La récente réforme des lycées en France de 2009 s'est accompagnée d'un changement de programmes en mathématiques. Relativement à la classe de seconde, deux sujets nous questionnent : d'une part, la nouvelle place de l'algèbre, désormais plongée dans le domaine fonctionnel, lui conférant un rôle essentiellement d'outil, et d'autre part l'introduction d'une familiarisation avec l'algorithmique. De par l'intérêt de lier ces deux sujets, ce travail de thèse propose une étude didactique de la reprise de l'algèbre élémentaire en classe de seconde, et plus particulièrement des objets gravitant autour du concept d'équation, objets dont nous cherchons à affiner le sens par le détour de l'algorithmique. Nous situant dans le cadre de la théorie anthropologique du didactique de Chevallard, nous étudions les conditions et les contraintes de cette reprise. Au travers d'une ingénierie didactique mise en place avec la collaboration de trois enseignants de lycée, nous montrons comment la reprise de concepts d'algèbre élémentaire par le biais de l'algorithmique induit pour les élèves un geste de généralisation, tout en réalisant une certaine matérialisation des objets algébriques, en les manipulant au sein d'un programme informatique. Pour les enseignants, cette ingénierie provoque un questionnement sur les praxéologies de leur enseignement de l'algèbre, suscité par des tâches non routinières de catégorisation et de modélisation des équations. Enfin, nous mettons en évidence la question de l'intégration du domaine de l'algorithmique dans la discipline des mathématiques et le besoin d'une formation des professeurs pour assurer la viabilité de cet enseignement. / The recent French schools reform (2009) was accompanied by a change in mathematics curriculum. With respect to the “classe de seconde” (9th-10th Grade in US High School), two subjects question us: first, the new positioning of algebra, now part of the functional domain, giving it a primary role of tool, and on the other hand, the introduction of algorithmic concepts.Through the value of combining these two subjects, this thesis proposes a didactic study of the resumption of elementary algebra in “classe de seconde”, and especially of the objects orbiting around equation concept, objects of which we search to refine the meaning, through the detour of algorithmics. Positioned within the anthropological theory of the didactic by Chevallard, we study the conditions and constraints of this resumption. Through a didactic engineering implementation in collaboration with three high school teachers, we show how the resumption of basic algebra concepts through algorithmics induced for students a gesture of generalization, while achieving some materialization of algebraic objects, manipulating them in a computer program. For teachers, this engineering induces a questioning of their praxeology teaching algebra, generated by non-routine tasks of equations categorization and modeling. Finally, we highlight the challenge of integrating algorithmics domain within mathematics discipline and the need for teachers to be trained, to ensure the viability of this teaching.

Made available in DSpace on 2016-06-02T20:02:57Z (GMT). No. of bitstreams: 1 6507.pdf: 2675622 bytes, checksum: 1fa2e1a89f27433883070d1a6ede575e (MD5) Previous issue date: 2014-10-17 / The difficulties in learning and teaching of algebra can be detected in the school cycle 4 (8th and 9th grades) of the Elementary School II and throughout High School, such difficulties being present in all Brazilian schools. These difficulties arise from an institutional failure, in others words, in the transition from arithmetic to algebra, in the phase of pre-algebra which occurs at cycle 3 ( 6th and 7th grades) of the Elementary School II. When this transition is unsatisfactory this compromises the subsequent studies making the students feel not motivated in learning the content of algebra. Therefore, in this research project we planned and executed 6 (six) activities based on the methodology of Problem Solving based on the phases proposed by George Polya, along with the methodology of the Bar Model from Singapore Mathematics. The activities were carried out in seventh grade classrooms of Elementary School II of Instituto Educacional Estilo , Campinas, SP. The results of this dissertation suggests to teachers of Elementary School II didactical sequences of activities that they can use and enjoy in classroom practices, so that they can improve also their teaching and learning, contributing to the development of the students. With the objective of achieving a satisfactory transition from arithmetic to algebra, the activities were developed and based on problems solving, and then analyzed critically using the Problem Solving steps. After 6 (six) activities, we applied a diagnostic evaluation in order to analyze the results and to check if the activities contributed to a meaningful learning of algebra. The dissertation presents a theoretical study about teaching and learning algebra as well as a study on the methodologies of Problem Solving in classroom practice and Bar Model from Singapore Mathematics. / As dificuldades na aprendizagem e no ensino da álgebra podem ser constatadas no ciclo 4 (8º Ano e 9º Ano) do Ensino Fundamental II e também em todo o Ensino Médio, tais dificuldades estão presentes em todas as escolas brasileiras. Essas dificuldades são decorrentes de uma falha na introdução, ou seja, na transição da aritmética para a álgebra, a pré-álgebra que ocorre no final do ciclo 3 (6º Ano e 7º Ano) do Ensino Fundamental II, já que feita de maneira não satisfatória pode comprometer as aulas seguintes fazendo com que os alunos se sintam desmotivados a aprenderem o conteúdo de álgebra. Deste modo foram planejadas e executadas 6 (seis) atividades utilizando a metodologia de Resolução de Problemas seguindo as etapas de George Polya, juntamente com a metodologia do Modelo de Barras segundo a Filosofia da Matemática de Singapura. As atividades foram aplicadas em duas turmas do sétimo ano do Ensino Fundamental II, no colégio Instituto Educacional Estilo, Campinas, SP. O trabalho desenvolvido nesta dissertação proporciona aos professores do Ensino Fundamental II e Ensino Médio uma sequência didática, que podem utilizar e aproveitar em suas aulas de forma que possam também melhorar em suas práticas de ensino e aprendizagens, de maneira a contribuir para o desenvolvimento de seus alunos. Com o objetivo de realizar uma transição satisfatória da aritmética para álgebra, as atividades foram elaboradas e baseadas na resolução de problemas, e depois analisadas criticamente por meio das etapas de resolução. Após as 6 (seis) atividades, aplicamos uma avaliação diagnóstica de forma a analisar os resultados para verificar se as atividades contribuíram com significado para uma aprendizagem da álgebra. O trabalho apresenta um estudo teórico sobre o ensino e aprendizagem da álgebra e também apresenta um estudo sobre as metodologias desenvolvidas no trabalho, Resolução de Problemas e Modelo de Barras segundo a Filosofia da Matemática de Singapura.

Research in early algebra is critical because a smooth transition from arithmetic to algebra will influence future algebra learning that is central to school mathematics. This study investigated learners’ interpretation of letters in different levels of generalised arithmetic activities. Thirty grade nine learners from one inner city school participated in this study. All learners engaged with seventeen paper and pencil tasks encompassing six different interpretations of letters and six learners were then interviewed. Analysis of the data showed that the overall performance of learners was very poor and most learners have not been successful in making the transition from arithmetic to algebra. Learner responses suggested a strong arithmetical influence and a poor understanding of algebraic letter and basic manipulative skills. Throughout the data a number of misconceptions surfaced which suggested that most learners in this sample were lacking ‘symbol sense’.

Made available in DSpace on 2016-04-27T16:57:48Z (GMT). No. of bitstreams: 1 EDM - Maria helena da silva.pdf: 815120 bytes, checksum: fef91164053cfd8045b3478b49e70ce0 (MD5) Previous issue date: 2006-10-20 / The aim of this study is to investigate the views on Algebra, in contents concerning Numbers and Operations, present in the Mathematics National Curricular Parameters (PCN) for Elementary Education. Therefore, in this documental analysis adopting qualitative approach, the content analysis proposed by Bardin (1977) was applied by means of enunciation analysis techniques. The studies by Lins and Gimenez (1997) on Arithmetic and Algebra and Lee s (2001) research on the views on Algebra were adopted as theoretical framework. A bibliographic review also brought elements of the conceptions of Algebra present in the discourse of teachers, student books and questions of the National Secondary Education Examination (ENEM). Supported by such framework, numbers and operations in the four cycles were then investigated. It became evident that, even though the document suggests the associated study of Algebra and Arithmetic, actions to make feasible such proposition are not contemplated in its guidelines and directions as a whole. The series of analysis revealed that the National Curricular Parameters (PCN) show, in their guidelines, conceptions of algebra as generalized arithmetic, as a tool, and of algebra as an activity both with the purpose of producing a symbolic language of the letters. Although it was not the main target of this investigation, results showed that, just like the Mathematics National Curricular Parameters (PCNs) suggest, it is necessary to create opportunities to reflect upon the teaching of algebra in the Elementary Education, that include the different segments involved in the teaching/learning process, such as teachers, researches, educational institutions, the community and the society, among others / Este estudo tem o objetivo de investigar as visões sobre Álgebra nos conteúdos que dizem respeito aos Números e Operações presentes nos Parâmetros Curriculares Nacionais (PCN) de Matemática do Ensino Fundamental. Nesta análise documental, com enfoque qualitativo, utilizamos a análise de conteúdo, referendada no estudo de Bardin (1977), por meio da técnica de análise da enunciação. O referencial teórico adotado foram os estudos de Lins e Gimenez (1997), sobre Aritmética e Álgebra, e a pesquisa de Lee (2001), sobre as visões da Álgebra. A revisão bibliográfica também trouxe elementos sobre concepções de álgebra presentes nos discursos de professores, livros didáticos e questões do ENEM, que contribuíram para o exame das visões. A partir desse referencial, investigamos os números e operações nos quatro ciclos. Ficou evidenciado que, apesar de o documento indicar o estudo associado de álgebra e aritmética, não estão contempladas no conjunto de suas orientações ações que possam concretizar essa indicação. As análises revelaram que os PCN trazem em suas orientações, visões da álgebra como aritmética generalizada, como ferramenta, e a álgebra como uma atividade todas com a finalidade de produzir a linguagem simbólica das letras. Embora não tenha sido foco de nossa investigação, os dados mostraram que, tal como sugerem os PCN de Matemática, faz-se necessário abrir espaços de reflexões sobre o ensino da álgebra no Ensino Fundamental, que englobe os diversos segmentos envolvidos no processo de ensino e da aprendizagem, como professores, pesquisadores, instituições afins, comunidade, sociedade, e outros

Made available in DSpace on 2016-06-02T20:02:48Z (GMT). No. of bitstreams: 1 3278.pdf: 1300698 bytes, checksum: 42d6e6a50f7a0f2127c6a2130f835199 (MD5) Previous issue date: 2010-03-13 / Difficulties in learning algebra found on high school, especially among students of the first year have motivated this research, in order to discover and understand the origins of the problem and consider proposals for possible solutions. The first target of the present research was to explore the possible causes of difficulties on the transition from arithmetic to algebra, which should be done in the second half of elementary school but occurs most notably in the eighth year / seventh grade. Activities configured as problem solving were planned and implemented to detect problems and to support the introduction to algebraic reasoning on three groups of seventh grade students at Escola Estadual Professor Euclides de Carvalho Campos , Botucatu, SP. The objectives of those activities are: 1 Search the steps involved in planning activities for teaching algebra; find the students difficulties in its learning; 2 Implement classroom activities in the seventh grade; collect and analyze the results in order to support the dissertation work and prepare proposals to assist the learning of algebra. In order to reach it, the problem solving methodology was used with proposals for contextual problems involving modeling problems with first-degree equations, linear systems, geometry and counting. In addition to explanative lessons, in which the results were synthesized, the group work and participatory learning were emphasized. As a result, the students shown their difficulties to discern the role of the unknowns in the equation s solving, the meaning of characters as variables in modeling problems and also a strong tendency of trying to solve only arithmetic s exercises, especially through the method of trial and error. The present research examined the difficulties found as by the points of view of the theoretical conceptual transition from arithmetic to algebra as by the contextual problem solving methodology, which involves the school planning and the social environment. / As dificuldades na aprendizagem de álgebra constatadas especialmente em alunos do primeiro ano do ensino médio motivaram esta pesquisa, que tem a finalidade de descobrir e entender as origens do problema e estudar propostas para possíveis soluções. O primeiro alvo do presente trabalho é explorar as possíveis causas das dificuldades na transição da aritmética para a álgebra, que deveria ser feita na segunda metade do Ensino Fundamental, porém ocorre com maior destaque no oitavo ano/sétima série. Foram planejadas e aplicadas atividades sob forma de resolução de problemas para detectar estas dificuldades e auxiliar na introdução ao raciocínio algébrico em três turmas de sétima série da Escola Estadual Professor Euclides de Carvalho Campos , Botucatu, SP. Os objetivos das atividades são: 1 Pesquisar as etapas do processo de planejamento de atividades matemáticas para o ensino de álgebra; detectar as dificuldades dos estudantes na sua aprendizagem; 2 Executar as atividades em salas de aula de sétima série; coletar os resultados e analisá-los de forma a subsidiar o trabalho de dissertação; elaborar propostas que contribuam para facilitar a aprendizagem de álgebra. Para isso foi utilizada a metodologia de resolução de problemas, com propostas de problemas contextualizados, envolvendo modelagem de problemas com equações do primeiro grau, sistemas lineares, geometria e contagem. Além de aulas expositivas nas quais se fez a síntese dos resultados obtidos, foi enfatizada a aprendizagem participativa do trabalho em grupo para a execução das atividades. Como resultado, foram detectadas dificuldades no discernimento do papel das incógnitas na resolução de equações, no significado das letras utilizadas como variáveis na modelagem de problemas e a forte tendência em tentar resolver exercícios apenas pela aritmética, especialmente pelo método da tentativa e erro. O presente trabalho analisou as dificuldades detectadas tanto do ponto de vista teórico-conceitual de transição da aritmética para a álgebra, quanto pela ótica contextual da metodologia de resolução de problemas que envolvem o planejamento escolar e o ambiente social.