Cette recherche propose d'étudier la place et le rôle des grandeurs dans la construction des différents domaines mathématiques et de leurs interrelations au collège. Elle s'inscrit dans le cadre de la théorie anthropologique du didactique développée par Yves Chevallard. Une analyse épistémologique des savoirs mathématiques relatifs aux grandeurs nous a permis d'examiner les choix institutionnels et leurs effets sur l'enseignement. Nous complétons ce travail par une étude écologique et praxéologique des programmes, et des documents institutionnels actuels pour caractériser les rapports institutionnels aux grandeurs. Une méthodologie du type clinique est mise en œuvre pour analyser le savoir enseigné concernant les grandeurs dans des classes de 6e et 5e en France. Pour cette étude, nous nous sommes inspirés des travaux de Bronner à propos du numérique pour élaborer un outil, le filtre des grandeurs, qui sert à décrire et analyser les pratiques relatives aux grandeurs au collège. Plus particulièrement, nous nous sommes intéressés aux interrelations entre les grandeurs, le fonctionnel et le numérique en prenant le cas de la proportionnalité, et au fonctionnement interne des grandeurs en étudiant l'espèce de grandeur aire.La recherche met en évidence que l'introduction des grandeurs en tant que domaine d'étude dans les programmes du collège de 2005 crée chez les professeurs des difficultés dans l'intégration des nouvelles technologies et théories pour un enseignement adéquat des grandeurs. De plus, elle montre une certaine stabilisation de la place et la fonction des grandeurs dans la construction d'autres domaines mathématiques, mais met en évidence des difficultés au niveau des interrelations entre ces domaines. / This research aims at studying the place and role of magnitudes in the development of different mathematics fields and in their interrelations at school. Our perspective is that of the Anthropological Theory of the Didactic developed by Yves Chevallard. An epistemological analysis of the mathematical knowledge related to magnitudes allowed us to examine certain institutional choices and their impact on teaching. This work is complemented by an ecological and praxeological study of the curricula, school textbooks and corporate documents to characterize the institutions' relations to magnitudes. A clinical type methodology was used to analyze the knowledge concerning magnitudes taught in grade 6 and 7 in France. This study was inspired by Bronner's work about the numeric to develop a tool - a magnitudes filter - which is used to describe and analyze the practices related to magnitudes at school. In particular, this research focused on the interrelations between the magnitudes, the functional and the numeric in the case of proportionality and on the internal functioning of magnitudes in the case of the area magnitude.As a result, this research shows that the introduction of magnitudes as a study domain in school programs of 2005 creates difficulties among professors in integrating new technologies and theories for an adequate teaching of magnitudes. Moreover, it shows a certain stabilization of the place and role of magnitudes in the development of different mathematics fields, but difficulties concerning the interrelations between these areas.

I den här studien åskådliggörs tre skilda perspektiv på hur undervisning kan planeras ochorganiseras i relation till matematikinnehåll, struktur och yttre resurser. Tidigare forskningbelyser hur lärarens arbete kring planering i stor utsträckning kan vara avgörande förelevernas lärande i matematik. Vidare belyser studien vikten av reflektioner och planering föratt medvetet uppmärksamma eleverna på de kritiska momentana inom matematiken.Skolinspektionens (2020) rapport benämner hur allt färre lärare reflekterar kring sin egenundervisning i syfte att utveckla lärandet. Resultaten pekar på ett liknande fenomen närdeltagarna talar om reflektionens roll för undervisningen. Vidare synliggör studienläromedlets roll för både planeringsarbetet, genomförandet av undervisningen samtreflektionen.

Made available in DSpace on 2016-04-27T16:58:53Z (GMT). No. of bitstreams: 1 Ana Maria Paias.pdf: 6600134 bytes, checksum: 1f7709c7d60aa2f33fdfc71732f6837e (MD5) Previous issue date: 2009-05-21 / Secretaria da Educação do Estado de São Paulo / Mathematics taught at school implies mainly in developing the mathematical thinking and the student s skills. These two items are necessary for the comprehension of different situations, including, those from the daily life and also, for support as a tool to other fields of knowledge. It is observed that since Elementary School, Mathematics is shown as an area where students have learning difficulties. Thus, the purpose of this research was to do a study and a diagnosis in respect to the power operation with 9th and 10th graders from a state school in São Paulo. For such, a study was done about the mistake and its importance in the teaching-learning process. It is a descriptive research, quanti-qualitative with the construction of a diagnosis about the students mistakes referring to the power operation, classifying and interpreting them. The theoretical foundation was based on the Anthropological Theory of the Didactic (1999) in the Register of Semiotic Representation from Duval (2003) and in the mistake studies from Cury (2007) The result of the analysis from the students answers indicated that, a great amount of the students does not know the power concept and as a result many understand the power operation as multiplication. Therefore, many facts aggravate the mistake in relation to this topic. The most important factors were the power cases that involved whole negative numbers and fractionary exponents. Categories were created for the analyzed mistakes. Zero also constitutes in a great mistake cause, manly, when it is an exponent because the student does not realize the convention in the right way. As for the exponent 1, it is perceived that the same problem appears in relation to the Mathematical convention, it is also observed that the student does not know how to justify it correctly / A matemática ensinada na escola implica sobretudo desenvolver o pensamento matemático e as habilidades do aluno. Estes dois itens são necessários para a compreensão de diferentes situações, inclusive, aquelas do cotidiano e também, para suporte como ferramenta a outros campos do conhecimento. Observa-se que desde o ensino básico, a Matemática mostra-se como uma área em que os alunos demonstram dificuldades de aprendizagem. Assim, esta pesquisa teve como objetivo realizar um estudo e um diagnóstico a respeito da operação potenciação com alunos da 8ª. Série do Ensino Fundamental e 1ª. Série do Ensino Médio de uma escola pública da rede estadual de ensino do Estado de São Paulo. Para tanto, realizamos um estudo sobre o erro e sua importância no processo de ensino e aprendizagem. Trata-se de uma pesquisa descritiva, quanti-qualitativa com a realização de um diagnóstico sobre os erros dos alunos referentes à operação potenciação, classificar e interpretá-los. A fundamentação teórica foi apoiada na Teoria Antropológica do Didático de Chevallard (1999); nos Registros de Representação Semiótica de Duval (2003) e nos estudos sobre o erro de Cury (2007). O resultado das análises das respostas dos alunos indicou que, grande parte dos alunos, não domina a concepção de potenciação, decorrendo disso muitos entendem a operação potenciação como multiplicação. Assim, vários fatos agravam o erro em relação a esse tópico. Os fatores mais relevantes foram os casos de potência que envolvem números inteiros negativos e expoentes fracionários. O zero e o um também se constituem em grande causa de erros, sobretudo quando eles são expoentes, pois o aluno não observa a convenção de modo correto

Esta pesquisa teve como objetivo investigar, com base na Teoria Antropológica do Didático (TAD), como educadores de um museu de ciência planejaram uma atividade educativa cujo tema central é uma controvérsia científica. O estudo aconteceu no Museu de Microbiologia do Instituto Butantan (MMB) em um contexto em que o educativo do museu participou de um curso de extensão intitulado Temas controversos e museus de ciências, a partir de uma parceria entre o Grupo de Estudo e Pesquisa em Educação Não Formal e Divulgação Científica/GEENF da Faculdade de Educação da USP, e o referido museu. O objetivo do curso foi estimular o debate nos educadores do museu acerca das potencialidades e dos desafios de se trabalhar com controvérsias em museus e exposições. A avaliação final do curso consistiu na produção de roteiros educativos que versassem sobre uma controvérsia científica. Dentre os temas propostos, o planejamento do roteiro As diferentes abordagens sobre a vacina contra o HPV foi selecionado como objeto de estudo dessa pesquisa, uma vez que o mesmo serviu de modelo experimental para esse tipo de ação educativa no museu. Os dados foram coletados durante os encontros de planejamento do grupo e foram constituídos pelas falas dos integrantes. Para análise nos apoiamos nos conceitos de praxeologia, de Percurso de Estudo e Pesquisa (PEP) e de níveis de co-determinação propostos por Yves Chevallard e colaboradores no contexto da Teoria Antropológica do Didático/TAD. Esta teoria auxilia a descrever como determinadas tarefas destinadas ao ensino de diferentes áreas de conhecimento são organizadas pelo didático e, desse modo, realizamos a análise praxeológica do planejamento explicitando os elementos que compõem o bloco logos e o bloco práxis. Na discussão, utilizamos o conceito de Percurso de Estudo e Pesquisa (PEP) e níveis de co-determinação em que a ação do didático é sistematizada em momentos de estudo. Foram identificados os momentos de estudo do planejamento dos educadores, possibilitando a compreensão sobre a forma com que os saberes do bloco logos praxeológico são acessados por eles. Foi possível também verificar quais foram as condições e as restrições que permitiram ou não que os educadores acessassem o bloco lógico da praxeologia. Os resultados da investigação revelaram a importância de promover ações de formação na rotina dos educadores de museus, uma vez que verificamos que nos quatro momentos de estudo vivenciados por eles durante o PEP, as situações relacionadas diretamente ao curso de extensão foram as que mais permitiram que os educadores acessassem o bloco lógico sobre a controvérsia do HPV. Além disso, ficou evidente que também houve um ganho institucional para o MMB, pois ao possibilitar que os educadores trabalhassem com novas teorias e metodologias, o museu ampliou seu escopo de atividades oferecidas, assim como em novas maneiras de abordar a relação entre ciência e sociedade para o seu público. / This research meant to investigate, based on Anthropological Theory of the Didactic (ATD), how educators in a science museum planned an educational activity whose central theme is a scientific controversy. The study occurred at the Museum of Microbiology of Butantan Institute (MMB) in a context in which the educational part of the museum participated in an extension course called Controversial Themes and the science museum, together with Group of Study and Research in Non-Formal Education and Scientific Dissemination/GEENF of the School of Education USP and the museum mentioned previously. The goal of the course was to encourage the debate among the educators of the museum concerning the potentialities and challenges of working with controversies in museums and exhibitions. The final assessment of the course consisted in the production of educational scripts about a scientific controversy. Among the proposed themes, the planning of the script \"The different approaches to the HPV vaccine\" was selected as the object of study of this research, since it served as an experimental model for this type of educational action in the museum. The data were collected during the planning meetings and they were formed by the speeches of the participants. The analysis was supported by concepts of praxeology of Research and Study Courses (RSC) and the levels of co-determination proposed by Yves Chevallard and collaborators in the context of Anthropological Theory of the Didactic (ATD). This theory helps to describe how certain tasks aimed at different areas of knowledge are organized by the didactics. This way, we do a praxeology analysis and planning explain the elements which form the block logos and the block praxis. In the discussion the concept Research and Study Courses (RSC) and the levels of co-determination are used in which the didactic action is systematized during periods of study. The periods of study and planning by the educators were identified, enabling the understanding about the way in which the knowledge of the praxeological block logos are accessed by them. It was also possible to verify which conditions and restrictions allowed or not the educators to access the logical block of praxeology. The results of the research revealed the importance of promoting training actions in the routine of museum educators, since we found that in the four moments of study experienced by them during the RSC, the situations directly related to the course of extension were the ones that most allowed educators to access the logical block on the HPV controversy. Moreover, it was evident that there was also an institutional gain for the MMB, because by enabling educators to work with new theories and methodologies, the museum expanded its scope of activities offered, as well as in new ways of addressing the relationship between science and society for your public.

Nas últimas décadas encontramos nos documentos oficiais (PCNEM e PCN+) orientações quanto à necessidade da inserção da Física Moderna e Contemporânea (FMC) nas aulas de física no Ensino Médio. Nesse mesmo período, professores e pesquisadores já sinalizavam quanto a esses aspectos e percebe-se que existem pontos em comum em suas pesquisas quanto às dificuldades de implementação dessa inserção. Nessa perspectiva, alguns conteúdos já foram inseridos a partir de projetos de inovação curricular nas escolas públicas e por meio de cursos de formação continuada, atendendo às necessidades dos professores de física, mostrando, por exemplo, que é possível levar tais conteúdos para a sala de aula. Contudo, quando procuramos analisar as pesquisas referentes ao professor nesse contexto, percebe-se uma escassez de trabalhos que se ocupam da formação e da prática docente no contexto da inovação curricular e, mais especificamente, da inserção da FMC. Dessa forma, este trabalho procura analisar a formação e a prática do professor nesse contexto, apresentando uma articulação possível entre a Teoria da Transposição Didática (TTD) e a Teoria Antropológica do Didático (TAD), ambas de Chevallard, na identificação das praxeologias adotadas pelo professor durante o processo de didatização dos saberes, na perspectiva de que ele desempenha um papel central e, ao mesmo tempo, sensível no momento de transformar o saber até que este chegue à sala de aula. Para isso, foi realizada uma pesquisa qualitativa, centrada em um estudo de caso, no qual é analisada a prática docente de um professor de física da rede pública que participou de um curso de formação continuada no contexto da inovação curricular, e que, posteriormente, trabalhou com uma sequência didática sobre a temática da dualidade onda-partícula no Ensino Médio. Assim, busca-se identificar as praxeologias transpostas do curso de formação continuada para a sala de aula, bem como o exercício da vigilância epistemológica ao longo desse processo. / In the official documents (PCNEM and PCN+) of recent decades an orientation on the need to insert Modern and Contemporary Physics (MCP) in high school physics classes was published. In the same period, teachers and researchers already signaled on these matters and it is noticed that there are common difficulties to implement this insertion, as related in their research about this subject. From this perspective, some contents on curriculum innovation projects in public schools have been suggested giving pedagogical updating courses focusing on the physics teachers needs, showing, for example, that it is possible to take such contents in a the classroom. However, when we analyzed the research on the teacher in this context, there is a perceived lack of studies dealing with training and teaching practice in the context of curriculum innovation and, more specifically, the insertion of the MCP. Thus, this paper analyzes the formation and practice of teacher in this context, indicating a possible link between the Didactic Transposition Theory (DTT) and the Anthropological Theory of the Didactic (ATD), both of Chevallard, in the identification of the praxeologies adopted by the teacher during a didactization process of knowledge, from the standpoint that it plays a central role. At the same time, it is analyzed how the teacher shows to be sensitive until the knowledge turns to reach the classroom. For this, a qualitative research was conducted, focusing on a case study, where the teaching practice of a physics professor at the public school who participated in a continual education course in the context of curriculum innovation, and subsequently worked with a teaching sequence on the topic of wave-particle duality in high school. Thus, we seek to identify the transposed praxeologies the continuing education course for the classroom, and how is the exercise of epistemological vigilance throughout this process.

La thèse étudie l'évolution sous contraintes du milieu que se donnent les élèves dans les cadres des premiers moments d'une activité d'étude et de recherche sur les équations du premier degré à une inconnue en classe de 4°. Le dispositif d'observation - analyses mathématique et didactique a priori soutenant la proposition d'activité, mise en place de groupes au sein des classes, films des interactions au sein du processus de recherche - recourt à une méthodologie de type clinique. Elle autorise une analyse micro-didactique qui, utilisant les récents développements de la théorie Anthropologique du didactique, conclut à la différenciation des milieux, des temporalités et des fonctions au sein des groupes observés et un enrichissement du concept de mémoire didactique. / The thesis studies the evolution under constraints of the milieu that pupils give themselves in the frames of the first moments of an activity of study and research on single-unknown equations of the first degree in class of 4 °.The observation device - a priori mathematical and didactical analyzes supporting the activity proposal, setting up groups within the classes, films of the interactions within the research process - resorts to a methodology of the clinical type. It allows a micro-didactic analysis which, using the most recent developments of the anthropological theory of didactics, concludes with the differentiation of the environments, the temporalities and the functions within the groups observed and an enrichment of the concept of didactic memory.

Made available in DSpace on 2016-04-27T16:57:36Z (GMT). No. of bitstreams: 1 Acylena Coelho Costa.pdf: 4495391 bytes, checksum: 036e84e7258e8b0ddf6bc886cce6222d (MD5) Previous issue date: 2015-04-24 / Universidade do Estado do Pará / This research aims to analyze how the authors of textbooks organized the activities proposed in relation to the study of Line and Planning for the teaching of Analytic Geometry in Space. Analyses of textbooks substantiate mainly on Anthropological Theory of the Didactic (TAD), with regard to praxeologias proposed by Chevallard (1999) and in didactic variables for the teaching of Analytic Geometry in Space established by Lebeau (2009). Based on the theoretical framework adopted we conducted a qualitative investigation of documentary type, based on a literature review on four textbooks Analytic Geometry devoted to higher education. The methodology used in our research was supported in manual analysis methodology developed by Chaachoua (2014a) analyzing the textbooks the following: time of writing, representation, structure, ecological analysis and praxeological analysis. For regards the praxeological analysis identified six types present in the analyzed manual tasks, namely to: determine the equation of the line in space, determining the straight condition of parallelism in space, to determine the alignment condition of points in space, present plane equation in space as a property of orthogonality, determine a plan characterized by two intersecting lines and characterize algebraically the parallelogram. Among the findings it can be inferred that the authors favor an algebraic modeling of mathematical objects, as well as the techniques adopted by them are situated in the field of Linear Algebra and Analytic Geometry / A presente pesquisa tem como objetivo analisar como os autores de livros didáticos organizaram as atividades propostas no que se refere ao estudo da Reta e do Plano para o ensino da Geometria Analítica no Espaço. As análises dos livros didáticos fundamentaram-se essencialmente na Teoria Antropológica do Didático (TAD), quanto às praxeologias, propostas por Chevallard (1999) e nas variáveis didáticas para o ensino da Geometria Analítica no Espaço estabelecidas por Lebeau (2009). Com base no referencial teórico adotado realizamos uma investigação de caráter qualitativo do tipo documental, partindo de um levantamento bibliográfico em quatro livros didáticos de Geometria Analítica destinados ao ensino superior. A metodologia adotada em nossa pesquisa foi subsidiada na metodologia de análise de manuais desenvolvida por Chaachoua (2014a) analisando nos livros didáticos os seguintes aspectos: momento da edição, representatividade, estrutura, análise ecológica e análise praxeológica. Em relação a análise praxeológica identificamos seis tipos de tarefas presentes nos manuais analisados, a saber: determinar a equação da reta no espaço, determinar a condição de paralelismo de retas no espaço, determinar a condição de alinhamento de pontos no espaço, apresentar a equação do plano no espaço como uma propriedade da ortogonalidade, determinar um plano caracterizado por duas retas secantes e caracterizar algebricamente o paralelogramo. Dentre os resultados encontrados é possível inferir que os autores privilegiam uma modelização algébrica dos objetos matemáticos, bem como as técnicas adotadas pelos mesmos encontram-se situadas no campo da Álgebra Linear e da Geometria Analítica

Nas últimas décadas encontramos nos documentos oficiais (PCNEM e PCN+) orientações quanto à necessidade da inserção da Física Moderna e Contemporânea (FMC) nas aulas de física no Ensino Médio. Nesse mesmo período, professores e pesquisadores já sinalizavam quanto a esses aspectos e percebe-se que existem pontos em comum em suas pesquisas quanto às dificuldades de implementação dessa inserção. Nessa perspectiva, alguns conteúdos já foram inseridos a partir de projetos de inovação curricular nas escolas públicas e por meio de cursos de formação continuada, atendendo às necessidades dos professores de física, mostrando, por exemplo, que é possível levar tais conteúdos para a sala de aula. Contudo, quando procuramos analisar as pesquisas referentes ao professor nesse contexto, percebe-se uma escassez de trabalhos que se ocupam da formação e da prática docente no contexto da inovação curricular e, mais especificamente, da inserção da FMC. Dessa forma, este trabalho procura analisar a formação e a prática do professor nesse contexto, apresentando uma articulação possível entre a Teoria da Transposição Didática (TTD) e a Teoria Antropológica do Didático (TAD), ambas de Chevallard, na identificação das praxeologias adotadas pelo professor durante o processo de didatização dos saberes, na perspectiva de que ele desempenha um papel central e, ao mesmo tempo, sensível no momento de transformar o saber até que este chegue à sala de aula. Para isso, foi realizada uma pesquisa qualitativa, centrada em um estudo de caso, no qual é analisada a prática docente de um professor de física da rede pública que participou de um curso de formação continuada no contexto da inovação curricular, e que, posteriormente, trabalhou com uma sequência didática sobre a temática da dualidade onda-partícula no Ensino Médio. Assim, busca-se identificar as praxeologias transpostas do curso de formação continuada para a sala de aula, bem como o exercício da vigilância epistemológica ao longo desse processo. / In the official documents (PCNEM and PCN+) of recent decades an orientation on the need to insert Modern and Contemporary Physics (MCP) in high school physics classes was published. In the same period, teachers and researchers already signaled on these matters and it is noticed that there are common difficulties to implement this insertion, as related in their research about this subject. From this perspective, some contents on curriculum innovation projects in public schools have been suggested giving pedagogical updating courses focusing on the physics teachers needs, showing, for example, that it is possible to take such contents in a the classroom. However, when we analyzed the research on the teacher in this context, there is a perceived lack of studies dealing with training and teaching practice in the context of curriculum innovation and, more specifically, the insertion of the MCP. Thus, this paper analyzes the formation and practice of teacher in this context, indicating a possible link between the Didactic Transposition Theory (DTT) and the Anthropological Theory of the Didactic (ATD), both of Chevallard, in the identification of the praxeologies adopted by the teacher during a didactization process of knowledge, from the standpoint that it plays a central role. At the same time, it is analyzed how the teacher shows to be sensitive until the knowledge turns to reach the classroom. For this, a qualitative research was conducted, focusing on a case study, where the teaching practice of a physics professor at the public school who participated in a continual education course in the context of curriculum innovation, and subsequently worked with a teaching sequence on the topic of wave-particle duality in high school. Thus, we seek to identify the transposed praxeologies the continuing education course for the classroom, and how is the exercise of epistemological vigilance throughout this process.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T14:51:38Z No. of bitstreams: 1 luiseduardoreyesperez.pdf: 2784061 bytes, checksum: f741a9cf532f95435bbd639a2f2df42f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-26T15:01:37Z (GMT) No. of bitstreams: 1 luiseduardoreyesperez.pdf: 2784061 bytes, checksum: f741a9cf532f95435bbd639a2f2df42f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-26T15:01:55Z (GMT) No. of bitstreams: 1 luiseduardoreyesperez.pdf: 2784061 bytes, checksum: f741a9cf532f95435bbd639a2f2df42f (MD5) / Made available in DSpace on 2017-05-26T15:01:55Z (GMT). No. of bitstreams: 1 luiseduardoreyesperez.pdf: 2784061 bytes, checksum: f741a9cf532f95435bbd639a2f2df42f (MD5) Previous issue date: 2015-11-26 / Este trabajo es el resultado de una investigación sobre las praxeologias matemáticas realizadas por alumnos de enseñanza media, sobre el concepto matemático de función exponencial, en la Institución Educativa Técnica de Tunia en el Departamento del Cauca, Colombia. En este documento las reflexiones fueron enfocadas hacia adolescentes colombianos de enseñanza media en Colombia y la propuesta fue investigar qué tipos de praxeologías puntuales surgen en el momento de realizar actividades acerca de función exponencial. Buscando responder esta pregunta, orientamos esta disertación mediante la metodología de Ingeniería Didáctica, con la intención de direccionar el trabajo del investigador. En ese sentido se cuenta también con los procedimientos metodológicos los cuales permitieron verificar si a partir de la intención de conocer e identificar las potencialidades de las praxeologías puntuales estas sean idóneas a fin de trasformarlas en praxeologías contextualizadas para una cierta institución de enseñanza I desarrollada por el profesor titular y en consecuencia ésta sea apropiada e interiorizada convirtiéndose en una herramienta potencialmente valida para el profesor. Por tanto, surgen preguntas en el contexto educativo, sobre la forma y construcción de lo que se enseña, en particular el concepto de función exponencial. La construcción de las praxeologías puntuales, así como su formulación, serán enfocadas a través de la teoría antropológica de lo didáctico que además de identificar, ayudará a orientar la construcción de secuencias didácticas relacionadas con la función exponencial que formarán parte del producto educacional de esta disertación y en especial a la contribución de nuevas investigaciones. / This work is the result of the research about the mathematic praxeologies made by students the middle education about the mathematic concept of exponential function, in the Technical Educational Institution from Tunia located in the department of Cauca. On this document the considerations were directed to colombians youngers on the middle education in Colombia and the proposal is to research which types of punctual praxeologies arise in the moment of effect questions about of exponential fuction. In order to answer this question, we focused this dissertation through the methodology of didactic engineering, with the intention of focussed of work de researcher. also included the methodological procedures, which allowed to assess whether, based on the intention to know and identify the potential of specific praxeologies in order to turn them into praxeologies contextualized in an educational institution I and developed by the main professor and as a consequence, that praxeology turn into an educational concept, about the form and construction itself of what we teach, in particular,the concept of exponential fuction. The construction of the puctual praxeologies as their formulation, will be focused by through the anthropological theory of didactic. That in addition to identifying, will help to guide the construction of teaching sequences relating with the exponential fuction that will take part on the educational product of this dissertation and in special the contribution of new researches.

Cette thèse s’intéresse à l’élaboration d’alternatives pour l’enseignement du calcul algébrique au collège et en particulier de la propriété de distributivité qui joue un rôle central.En appui sur des recherches antérieures en didactique de l’algèbre, nous analysons les spécificités des savoirs à enseigner et enseignés sur le calcul algébrique, au regard de difficultés protomathématiques (Chevallard 1985) prégnantes du côté des élèves. Ceci conduit à appréhender de nouvelles formes de savoirs à enseigner accompagnant les savoirs mathématiques liés aux aspects sémantiques et syntaxiques des écritures symboliques algébriques. La notion de transformation de mouvement (Drouhard 1992) et l’exploration des caractères formalisateur, unificateur et généralisateur (ou FUG, Robert 1998) amène à envisager la distributivité au regard d’un domaine d’étude plus large, à la fois numérique et algébrique. L’étude d’une transposition possible des savoirs à enseigner permet de dégager des conditions et des contraintes pour élaborer une ingénierie didactique. Les résultats d’une expérimentation en classe de 5e (élèves de 12-13 ans) à partir d’analyses a priori et a posteriori, concernent les discours dont les élèves parviennent à s’emparer, justifiant et soutenant leurs techniques de calcul, ainsi que les organisations de connaissances qui émergent. Une nouvelle étude didactique et épistémologique relative à la notion de substitution vient clore la thèse afin de déterminer en quoi elle pourrait fonder un prolongement possible aux enjeux FUG pour l’enseignement de la distributivité et poursuivre l’ingénierie didactique amorcée visant à enseigner le calcul algébrique tout au long du collège. / This thesis seeks to explore alternatives for the teaching of algabraic calculus in second grade, and more specifically of the distributive law that plays a central role.Drawing on prior researches on didactic of algebra, characteristics of the knowledge to be taught and the knowledge taught about algebraic calculus are analyzed towards protomathematics difficulties (Chevallard 1985) constantly arising in students’work. This leads to consider new forms of knowledge, along with mathematical knowledge, that would be linked to semantic and syntactic aspects of symbolic algebraic expressions.Exploring the notion of movement transformation (Drouhard 1992) and the potential of formalizing, unifying, and generalizing (or FUG, Robert 1998), brings out the distributive law in a larger study field both numerical and algebraic.The study of a possible transposition of the knowledge to be taught yields a set of conditions and constraints to design a didactic situation. The results from a first experiment in a 5th grade class (12-13 year-olds) are based on a priori and a posteriori analysis. They focus on the discourses built and used by the students, justifying and supporting their manipulations, along with the knowledge organizations arising out.The last chapter addresses a new didactic and epistemological study of the notion of substitution aiming at discussing its potential to extend the FUG point of view on the teaching of the distributive property, and further on to provide a new perspective of research to carry on with our didactic design to teach algebraic calculus all along secondary school.