In this work, is presented the result of a research that had as main theme strategies used by the students of 8th and 9th for solving algebraic problems of the Municipal Aracaju / SE school. In another way, the objective of the research was identify and examine the strategies used by these students in solving algebraic problems. To achieve this purpose, students from schools in different geographic regions of Aracaju / SE were selected initially to respond a selection of mathematical problems taken from the textbook The Conquest of Mathematics by Geovanni and Castrucci Jr (2009). After an examination with one hundred eighty-two instruments, were conducted semi structured with twenty percent of the students. The main theoretical support was clamped in the work of George Polya (1978) entitled The Art of Problem Solving for the understanding of mathematical, algebraic, problem type and problem solving strategies. The author defends as the algebraic problem those which use them to solve algebraic content. In according of the statement of mathematical problems, Polya (1978) classifies as routine, practical and enigma, and the solution in determination and demonstration. It possible says that most students use arithmetic strategies in solving algebraic problems, because in routine problems, students didn t need to make a plan, only to solve questions. Moreover, in relation to the practical problems students needed to understand the problem and develop a solution strategy, they were compelled to understand the problems to formulate a plan, most showed doubts of statements and appealed to arithmetic strategies. / Neste trabalho, é apresentado o resultado de uma pesquisa que teve como temática principal as estratégias utilizadas pelos alunos do 8º e 9º ano para a resolução de problemas algébricos da Rede Municipal de Aracaju/SE. Dito de outra forma, o objetivo da pesquisa foi identificar e examinar as estratégias utilizadas por esses alunos na resolução dos problemas algébricos. Para alcançar esse propósito, foram selecionados alunos de escolas de diferentes regiões geográficas de Aracaju/SE para responderem inicialmente a uma seleção de problemas matemáticos retirados do livro didático A Conquista da Matemática de autoria de Geovanni Jr e Castrucci (2009). Depois de um exame dos cento e oitenta e dois instrumentos, foram realizadas entrevistas semiestruturadas com vinte por cento dos alunos. O principal suporte teórico foi pinçado da obra de George Polya (1978) intitulada A Arte de Resolver Problemas para o entendimento sobre problema matemático, problema algébrico, tipologia e estratégias de resolução. O autor defende como problema algébrico os que recorrem a conteúdos algébricos para resolvê-los. Em relação ao enunciado dos problemas matemáticos, Polya (1978) classifica como rotineiro, prático e enigma, e pela sua solução em de determinação e de demonstração. A partir dos dados coletados, é possível afirmar que a maioria dos alunos utiliza estratégias aritméticas na resolução dos problemas algébricos, pois, nos problemas rotineiros, o aluno não necessitava elaborar um plano; era só resolver. Já nos problemas práticos que os alunos precisariam compreender o problema e elaborar uma estratégia de solução, ou seja, eram impelidos a compreender os problemas para formular um plano, a maioria mostrou equívocos de interpretação dos enunciados e recorreram a estratégias aritméticas.

Le problème d'intersection d'automates consiste à vérifier si plusieurs automates finis déterministes acceptent un mot en commun. Celui-ci est connu PSPACE-complet (resp. NL-complet) lorsque le nombre d'automates n'est pas borné (resp. borné par une constante). Dans ce mémoire, nous étudions la complexité du problème d'intersection d'automates pour plusieurs types de langages et d'automates tels les langages unaires, les automates à groupe (abélien), les langages commutatifs et les langages finis. Nous considérons plus particulièrement le cas où chacun des automates possède au plus un ou deux états finaux. Ces restrictions permettent d'établir des liens avec certains problèmes algébriques et d'obtenir une classification intéressante de problèmes d'intersection d'automates à l'intérieur de la classe P. Nous terminons notre étude en considérant brièvement le cas où le nombre d'automates est fixé. / The automata non emptiness intersection problem is to determine whether several deterministic finite automata accept a word in common. It is known to be PSPACE-complete (resp. NL-complete) whenever the number of automata is not bounded (resp. bounded by a constant). In this work, we study the complexity of the automata intersection problem for several types of languages and automata such as unary languages, (abelian) group automata, commutative languages and finite languages. We raise the issue of limiting the number of final states to at most two in the automata involved. This way, we obtain relationships with some algebraic problems and an interesting classification of automata intersection problems inside the class P. Finally, we briefly consider the bounded version of the automata intersection problem.

Le problème d'intersection d'automates consiste à vérifier si plusieurs automates finis déterministes acceptent un mot en commun. Celui-ci est connu PSPACE-complet (resp. NL-complet) lorsque le nombre d'automates n'est pas borné (resp. borné par une constante). Dans ce mémoire, nous étudions la complexité du problème d'intersection d'automates pour plusieurs types de langages et d'automates tels les langages unaires, les automates à groupe (abélien), les langages commutatifs et les langages finis. Nous considérons plus particulièrement le cas où chacun des automates possède au plus un ou deux états finaux. Ces restrictions permettent d'établir des liens avec certains problèmes algébriques et d'obtenir une classification intéressante de problèmes d'intersection d'automates à l'intérieur de la classe P. Nous terminons notre étude en considérant brièvement le cas où le nombre d'automates est fixé. / The automata non emptiness intersection problem is to determine whether several deterministic finite automata accept a word in common. It is known to be PSPACE-complete (resp. NL-complete) whenever the number of automata is not bounded (resp. bounded by a constant). In this work, we study the complexity of the automata intersection problem for several types of languages and automata such as unary languages, (abelian) group automata, commutative languages and finite languages. We raise the issue of limiting the number of final states to at most two in the automata involved. This way, we obtain relationships with some algebraic problems and an interesting classification of automata intersection problems inside the class P. Finally, we briefly consider the bounded version of the automata intersection problem.

Made available in DSpace on 2018-06-27T19:13:43Z (GMT). No. of bitstreams: 3 Marcelo de Freitas Bortoli.pdf: 1855849 bytes, checksum: aae4e2652fdd6a0b6dcd531b73f396c2 (MD5) Marcelo de Freitas Bortoli.pdf.txt: 180359 bytes, checksum: 54e34afa21ecad8515685caddd47cf0a (MD5) Marcelo de Freitas Bortoli.pdf.jpg: 3655 bytes, checksum: 3be54c22b4fca70ec1c2251e57637c2f (MD5) Previous issue date: 2011-04-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This study aimed to analyze the mistakes made by students of Higher Education courses in Administration, Accounting, Agricultural Engineering, Chemistry and Information Systems in a test solution on the discipline of Pre-Calculus, and use them to plan educational strategies which improve their learning in this subject, as well as in the subsequent mathematical subjects in each course. The work was developed with 31 university students of the Instituto Federal do Paraná (IFPR), campus Palmas. The literature review was based on books, articles, dissertations and theses on students' difficulties in solving problems, on algebra and its teaching and on the algebraic thinking. The research is qualitative in nature and were employed, as instruments, questionnaires, tests and notes from classroom observations. The questionnaire provided data on students, all freshmen in IFPR through external transfer or having a bachelor's degree. Categories of errors found in this research were compared with those that Movshovitz-Hadar and colleagues employed in research with high school students; it was possible to note that the technical and computational errors, as well as from algebraic manipulation and incorrect use of algorithms, show the greatest difficulties of students in solving questions. As a product, we elaborated a didactic sequence to help students overcome their difficulties on algebraic operations, particularly in reducing similar terms. / Esta pesquisa teve como objetivo geral analisar erros cometidos por alunos de Ensino Superior, de cursos de Administração, Ciências Contábeis, Engenharia Agronômica, Química e Sistemas de Informação, na resolução de testes da disciplina de Pré-Cálculo, e utilizá-los para planejar estratégias de ensino que propiciem uma melhoria de sua aprendizagem nessa disciplina, bem como nas subsequentes disciplinas matemáticas de cada curso. O trabalho desenvolveu-se com 31 estudantes de cursos superiores do Instituto Federal do Paraná (IFPR), campus Palmas. A revisão de literatura baseou-se em livros, artigos, dissertações e teses sobre dificuldades dos estudantes na resolução de problemas, sobre a Álgebra e seu ensino e sobre o pensamento algébrico. A pesquisa é de caráter qualitativo e nela empregaram-se, como instrumentos, questionários, testes e anotações de observações de sala de aula. O questionário socioeducacional forneceu dados sobre os alunos, todos ingressantes no IFPR por meio de processo de transferência externa ou por posse de diploma de graduação. Compararam-se as categorias de erros encontrados às que Movshovitz-Hadar e colaboradores empregaram em pesquisa com alunos de high school, sendo possível notar que os erros técnicos, computacionais, de manipulação algébrica e de uso incorreto de algoritmos, evidenciam as maiores dificuldades dos estudantes na resolução das questões. Como produto, elaborou-se uma sequência didática para auxiliar os estudantes a superarem suas dificuldades em operações algébricas, em especial na redução de termos semelhantes.