. Este estudo pretendeu explorar o raciocínio lógico-matemático de uma criança com Trissomia 21 (T21), durante o seu primeiro ano escolar do primeiro ciclo do ensino básico. Esta criança é do sexo feminino, tem sete anos de idade e tem acompanhamento técnico ao nível da reabilitação e ensino especial desde os 15 dias de idade. Este estudo de caso é essencialmente exploratório e descritivo, acompanhando a evolução cognitiva das competências lógico-matemáticas da criança através da gravação em vídeo de sessões de ensino-aprendizagem, nas quais a criança realizou um conjunto de baterias de exercícios, elaborados de acordo com os conteúdos programáticos propostos para este nível de ensino e verbalizando os seus raciocínios ao longo destas sessões. Foram aplicadas três baterias de exercícios, desenvolvidas de acordo com o nível cognitivo da criança em estudo, tendo a primeira sido utilizada mais duas vezes, como teste de retenção, no fim do ano letivo e após três meses do fim do estudo investigativo. O estudo realizou-se durante onze meses e ao longo das sessões de apoio de reabilitação e ensino especial, em contexto domiciliário, com frequência semanal. Houve, igualmente, a realização de entrevistas à mãe da criança, à professora do ensino regular e à técnica que acompanhou o processo investigativo, de forma a recolher informação sobre o desenvolvimento cognitivo da criança em vários contextos da sua vida diária. Este estudo decorreu fora do contexto de sala de aula, mas as conclusões apresentadas são generalizadas e direcionadas a todos os contextos onde haja desenvolvimento do raciocínio lógico-matemático da criança. O estabelecimento de regras, através da memorização e do acompanhamento visuo-espacial dos conteúdos, assim como a atitude do elemento educativo perante a criança, com elevadas expectativas e tentando não limitar a evolução da mesma, são duas estratégias determinantes para uma evolução cognitiva consolidada. Ao nível da matemática, a repetição e estruturação das sequências de resolução de exercícios é essencial para a memorização da regra. Ao adquirir esta capacidade, a criança é então capaz de a aplicar, sempre que necessário, e também adaptar-se a diferentes cenários, mesmo quando a questão colocada não é exatamente a mesma. Esta capacidade poderá potenciar o desenvolvimento de futuras competências profissionais, autonomizando-a e permitindo a sua participação plena na sociedade.

The acquisition of mathematical skills is essential for the development of our autonomy. This study aims to explore the logical-mathematical reasoning of a child with Down syndrome, during the first school year of primary education. The child that participated in the study is female, has seven years old and has technical support in terms of rehabilitation and special education since she was 15 days old. This case study is essentially exploratory and descriptive and followed the child’s cognitive…

Advisors/Committee Members: Morato, Pedro Jorge Moreira de Parrot.

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A dissertação tem por objetivo problematizar como professoras de uma escola estadual do município de Esteio, situado na região metropolitana da capital do Estado do Rio Grande do Sul, descrevem a aprendizagem da matemática escolar e, em particular, o raciocínio lógico de seus alunos. Os aportes teóricos vinculam-se ao pensamento pós-estruturalista, especialmente a algumas formulações do filósofo Michel Foucault e a ideias presentes na obra Discurso do Método, de René Descartes, principalmente aquelas ligadas à ordenação do pensamento. O material de pesquisa está constituído por documentos avaliativos da escola, em especial por 58 pareceres descritivos elaborados por cinco professoras dos anos iniciais do Ensino Fundamental da instituição estudada, e por entrevistas que com elas foram realizadas. As entrevistas foram gravadas e posteriormente transcritas. A análise do material de pesquisa possibilitou: mostrar as ressonâncias das ideias cartesianas como condutoras das condutas dos sujeitos escolares na contemporaneidade; refletir sobre as relações existentes entre a Matemática como fundamento da racionalização e suas marcas deixadas na constituição dos sujeitos escolares, identificando-os como sujeitos aprendentes ou não-aprendentes nessa área do conhecimento; entender a subjetivação desses indivíduos a partir de narrativas que os posicionam hierarquicamente, com base em níveis do desenvolvimento infantil, construindo posições de sujeito ao caracterizar como deve ser o aluno que possui um bom raciocínio lógico.

This dissertation aims at problematizing how teachers from a state school in Esteio, a town located in the metropolitan region surrounding the capital of Rio Grande do Sul, describe the learning of school mathematics, particularly their students logical reasoning. The theoretical bases of this investigation have been linked to the poststructuralist thought, following some formulations of Michel Foucault, and ideas from Discourse on Method, by René Descartes, mainly those linked to thought ordering. Evaluation documents of the school studied, especially 58 descriptive evaluations written by five teachers of elementary school, as well as interviews with these teachers have constituted the research material. The interviews were recorded and later transcribed. The analysis of the research material has allowed for: showing the resonances of Cartesian ideas as conducting the conducts of school subjects in contemporaneity; reflecting upon relationships between Mathematics as a foundation of reasoning and its marks on the constitution of school subjects, identifying them as learners or non-learners in this area of knowledge; understanding subjectivation of those individuals from narratives that position them hierarchically, according to levels of children development, thus constructing subject positions while characterizing how a student with good logical reasoning should be.

Advisors/Committee Members: Gelsa Knijnik.

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Os pressupostos teóricos que embasam o presente estudo abordam o pensamento lógico como um dos processos mais importantes que ocorrem na formação da consciência. A escola é vista como um dos principais meios de transmissão da cultura historicamente construída pela humanidade. O estudo analisou como se apresenta o raciocínio lógico em quarenta e duas crianças provenientes de famílias de renda média e média-alta, com idades entre quatro, cinco e seis anos, que freqüentam uma pré-escola com projeto pedagógico de qualidade. Como instrumento, foram utilizadas provas silogísticas de aplicação individual. As respostas foram anotadas atribuindo-se pontos e as justificativas dadas às respostas consideradas corretas foram analisadas qualitativamente. Os resultados mostraram aumento na freqüência de acertos e de justificativas consideradas teóricas, acompanhando a faixa etária e a conseqüente progressão de série escolar da criança. A escolarização desempenha papel importante no desenvolvimento de formas lógicas de pensamento que dificilmente se desenvolveriam sem a cultura escolar

The theory, which the present study is, based broaches the logical thought as one of the most important process that occurs in the conscience formation. The school is seen as one of the main means of culture transmission historically built by humanity. The study analyses the way of the logical reasoning is reported in 42 children from middle and middle-high income, with ages between four, five and six years old that used to go to a pre school with a quality pedagogical project. By instrument were used syllogistics tests of individual application. The answers were noted down. Points were given to them and the justifies for the answers considered corrects were quality analyses. The results show increase of rights and teorical justifies frequency following the ages and the consequent school series progression of the child. The schooldays are seen with an important place for the logical forms of thought development that hardly would happen without the school culture

Advisors/Committee Members: Maria Regina Maluf.

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Dans ce mémoire, on s’inspire des sciences cognitives pour aborder la question de la modélisation logique du raisonnement et de l’apprentissage. Notre principale conviction est qu’il faudrait, pour traiter ce problème, prendre modèle sur la manière dont les agents naturels (c’est à dire les humains et les animaux) procèdent lorsqu’ils raisonnent ou apprennent. Considérant que le raisonnement fait appel à un grand nombre de facultés cognitives distinctes, et qu’il ne serait donc pas raisonnable d’espérer modéliser d’un seul coup l’ensemble du raisonnement humain, on se concentre ici sur un type d’inférences très simples dont on soutient qu’elles constituent le coeur du raisonnement chez tous les animaux à cerveau. On identifie un processus sous-jacent plausible pour ces inférences, d’abord au niveau mental de description, puis au niveau neuronal, et on développe une famille de modèles logiques permettant de le simuler. On s’attache ensuite à produire un ensemble de règles d’inférence caractérisant les relations d’inférence induites par ces modèles. Ces règles résultent du processus suggéré, et doivent donc être vues comme des règles qui, d’après le modèle, émergent fonctionnement des cerveaux. Enfin, on analyse les processus d’apprentissage attachés aux inférences considérées, et on montre comment le formalisme proposé permet de les modéliser. Pour conclure on évoque brièvement les possibles développements futurs du modèle, et notamment on donne quelques indications quant à la manière dont la modélisation d’un certain nombre de facultés additionnelles pourrait être envisagée.

In this dissertation, we take inspiration in cognitive sciences to address the issue of the logical modelling of reasoning and learning. Our main thrust is that to address these issues one should take inspiration in the way natural agents (i.e., humans and animals) actually proceed when they draw inferences and learn. Considering that reasoning incorporates a wide range of cognitive abilities, and that it would thus be unreasonable to hope to model the whole of human’s reasoning all at once, we focus here on a very basic kind of inferences that, we argue, can be considered as the primary core of reasoning in all brained animals. We identify a plausible underlying process for these inferences, first at the mental level of description and then at the neural level, and we develop a family of logical models that allow to simulate it. Then we tackle the issue of providing sets of rules to characterise the inference relations induced by these models. These rules are a by-product of the posited process, and should thus be seen as rules that, according to the model, result from the very functioning of brains. Finally we examine the learning processes attached to the considered inferences, and we show how to they can be modelled within our framework. To conclude we briefly discuss possible further developments of the framework, and in particular we give indications about how the modelling of some other…

Advisors/Committee Members: Rahman, Shahid (thesis director).

▼ Hybrid is a two-level logical framework that supports higher-order abstract syntax (HOAS), where a specification logic (SL) extends the class of object logics (OLs) we can reason about. We develop a new Hybrid SL and formalize its metatheory, proving weakening, contraction, exchange, and cut admissibility; results that greatly simplify reasoning about OLs in systems providing HOAS. The SL is a sequent calculus defined as an inductive type in Coq and we prove properties by structural induction over SL sequents. We also present a generalized SL and metatheory statement, allowing us to prove many cases of such theorems in a general way and understand how to identify and prove the difficult cases. We make a concrete and measurable improvement to Hybrid with the new SL formalization and provide a technique for abstracting such proofs, leading to a condensed presentation, greater understanding, and a generalization that may be instantiated to other logics.

▼  Many Automated Theorem Prover (ATP) systems for different logical forms, and translators for translating different logical forms from one to another, have been developed and are now available. Some logical forms are more expressive than others, and it is easier to express problems in those forms. On the other hand, the ATP systems for less expressive forms have been tested for many years, and are more powerful and reliable. There is a trade-off between expressivity of a logical form, and power and reliability of the available ATP systems. Different ATP systems and translators can be combined to solve a problem expressed in a logic. In this research, an experiment has been designed and carried out to compare all different possible ways of trying to solve a problem. The possibilities are either to use an ATP system for the original form, or translate the problem to a less expressive form. If the problem is translated to a less expressive form, then again the same two possibilities are available, until no further translation is possible. No translator was available to translate from Conjunctive Normal Form to Description Logic, which sits between Effectively Propositional Logic and Propositional Logic in terms of expressivity. A translation procedure for translating Conjunctive Normal Form to Description Logic, and its implementation as Saffron are presented. Additionally, this research includes a survey of different logical forms. Propositional Logic, Description Logic, First Order Form, Conjunctive Normal Form, Effectively Propositional, Typed First order Form - monomorphic, Typed First order Form - polymorphic, Typed Higher order Form - monomorphic. The properties, syntax, and semantics of each logical form are briefly described. For each form, the most popular ATP systems and translators for translating to a less expressive forms are introduced. Advisors/Committee Members: Geoff Sutcliffe, Ubbo Visser, Peter Lewis.

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D’Ambrósio (1996, p.29) aponta que “os programas de Matemática consistem em coisas acabadas, mortas e absolutamente fora do contexto e com isso, torna-se casa vez mais difícil motivar alunos para uma ciência tão cristalizada.” No presente trabalho, buscamos apresentar propostas de atividades desafiadoras, em que os alunos são encorajados a pensar de maneira autônoma, a criar, a experimentar, a estabelecer as estratégias para chegar às soluções. Diferente da sala de aula onde, normalmente, se apresenta conhecimentos prontos e acabados, tornando-o apenas reprodutor de métodos e técnicas.

D'Ambrósio (1996, p.29) points out “that Mathematic programs are often outdated, obsolete and out of context. Due to this more and more students find it difficult to be motivated about this crystallized science.” In the present work, we seek to present proposals for challenging activities, where students are encouraged to think in an autonomous way. To create, to experiment, to establish strategies and to reach solutions. Different from the classroom, where one usually presents ready and finished work, making it only a copy of the method or the technique.

Advisors/Committee Members: Nascimento, Carlos Wagner Marques do, 19235828287, http://lattes.cnpq.br/5720694292404145, Miranda, Juliana Ferreira Ribeiro de, Alcântara, Marcos Aurélio de, [email protected].

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21-98715-0469

D'Ambrósio (1996, p.29) points out “that Mathematic programs are often outdated, obsolete and out of context. Due to this more and more students find it difficult to be motivated about this crystallized science.” In the present work, we seek to present proposals for challenging activities, where students are encouraged to think in an autonomous way. To create, to experiment, to establish strategies and to reach solutions. Different from the classroom, where one usually presents ready and finished work, making it only a copy of the method or the technique.

D’Ambrósio (1996, p.29) aponta que “os programas de Matemática consistem em coisas acabadas, mortas e absolutamente fora do contexto e com isso, torna-se casa vez mais difícil motivar alunos para uma ciência tão cristalizada.” No presente trabalho, buscamos apresentar propostas de atividades desafiadoras, em que os alunos são encorajados a pensar de maneira autônoma, a criar, a experimentar, a estabelecer as estratégias para chegar às soluções. Diferente da sala de aula onde, normalmente, se apresenta conhecimentos prontos e acabados, tornando-o apenas reprodutor de métodos e técnicas.

Advisors/Committee Members: [email protected], Nascimento, Carlos Wagner Marques do, Miranda, Juliana Ferreira Ribeiro de, Alcântara, Marcos Aurélio de.

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CAPES

A relação entre música e matemática perpassa pela história desde Pitágoras até os dias de hoje, entretanto tal proximidade perdeu força ao longo do tempo com diferentes acontecimentos na filosofia, educação, artes, política e etc. O presente estudo tem como objetivo o estudo correlacional entre os aspectos cognitivos do raciocínio lógico-matemático e da atividade musical instrumental em adolescentes, utilizando-se do Teste de Avaliação Psicológica das Matrizes do Raciocínio Lógico-Matemático para Adolescentes (TRMa). O TRMa foi construído especificamente nesta investigação cuja fundamentação teórica está na psicogenética piagetiana. Participaram, do primeiro estudo, 41 adolescentes de 14 a 18 anos de idade. Do segundo estudo participaram da amostra 39 adolescentes entre 14 e 18 anos de idade devidamente matriculados na escola regular e em escolas de músicas. Nesta faixa etária são requisitados dos adolescentes, na escola regular, a capacidade de resolução de problemas que mobilizam estruturas cognitivas típica do estágio das operações formais, estágio em que, cognitivamente, ocorre o ápice da capacidade de abstração e maturação do raciocínio hipotético dedutivo. Foram analisados os resultados da prova de Raciocínio Abstrato (RA) da Bateria de Provas de Raciocínio (BPR-5), do teste de Raciocínio Auditivo (RAu) e do Teste de Avaliação Psicológica das Matrizes do Raciocínio Lógico-Matemático para Adolescentes (TRMa). O TRMa foi inspirado na estrutura de avaliação da Olimpíada Brasileira de Matemática de diversas edições com os três eixos principais: a aritmética, a geometria e a análise combinatória. Foram utilizados, na primeira parte, estudo de validade e resultados de análises oriundos Teoria de Resposta ao Item (TRI) no estudo das propriedades psicométricas. Na segunda parte foi utilizado estudo de validade incremental, controlando-se a Inteligência Fluida na correlação entre o raciocínio matemático e a atividade musical instrumental. Os resultados apontaram para a viabilidade da construção de um instrumento de avaliação psicológica específico do raciocínio matemático, foi apresentado um índice de precisão pelo Coeficiente de Kuder-Richardson (fórmula 20) em 0,677 e foi apresentada adequação dos itens diante da habilidade dos adolescentes pela TRI. Na análise incremental verificou-se uma correlação positiva (0,421) e significativa (p <0,01) entre o raciocínio matemático e a atividade musical, contudo não foram apresentadas evidências seguras quanto a capacidade de previsibilidade entre os aspectos cognitivos destas atividades quando se controla a Inteligência Fluida. Estes resultados apresentam bons indícios para o aprofundamento da temática em estudos futuros aumentando-se o tamanho da amostra e incluindo-se outros instrumentos de avaliação psicológica que contemplem os mesmos fatores.

The relation between music and Math crosses history from Pitagoras to these days, however such proximity has lost its strength overtime with different events on Philosophy, education, Arts, Politics ,etc. The present…

Advisors/Committee Members: BUENO, José Mauricio Haas.

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Com maiores publicações no início do século XXI, o tema superimitação, ou overimitation, tem despertado interesse de alguns psicólogos e biólogos por se tratar de um possível mecanismo social que leva os humanos a copiarem comportamentos irrelevantes na resolução de uma situação problema a partir de um modelo. Neste contexto, a presente dissertação de mestrado objetivou uma análise do comportamento de superimitação com mecanismos das funções executivas e cognição social de crianças da educação infantil, de 4 a 5 anos de idade. Participaram 36 crianças da educação infantil selecionadas de uma EMEI em São Paulo, segundo critérios pré-estabelecidos, avaliadas com os testes: superimitação (resolução da caixa problemas), teoria da mente - ToM (teste de falsa crença), testes para funções executivas de memória operacional (Missing Test), flexibilidade mental (DCCS), e raciocínio lógico (Teste de Matrizes Coloridas Progressivas de Raven). Os resultados indicaram que a superimitação não apresentou correlação com as funções executivas examinadas. Foram verificados indícios de tratar-se de uma atividade social, por consequência da associação com a função de teoria da mente, em detrimento a uma atividade isolada de resolução de problema cognitivo

Having experienced a growth in number of studies at the beginning of the 21st century, the subject of overimitation has caught the interest of psychologists and biologists in referring to a possible social mechanism that makes humans copy irrelevant behavior during the resolution of problem situations from a model. In that context, the present dissertation has as its objective an analysis of the overimitation behavior in relation to executive function and social cognition mechanisms in preschool children, 4 to 5 years of age. Thirty-six children participated in the study. They wereselected from a public municipal school (EMEI) in São Paulo, Brazil, according to preestablished criteria and evaluated using the problem box resolution task (overimitation), false belief test (ToM), and tests for executive functions of working memory (Missing Test), mental flexibility (DCCS), and logical thinking (Ravens Progressive Matrices Test). The results indicate that overimitation presented no significant correlation to executive functions. The findings provide evidence that overimitation is a result of social activity, since it showed possible association to theory of mind, instead of signs of an isolated activity cognitive problem solving

Advisors/Committee Members: Elaine Cristina Zachi, Patricia Izar Mauro, Rita Margarida Toler Russo, Rosaní Aparecida Antunes Teixeira.

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Este trabalho revisa e problematiza a usual associação entre a matemática escolar e o desenvolvimento do raciocínio lógico, introduzindo a argumentação e a prova em matemática como um elo de tal associação. Assim, desenvolve-se uma investigação tanto teórica quanto de campo, visando compreender por que, embora central na matemática acadêmica, a prova é tão pouco explorada na matemática escolar. A investigação teórica reviu diversas referências, em geral oriundas da Educação Matemática, mas também da História da Matemática, com vistas a: (1) compreender o papel de destaque da prova no contexto atual da Matemática, a partir do entrelaçamento entre a noção de demonstração e o próprio desenvolvimento dessa ciência; (2) discutir aspectos conceituais e epistemológicos da prova matemática que podem contribuir para um trabalho significativo com o tema na educação básica; e (3) compreender as razões da ausência de um trabalho significativo com a prova matemática na educação básica do Brasil nas últimas décadas. A investigação de campo complementa a investigação teórica apresentando um estudo de caso sobre conhecimentos, crenças e concepções de um pequeno grupo de professoras de matemática da educação básica a respeito do raciocínio lógico e da prova matemática. Essas investigações permitiram concluir que são diversas as causas do suposto abandono do trabalho com a prova matemática na educação básica, entre as quais destacamos a falta de uma discussão aprofundada sobre o tema na formação inicial de professores e a presença de práticas de ensino de matemática pautadas numa concepção tecnicista

In this dissertation, we review and problematize the usual association between school mathematics and the development of logical reasoning, introducing the proof in mathematics as a link of this association. Therefore we conducted both theoretical research and field research, in order to understand why the mathematical proof, although central to academic mathematics, is so little emphasized in school mathematics. In theoretical research, we reviewed several references, usually arising from the Mathematics Education, but also from the History of Mathematics, in order to: (1) understand the important role of proof in the current context of mathematics, from the entanglement between the notion of demonstration and even the development of this science; (2) discuss conceptual and epistemological aspects of the mathematical proof that can contribute to meaningful work with this theme in basic education; and (3) understand the reasons for the absence of meaningful work with mathematical proof in basic education in Brazil in recent decades. The field research complements the theoretical research presenting a case study on knowledge, beliefs and conceptions of a small group of mathematics teachers of basic education about logical reasoning and mathematical proof. These investigations allowed us to conclude that there are various causes for the alleged abandonment of work in mathematical proof in basic education, among which we…

Advisors/Committee Members: Claudia Cueva Candido, Saddo Ag Almouloud, Nilson Jose Machado.

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Exímio professor de matemática, Lewis Carroll, pseudônimo de Charles Lutwidge Dodgson (18321898), fez da mistura da matemática com a literatura um ambiente lúdico para a aprendizagem dessa disciplina. Autor dos conhecidos Alice no país das Maravilhas e Alice através do espelho, acabou criando um universo real e complexo no qual se utiliza do que chamamos lógica do nonsense como elemento para motivar o desenvolvimento do pensamento matemática do leitor, levando-o, assim, a aprender, estabelecendo uma ligação entre o concreto (matemática) e o imaginário (seu universo). Com o objetivo de investigar e discutir as potencialidades didáticas de suas obras e de elencar alguns elementos que possam contribuir para uma educação matemática descentralizada da tradicional metodologia de seguir os modelos e decorar fórmulas, visitamos suas obras tendo por base os estudos sobre arqueologia do saber (FOUCAULT, 2007), o pensamento racional e o pensamento simbólico (VERGANI, 2003) e sobre a importância das histórias e narrativas para o desenvolvimento da cognição humana (FARIAS, 2006). Por meio de um estudo descritivo-analítico, utilizamos a construção literária, apresentamos parte de nosso estudo na forma de um romance matemático, visando conferir à matemática escolar um encanto particular, sem privar-lhe de suas propriedades básicas enquanto disciplina e conteúdo. Nosso estudo mostrou o quanto as obras de Carroll possuem uma forte vertente didática que pode se desdobrar nas mais variadas atividades de estudo e ensino para as aulas de matemática

Notable mathematics teacher, Lewis Carroll, pseudonym of Charles Lutwidge Dodgson (1832-1898), made the mixture of mathematics with literature a ludic environment for learning that discipline. Author of Alices Adventures In Wonderland and its sequel Alice Through The Looking Glass, he eventually created a real and complex universe which uses what we call the logic of the nonsense as an element to motivate the development of mathematical thinking of the reader, taking it as well, learn by establishing a link between the concrete (mathematics) and the imaginary (their universe). In order to investigate and discuss the educational potential of their works and state some elements that can contribute to a decentralized math education from the traditional method of following the models and decorate formulas, we visited his works based on the studies of archeology of knowledge (FOUCAULT, 2007), the rational thought and symbolic thinking (VERGANI, 2003) and about the importance of stories and narratives to the development of human cognition (FARIAS, 2006). Through a descriptive, analytical study, we used the literary construction and presented part of our study in form of a mathematical novel, to give the mathematical school a particular charm, without depriving it of its basics properties as discipline and content. Our study showed how the works of Carroll have a strong didactic element that can deploy in various activities of study and teaching for mathematics classes

Advisors/Committee Members: Maria da Conceição Xavier de Almeida, Iran Abreu Mendes, John Andrew Fossa, Ligia Arantes Sad.

▼ Notable mathematics teacher, Lewis Carroll, pseudonym of Charles Lutwidge Dodgson (1832-1898), made the mixture of mathematics with literature a ludic environment for learning that discipline. Author of Alice s Adventures In Wonderland and its sequel Alice Through The Looking Glass, he eventually created a real and complex universe which uses what we call the logic of the nonsense as an element to motivate the development of mathematical thinking of the reader, taking it as well, learn by establishing a link between the concrete (mathematics) and the imaginary (their universe). In order to investigate and discuss the educational potential of their works and state some elements that can contribute to a decentralized math education from the traditional method of following the models and decorate formulas, we visited his works based on the studies of archeology of knowledge (FOUCAULT, 2007), the rational thought and symbolic thinking (VERGANI, 2003) and about the importance of stories and narratives to the development of human cognition (FARIAS, 2006). Through a descriptive, analytical study, we used the literary construction and presented part of our study in form of a mathematical novel, to give the mathematical school a particular charm, without depriving it of its basics properties as discipline and content. Our study showed how the works of Carroll have a strong didactic element that can deploy in various activities of study and teaching for mathematics classes Advisors/Committee Members: Mendes, Iran Abreu (advisor), CPF:12432962249 (advisor), http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4704236U8 (advisor).

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Relatório de Estágio apresentado à Escola Superior de Educação do Instituto Politécnico de Castelo Branco para cumprimento dos requisitos necessários à obtenção do grau de Mestre em Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico.

No âmbito da Prática de Ensino Supervisionada (PES) desenvolveu-se o presente relatório de estágio, inserido no plano de estudos do Mestrado em Educação Pré-Escolar e Ensino do 1.º Ciclo do Ensino Básico. Este destina-se à obtenção do grau de Mestre em Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico, da Escola Superior de Educação do Instituto Politécnico de Castelo Branco. O presente relatório está dividido em três partes: PES em Educação Pré-Escolar, PES no 1º Ciclo do Ensino Básico e a realização de uma investigação implementada na área da Matemática sobre resolução de problemas de dedução lógica. O nosso trabalho baseou-se no programa de Matemática do Ensino Básico que refere três grandes finalidades do ensino da disciplina: a estruturação do pensamento, a análise do mundo natural e a interpretação da sociedade. A investigação foi desenvolvida com 10 alunos do 1º ano de escolaridade do 1º Ciclo de Ensino Básico da Escola Quinta da Granja em Castelo Branco, com idades compreendidas entre os 5 e os 6 anos. O objetivo deste estudo é analisar, comparativamente, a Comunicação Oral de alunos com níveis de desempenho diferentes. Para a implementação do estudo foram aplicados três problemas que tinham que ser resolvidos, de forma individual, para poderem evidenciar o raciocínio lógico matemático envolvido na sua resolução. Na elaboração deste trabalho os problemas propostos foram orientados no sentido de se resolverem através da dedução lógica inerente às premissas dadas. Esta ideia do Pensar Alto foi incentivar os alunos a falar para si mesmo, de forma audível, no intuito de expressar as suas ideias de como resolver os problemas. Em termos metodológicos, optámos por uma abordagem de tipo investigação-ação, com recurso a várias técnicas e instrumentos para recolha de dados. Para avaliar os resultados obtidos foi utilizada a Escala Holística Focada adaptada e assim ser possível classificá-los com o objetivo de avaliar a sua evolução. A análise de dados demonstrou uma melhoria na capacidade de Comunicação Oral Matemática em alguns alunos, nomeadamente no grupo A. Esses alunos têm mais autoconfiança e autoestima que os do grupo B. O grupo A também se carateriza por ter alunos com melhor preparação para este novo ciclo de estudo, realizando os trabalhos para casa e por vezes até trabalhos autónomos. Isto demonstra responsabilidade na própria aprendizagem. Uma das maiores caraterísticas que os diferencia do grupo B é que emocionalmente são mais estáveis. Concluímos, ainda, que lançar o desafio de os alunos terem que explicar oralmente o raciocínio feito para dar uma resposta não é tarefa fácil, pois a comunicação oral é mais complexa e requer uma maior destreza ao nível do próprio raciocínio.

Under the Supervised Teaching Practice (PES) developed this training…

Advisors/Committee Members: Afonso, Paulo José Martins.

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Cette thèse vise à définir des techniques pour mieux exploiter les connaissances du domaine dans l’objectif de rendre compte de la réalité de systèmes qualifiés de complexes et critiques. La modélisation est une étape indispensable pour effectuer des vérifications et exprimer des propriétés qu’un système doit satisfaire. La modélisation est une représentation simplificatrice, mais réductionniste de la réalité d’un système. Or, un système complexe ne peut se réduire à un modèle. Un modèle doit s’intégrer dans sa théorie observationnelle pour rendre compte des anomalies qu’il peut y contenir. Notre étude montre clairement que le contexte est la première problématique à traiter car principale source de conflits dans le processus de conception d’un système. L’approche retenue dans cette thèse est celle d’intégrer des connaissances du domaine en associant le système à concevoir à des formalismes déclaratifs qualifiés de descriptifs appelés ontologies. Notre attention est portée au formalisme Event-B dont l’approche correct-par-construction appelée raffinement est le principal mécanisme dans ce formalisme qui permet de faire des preuves sur des représentations abstraites de systèmes pour exprimer/vérifier des propriétés de sûreté et d’invariance. Le premier problème traité concerne la représentation et la modélisation des connaissances du contexte en V&V de modèles. Suite à l’étude des sources de conflits, nous avons établi de nouvelles règles pour une extraction de connaissances liées au contexte par raffinement pour la V&V. Une étude des formalismes de représentation et d’interprétation logiques du contexte a permis de définir un nouveau mécanisme pour mieux structurer les modèles Event-B. Une deuxième étude concerne l’apport des connaissances du domaine pour la V&V. Nous définissons une logique pour le formalisme Event-B avec contraintes du domaine fondées sur les logiques de description, établissons des règles à exploiter pour l’intégration de ces connaissances à des fins de V&V. L’évaluation des propositions faites portent sur des études de cas très complexes telles que les systèmes de vote dont des patrons de conception sont aussi développés dans cette thèse. Nous soulevons des problématiques fondamentales sur la complémentarité que peut avoir l’intégration par raffinement des connaissances du domaine à des modèles en exploitant les raisonnements ontologiques, proposons de définir de nouvelles structures pour une extraction partiellement automatisée

This thesis aims at defining techniques to better exploit the knowledge provided from the domain in order to account for the reality of systems described as complex and critical. Modeling is an essential step in performing verifications and expressing properties that a system must satisfy according to the needs and requirements established in the specifications. Modeling is a representation that simplifies the reality of a system. However, a complex system can not be reduced to a model. A model that represents a system must always fit into its observational theory to…

Advisors/Committee Members: Méry, Dominique (thesis director).

▼ The impetus for this study evolved out of my scholarship in the areas of rhetoric and business management. In spite of the overall good feeling that in the twenty-first century women are working and procuring jobs in management, the 2005 Catalyst Census of Women Corporate Officers and Top Earners of the Fortune 500 revealed that women held only 16.5 percent of the Fortune 500 corporate officer positions. Of additional concern to me was that this gender inequity in top leadership positions was not just confined to the Fortune 500 corporate arena. In The Classic Touch: Lessons in Leadership from Homer to Hemingway, John K. Clemens and Douglas F. Mayer assert the source for "timeless and time-tested advice" on leadership is the classics. I agreed with these authors. I returned to Aristotle to uncover the real barrier as to why more women are not holding positions in top management. Aristotle valued logos as the most critical artistic appeal to effect persuasion. This study began by establishing that there is a rhetorical system of logical reasoning historically embedded with the male sphere of top management. Then, this study designed and assembled a case study on Condoleezza Rice, the 66th Secretary of State and number one on Forbes 2004list of"The World's Most Powerful Women." My contention was that Condoleezza Rice had been socialized, educated, and mentored in the art of logical reasoning in high contrast to the emotional type of reasoning stereotypically associated with the female persona. I took Condoleezza Rice out of the margins and placed her front and center as a strong female leader in the predominantly male sphere of top management. I demonstrated that Condoleezza Rice has reasoning capabilities that are viewed by the polis as having the same power and authority as that found in men. The end goal of this dissertation was to remap rhetorical history by locating the cultivation of the art oflogical reasoning into the female praxis for empowerment and advancement as leaders in the workplace. Advisors/Committee Members: Burns, Hugh (Committee Chair), Marsh, Kay (committee member), Hughes, Paula (committee member).

▼ The primary objective of an automated visual surveillance system is to observe and understand human behavior and report unusual or potentially dangerous activities/events in a timely manner. Automatically understanding human behavior from visual input, however, is a challenging task. The research presented in this thesis focuses on designing a reasoning framework that can combine, in a principled manner, high level contextual information with low level image processing primitives to interpret visual information. The primary motivation for this work has been to design a reasoning framework that draws heavily upon human like reasoning and reasons explicitly about visual as well as non-visual information to solve classification problems. Humans are adept at performing inference under uncertainty by combining evidence from multiple, noisy and often contradictory sources. This thesis describes a logical reasoning approach in which logical rules encode high level knowledge about the world and logical facts serve as input to the system from real world observations. The reasoning framework supports encoding of multiple rules for the same proposition, representing multiple lines of reasoning and also supports encoding of rules that infer explicit negation and thereby potentially contradictory information. Uncertainties are associated with both the logical rules that guide reasoning as well as with the input facts. This framework has been applied to visual surveillance problems such as human activity recognition, identity maintenance, and human detection. Finally, we have also applied it to the problem of collaborative filtering to predict movie ratings by explicitly reasoning about users preferences. Advisors/Committee Members: Davis, Larry S (advisor).

▼ This thesis proposes a model-theoretic approach to address two foundational issues: the semantic conception of indicative conditionals, and that of causation. For many years both issues have remained a tangled series of open problems that attracted great efforts and heated debates among logicians and analytic philosophers, and it is no surprise that these open problems are very fundamental challenges for Artificial Intelligence. The solution proposed in this thesis includes two formal theories introduced in separate chapters. The first theory presents a logic that has a type of conditional for which the exact truth condition is defined, and we argue that using such conditionals enables the appropriate representation of indicative conditionals, or, more assertively, they are indicative conditionals. The second theory is an extension of the first one, where another type of conditional is defined to capture a unified notion of both general causation and specific actual causation.Both theories provide the semantic constructions that realise the desirable properties for representing and reasoning with implicational and causal knowledge without the traditional dilemma of sacrificing some other desirable properties for a formal logic. The Tarskian notion of entailment is preserved in the model theory. Some of these properties are expressed as theorems with proofs provided; other properties are shown by using the semantics to model a series of scenarios in the benchmark examples where we analyse how the difficulties for traditional approaches are resolved in this approach.The logical tools and fundamental claims resulting from this pair of semantic theories may bring a fresh viewpoint or even a general solution to the open problems mentioned above, and contribute to laying a better foundation for the formalisation of commonsense reasoning. Advisors/Committee Members: Wobcke, Wayne, Computer Science & Engineering, Faculty of Engineering, UNSW, Foo, Norman, Computer Science & Engineering, Faculty of Engineering, UNSW.

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Un nombre croissant d'ontologies de grandes tailles ont été développées et mises à disposition dans des référentiels tels que le NCBO Bioportal. L'accès aux connaissances les plus pertinentes contenues dans les grandes ontologies a été identifié comme un défi important. À cette fin, nous proposons dans cette thèse trois notions différentes : modules d’ontologie minimale (sous-ontologies conservant toutes les implications sur un vocabulaire donné), meilleurs extraits ontologiques (certains petits nombres d’axiomes qui capturent le mieux les connaissances sur le vocabulaire permettant un degré de perte sémantique) et un module de projection (sous-ontologies d'une ontologie cible qui impliquent la subsomption, les requêtes d'instance et les requêtes conjonctives issues d'une ontologie de référence). Pour calculer le module minimal et le meilleur extrait, nous introduisons la notion de justification de subsomption en tant qu'extension de la justification (ensemble minimal d'axiomes nécessaires pour conserver une conséquence logique) pour capturer la connaissance de subsomption entre un terme et tous les autres termes du vocabulaire. De même, nous introduisons la notion de justifications de projection qui impliquent une conséquence pour trois requêtes différentes afin de calculer le module de projection. Enfin, nous évaluons nos approches en appliquant une implémentation prototype sur de grandes ontologies.

An increasing number of ontologies of large sizes have been developed and made available in repositories such as the NCBO Bioportal. Ensuring access to the most relevant knowledge contained in large ontologies has been identified as an important challenge. To this end, in this thesis, we propose three different notions: minimal ontology modules (sub-ontologies that preserve all entailments over a given vocabulary), best ontology excerpts (certain, small number of axioms that best capture the knowledge regarding the vocabulary by allowing for a degree of semantic loss) and projection module (sub-ontologies of a target ontology that entail subsumption, instance and conjunctive queries that follow from a reference ontology). For computing minimal module and best excerpt, we introduce the notion of subsumption justification as an extension of justification (a minimal set of axioms needed to preserve a logical consequence) to capture the subsumption knowledge between a term and all other terms in the vocabulary. Similarly, we introduce the notion of projection justifications that entail consequence for three different queries in order to computing projection module. Finally, we evaluate our approaches by applying a prototype implementation on large ontologies.

Advisors/Committee Members: Dague, Philippe (thesis director).

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This work aims to propose a method of teaching the fundamentals of mathematical logic for high school students, with a view to the development of critical and analytical reasoning, enabling them to exams for selection, especially for competitions of public agencies. The inclusion of exams on logical and analytical reasoning in several public examinations at the middle and higher levels shows that the capacity to think critically has been widely recognized in Brazil. However, critical thinking in general is not taught in Brazilian schools, unfortunately, this subject is not even part of the curricular matrix of high school mathematics, for public schools. The work presents, in a language accessible and adapted to the average students level, the logical bases for the construction and critical analysis of arguments, not only the deductive ones, of the formal logic, but also the inductive ones, so present in the daily life of the people. For this, from the notion of sets studied in high school, we arrived at the formal system of logic, where we presented the alphabet, operators and equivalence relations, with their definitions and theorems. Progressively, we came to the study of arguments. For the deductive arguments, from the validity theorem, a set of techniques, including an innovative diagramming technique, was presented for the analysis of these arguments. In sequence, a chapter was devoted to the study of Aristotle's categorical propositions and their syllogisms. Finally, the arguments and fallacies of informal logic were presented through the study of analogic, enumerative and hypothetical inductions.

Este trabalho visa propor um método de ensino dos fundamentos da lógica matemática para os alunos do ensino médio, com vistas ao desenvolvimento do raciocínio crítico e analítico, capacitando-os para exames de seleção, em especial para concursos de órgãos públicos. A inclusão da prova de raciocínio lógico e analítico em diversos concursos públicos, de níveis médio e superior, mostra que a capacidade de pensar criticamente tem sido amplamente reconhecida no Brasil. Entretanto, o pensamento crítico, no geral, não é ensinado nas…

Advisors/Committee Members: Santee, Donald Mark, Santee, Donald Mark, Bergamachi, Paulo Roberto, Guimarães, Tobias Anderson.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico

In education, there has been concern in developing problem-solving behavior as one of the responsibilities of the school, whose task includes that of providing students with procedures to enable them to meet the challenges of contemporary life. The present study had problem solving as its central theme and aimed at identifying and understanding possible relationships between performance on tasks involving logical thinking and mathematics performance. To this end, subjects were 12 children aged 8 and 9 years old which attended the third year of elementary school - of a public school in the city of São Paulo. The instrument for data gathering was composed by12 questions involving problems without number, 12 problems with and without possible solution, 36 problems with some sort of logical violation (12involvingcausal violations, 12 implied temporal violations and 12 consisted of general violations). The subjects responded to the 60 problems whose correctness criterion required that the answer was in the predicted direction and was accompanied by proper justification, obtained by the inquiry provoked by the researcher. The following procedures for data analysis were performed by: 1. Individual performance on each problem; 2.Analysis of the relative difficulty of each question in each type of problem (depending on the number of hits obtained in them); 3.Analysis of the justifications; 4.Analysis of performance for classification of students (previously classified as strong or weak performers on mathematics), and 5. Performing Statistical using Minitab Statistical Software testing. It was possible to identify relationship between mathematics performance and behavior in solving logical problems

Na educação, constata-se a preocupação em desenvolver o comportamento de solucionar problemas como uma das responsabilidades da escola, à qual caberia a tarefa de prover os estudantes com procedimentos para que sejam capazes de responder aos desafios da vida contemporânea. O presente estudo teve como tema central a resolução de problemas, buscando identificar e compreender possíveis relações entre o desempenho em tarefas envolvendo raciocínio lógico e o desempenho em matemática. Para tanto, foram avaliadas 12 crianças de 8 e 9 anos de idade, frequentando o 3o ano do ensino fundamental de uma escola da rede pública de ensino na cidade de São Paulo. O instrumento para a realização deste estudo continha 12 questões que envolviam problemas sem número, 12 problemas com e sem solução possível e 36 problemas com algum tipo de violação lógica, sendo que 12 delas abordavam violações causais, 12 implicavam violações temporais e 12 eram compostas por violações gerais. Os sujeitos responderam aos 60 problemas, cujo critério de acerto implicava que a resposta estivesse na…

Advisors/Committee Members: Luna, Sergio Vasconcelos de.