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The framework of the theory of didactical situations is based on "experimental epistemology". It allows us to question mathematics and learning situations. This text will present mathematical and experimental tools derived from this theory for thinking about a typology of situations for didactic use. In particular, it will focus on didactic variables and proof situations. Two examples will illustrate these aspects. From the fields of game theory on the one hand and number theory on the other, these examples will be discussed and revisited: the Race to 20 and the Frobenius problem., Le cadre de la théorie des situations didactiques s’est construit dans des travaux « d’épistémologie expérimentale ». Il permet d’interroger les mathématiques et les situations d’apprentissage. Ce texte présentera les moyens mathématiques et expérimentaux issus de cette théorie pour penser une typologie de situations à usage didactique. Un focus sera fait en particulier sur les variables didactiques et les situations de preuve. Deux exemples permettront d’illustrer ces aspects. Issus des domaines de la théorie des jeux d'une part et de la théorie des nombres d'autre part, ils seront discutés et revisités : la Course à 20 et le problème de Frobenius.

El objeto matemático función de primer grado es tratado en la enseñanza con el uso excesivo de representaciones algebraicas o presentando distintas representaciones sin ninguna articulación entre ellas. En esta propuesta de trabajo se usó del software de geometría dinámica DGPad-Colombia con el fin de vincular la coordinación entre tres registros de representación de la función de primer grado. El marco teórico de la propuesta está enfocado en la Teoría de las Situaciones Didácticas de Guy Brousseau y la Teoría de las representaciones semióticas de Duval. La metodología utilizada se fundamentó en las primeras etapas de la Ingeniería didáctica desde la propuesta de Artigue. The mathematical issue first degree function is based on the teaching with the excessive use of algebraic representations or presenting different representations without any articulation among them. In the present proposal, the dynamic geometry software DGPad-Colombia was used in order to link the coordination among three first degree function representation recordings. The theoretical framework of the present proposal is focused on the Guy Brousseau’s theory “Theory of Didactical Situations” and on the Duval’s theory “Semiotic Representations”. The methodology implemented was based on the Didactic engineering first stages from Artigue’s proposal.

Este trabalho detalha a elaboração de uma sequência didática sobre o Sistema Internacional de Unidades (SI). Um dos maiores problemas enfrentados no século 21, a poluição foi problematizado e, como estratégia de ensino optou-se pela dialogação, importante para criar níveis de interação significativos entre os estudantes e o professor. O desafio de articular essa temática às sete unidades de base do SI, contextualizando-as e relacionando-as entre si, motivou que fosse fundamentada na Teoria das Situações Didáticas de Guy Brousseau. As cinco fases preconizadas por essa teoria são devolução, ação, formulação, e institucionalização, alternando situações didáticas e adidáticas. Os estudantes atuaram como agentes da estruturação dos seus pensamentos, participando ativamente na construção do conhecimento. A sequência foi aplicada em uma turma do terceiro ano do ensino médio de uma escola estadual na cidade de Franca-SP. Elenca situações em que, partindo de problemas idealizados subsidiados por dados reais, os alunos escolheram e testaram estratégias de resolução relacionadas à medição de grandezas de interesse físico e a tratamento de dados. Na fase de institucionalização, a padronização das unidades básicas do SI e suas definições atuais em termos de constantes universais foram sistematizadas e a questão problematizadora foi retomada, especificamente a poluição nos oceanos causada pelo descarte inapropriado de plásticos e a poluição luminosa. Essas atividades foram organizadas no produto educacional intitulado Quantificação e o Novo Sistema Internacional de Unidades: pensando em algumas faces da poluição, que apresenta também um breve histórico do SI, as regras oficiais de nomes e símbolos. Os resultados obtidos apontam uma boa adesão dos estudantes na execução das tarefas e discussões, com ampla utilização do raciocínio lógico e resultante das estratégias de questionamento implementadas. A necessidade de boas práticas contextualizadas, angariando o interesse e participação dos estudantes, concorrem para um processo de ensino e aprendizagem mais eficaz e devem ser amplamente divulgadas. This work describes elaboration of a didactic sequence about the International System of Units (SI). In this century, pollution plays a central role and was problematized through didactical dialogue was chosen as a teaching strategy, whose main goal is to improve significant levels of interaction between students and the teacher. Link this theme to the seven basic SI units, contextualizing and relating them to each other was our challenge, motivated us to base it on Guy Brousseau's Theory of Didactic Situations. The five phases recommended by this theory are devolution, action, formulation, and institutionalization, alternating didactic and adidactic situations. Students acted as agents to their thoughts structuring, actively participating in the knowledge construction process. The sequence was applied to a third-year high school class at a public school in Franca-SP city. It catalogs situations in which, starting from idealized problems supported by real data, students have chosen and tested resolution strategies related to measurement of physical interest quantities and data treatment. In the institutionalization phase, basic units of SI standardization and its current definitions in universal constants terms were systematized and the problematic question was taken up, specifically oceans pollution caused by inappropriate plastics disposal and light pollution. These activities were organized in the educational product entitled Quantificação e o Novo Sistema Internacional de Unidades: pensando em algumas faces da poluição, which join also a brief history of SI, and the names and symbols official rules. The results obtained point to a good students approval to tasks execution and discussions, resulting from questioning strategies implemented, with wide logical reasoning use. The need for good contextualized practices, appeal to students' interest and participation, contributing to a more effective teaching and learning process and must be widely disseminated. Dissertação (Mestrado)

Acompanha produto educacional: "Geosmart: geometria do cotidiano através do suíte GeoGebra calculadora". This paper presents a didactic proposal based on the use of Olympic Problems, specifically the International Mathematical Olympiad, in a complementary way, with the Didactic Situation Theory for the teaching of plane geometry. Thus, its main objective is to investigate whether the Olympic Didactic Situations work the processes of plane geometry teaching with the help of GeoGebra software in a classroom context with international mathematics issues. The investigation was aligned to the Didactic Engineering research methodology, with the use of GeoGebra software to aid in teacher training and student Olympic preparation. The Experimentation phase took place in a state school in Ceará and was organized in four meetings, with a study team formed by eight student Olympiad coaches. First, the description and application of the three Olympic Teaching Situations were performed, by the web conference virtual platform, remotely, besides the instant messaging application, which helped in the resolutions, by the students, and in the elaborations, by the researcher, with the GeoGebra software, which meets the new strategies of the teaching and learning processes of the geometry, algebra, calculus and statistics contents, allowing the interested parties the possibility of exploring, conjecturing, investigating such contents in the construction of the mathematical knowledge of the Olympic Situation. Data were collected remotely, viewing the students' performance in preparation, recorded by writing, photos and audios, based on the four stages proposed in the Theory of Didactic Situations: Action, Formulation, Validation and Institutionalization, validating internally by confronting the phases of Didactic Engineering, which are Preliminary or Prior Analysis, Design and a priori Analysis, Experimentation, a posteriori Analysis and (internal) Validation. We analyzed some dissertations from the repository of the Professional Masters in Mathematics in National Network with the topics incenter and circumcenter applied in Olympic competitions. In this situation, it was identified that there is little research that uses Olympic problems using GeoGebra. Finally, there is the Educational Product constituted by the data collected through this research, which includes five teaching situations that were elaborated, explore the concept of geometry and were mediated by a GeoGebra Calculator Suite application and a website with videos and application suggestions for the high school teacher. Este trabalho apresenta uma proposta didática baseada na utilização de Problemas Olímpicos, em específico, da Olimpíada Internacional de Matemática, de maneira complementar, com a Teoria das Situação Didáticas para o ensino de geometria plana. Dessa forma, tem como objetivo principal investigar se as Situações Didáticas Olímpicas trabalham os processos do ensino de geometria plana com o auxílio do software GeoGebra em um contexto da sala de aula com questões internacionais de matemática. A investigação foi alinhada à metodologia de pesquisa da Engenharia Didática, com o uso do software GeoGebra para auxiliar na formação de professores e preparação olímpica de estudantes. A fase de Experimentação ocorreu em uma escola estadual do Ceará e foi organizada em quatro encontros, com uma equipe de estudo formada por oito estudantes treinadores de olimpíadas. Primeiro, foram realizadas a descrição e aplicação das três Situações Didáticas Olímpicas, pela plataforma virtual de web conferência, de forma remota, além do aplicativo de mensagens instantâneas, que auxiliou nas resoluções, pelos alunos, e nas elaborações, pelo pesquisador, junto ao software GeoGebra, que vem ao encontro das novas estratégias dos processos de ensino e de aprendizagem dos conteúdos de geometria, álgebra, cálculo e estatística, permitindo aos interessados a possibilidade de explorar, conjecturar, investigar tais conteúdos na construção do conhecimento matemático da Situação Olímpica. Os dados foram coletados de forma remota, visualizando o desempenho dos alunos em preparação, registrados pela escrita, fotos e áudios, com base nas quatro etapas propostas na Teoria das Situações Didáticas: Ação, Formulação, Validação e Institucionalização, validando internamente pelo confronto das fases da Engenharia Didática, que são Análise Preliminar ou Prévia, Concepção e Análise a priori, Experimentação, Análise a posteriori e Validação (interna). Foram analisadas algumas dissertações do repositório do Mestrado Profissional em Matemática em Rede Nacional com os tópicos incentro e circuncentro aplicados em competições olímpicas. Nessa situação, foi identificado que há poucas pesquisas que utilizam problemas olímpicos com uso do GeoGebra. Por fim, tem-se o Produto Educacional constituído pelos dados colhidos por meio desta pesquisa, o qual inclui cinco situações didáticas que foram elaboradas, exploram o conceito de geometria e foram mediadas por um aplicativo Suíte GeoGebra Calculadora e um site com vídeos e sugestões de aplicação para o professor do ensino médio.

International audience; In this course, I adopt a curricular point of view on the research carried out with the framework of the theory of didactic situations (TSD) since its origin. To do that, I identify three dynamics of curriculum criticism, production of tools to analyze them and the proposals to make them evolve during the development of the framework, in connection with the evolution of research issues. For each of these dynamics, I explain the curricular dimension of the issues addressed and the results obtained. I place this research as a micro-didactic and a macro-didactic research. This study allows me to draw up in the last part of the chapter some perspectives of curricular research with TSD.; Dans ce cours, j’adopte un point de vue curriculaire sur les recherches menées avec le cadre de la théorie des situations didactique (TSD) depuis son origine, à partir de l’identification de trois dynamiques de critique de curriculum, production d’outils pour les analyser et propositions pour les faire évoluer au cours du développement du cadre, en lien avec l’évolution des problématiques de recherche. Pour chacune de ces dynamiques, j’explicite la dimension curriculaire des questions traitées et des résultats obtenus. Je situe ces recherches comme relevant de la micro et de la macro didactique. Cette étude me permet de dresser dans la dernière partie du cours des perspectives de recherche curriculaires avec la TSD.

In this article we study the teacher's language interactions with his students during two implementations in first year of primary school (1 st grade, 6-7 years old pupils) of the same situation of action (in the sense of the Theory of Didactic Situations). More specifically we question the impact of the teacher's language on the interactions between pupils and the milieu, in connection with the possible apprehensions of a figure. This research is thus connected with works in geometry about construction and observation of teaching situations designed to develop the cognitive apprehension of figures., Dans cet article, nous étudions les interactions langagières de l’enseignant avec ses élèves au cours de deux mises en oeuvre en CP (première année de primaire, 6-7 ans) d’une même situation d’action (au sens de la Théorie des Situations Didactiques). Plus précisément nous interrogeons l’impact du langage de l’enseignant sur les relations entre les élèves et le milieu, en lien avec les différentes appréhensions d’une figure. Cette recherche entre ainsi en résonance avec les travaux en géométrie portant sur la construction et l'observation dans des classes de situations d'enseignement visant à faire évoluer le rapport aux figures.

Há várias décadas que a Neurociência vem se destacando no meio científico através de resultados auspiciosos para a Educação. Porém, levar esses resultados para a sala de aula está sendo a grande dificuldade a ser superada. Pensando nisso, vários pesquisadores propuseram modelos de pesquisa multi ou interdisciplinar na tentativa de resolver esse problema e, não apenas isto, alguns sugerem caminhos de mão dupla, isto é, que educadores possam também levar os resultados de pesquisa relativos ao processo ensino-aprendizagem à Neurociência. Com base nestes modelos, surgiram várias propostas de criação de uma nova área das ciências, cuja denominação apresenta vários rótulos, dentre os quais se destacam “Neuroeducação” e “Ciência da Mente, Cérebro e Educação”. Encontramos nesses modelos quase um consenso quanto às áreas das ciências que deverão compor este novo campo científico, a saber: Educação, Psicologia e Neurociência. Escolhemos usar a denominação Neuroeducação para nos referirmos a esse campo emergente. Nesta pesquisa, propomos uma aproximação entre a Neuroeducação e a Teoria das Situações Didáticas (TSD) de Guy Brousseau, através de um Modelo de Pesquisa Neuroeducacional Multidisciplinar. Este Modelo possibilitou a realização de discussões a respeito da viabilidade de inserção de resultados da Abordagem Neopiagetiana de Bidell e Fischer (2017) e da Neurociência Socioafetiva de Immordino-Yang na TSD. Acreditamos que uma das causas da dificuldade de aprendizagem em matemática escolar está no fato de o professor não considerar a diversidade encontrada em sala de aula, que nesta pesquisa está representada por fatores contextual (contexto de aprendizagem em sala de aula), socioeconômico (status socioeconômico dos alunos) e socioemocional (ansiedade matemática). As discussões nos levaram a refletir sobre estratégias de ensino que contemplem a mudança do contexto de aprendizagem em sala de aula (através de apoio contextual) e a criação de significado para o conhecimento matemático ( através de trabalhos extraclasses que proporcionem aos alunos oportunidade de aplicar os conhecimentos matemáticos adquiridos na escola em atividades cotidianas, tais como o esporte, as profissões, etc.). Essas estratégias podem ajudar na elevação do nível de desempenho matemático dos alunos, na adaptação de alunos menos favorecidos socioeconomicamente à matemática escolar e na identificação, remediação ou reversão da ansiedade matemática, além de atuarem como agentes motivacionais na aprendizagem matemática. A viabilidade de inserção dessas estratégias de ensino na TSD e as suas possíveis contribuições na aplicação dessa teoria são discutidas ao longo do texto e apresentadas ao final como uma Proposta Neuroeducacional de aplicação da TSD. Nesse sentido, nossa Proposta Neuroeducacional sugere complementações na atuação do professor em cada momento didático da TSD para que possamos refletir sobre a possibilidade de amenizarmos ou revertermos os casos de dificuldade de adaptação à matemática escolar e de ansiedade matemática. In some decades the Neuroscience has been getting more space in the scientific context through auspicious results for the Education field. However, bringing these results to the classroom has also been a difficult step to overcome. Then, worried about that, many researchers proposed multi or interdisciplinary research in a try to solve this issue, and also some of these researchers suggest a both-handed path, which means that educational researchers could either bring their results related to the teaching-learning process to the Neuroscience field. Based on these models a lot of new proposals for the creation of a new science area came out, whereupon the denomination present so many labels, and among these “Neuroeducation” and “Mind science, Brain and Education” stand out. It was launched in these models a rough consensus about the science areas which must be part of this new scientific field, videlicet: Education, Psychology, and Neuroscience. And, here it will be using the term Neuroeducation to refer to this emerging field. In this thesis is proposed an approach within the Neuroeducation and the Theory of Didactical Situations (TDS), by Guy Brousseau, through a Multidisciplinary Neuroeducational Research Model. This Model enabled discussions about the viability of the insertion of results into the Neo-Piagetian Approach by Bidell and Fischer (2017), and the Socio-Affective Neuroscience by Immordino-Yang in TDS. It presupposed that one cause of the learning difficulty in scholar mathematics is the fact that the teacher does not consider the diversity usually found in the classrooms, which in this work is represented by factors: contextual (learning context in the classroom), socioeconomic (socioeconomic status of the students) and socioemotional (math’s anxiety). These discussions point out the possibility of reflecting about teaching strategies which contemplate the learning context change in the classroom (through contextual support) and creation of meaning for the mathematics knowledge (through extra class works which afford the students an opportunity to apply their mathematics knowledge acquired at school into their daily activities, namely: sports, professions, etc.). These strategies could support the growing status of the students’ mathematics accomplishment, also in the adaptation of less socioeconomically favored students to scholar mathematics and in the identification, remediation, or reversion of the math’s anxiety, further, acting as motivational agents in mathematical learning. This insertion viability of those teaching strategies into the TDS and its possible contributions in this theory application are discussed along with this work and presented at the end as a Neuroeducational Proposal of application of the TDS. In this sense, the Neuroeducational Proposal made out in this work, suggests complements in the teachers’ acting in classroom in each didactic moment in the TDS, enabling the reflection over the possibility to ease or revert difficult cases of mathematics scholar adaptation and the case of mathematics anxiety. Universidade Federal do Pará

El postulado del aprendizaje por adaptación a un medio antagonista asumido por la teoría de situaciones didácticas en matemáticas se corresponde con el papel central que esta teoría atribuye a las situaciones adidácticas (momentos en los que tiene lugar la producción autónoma de conocimientos por parte de los estudiantes). Desde el punto de vista de las teorías socioculturales del aprendizaje se cuestiona la pertinencia de los planteamientos constructivistas cuando se trata del aprendizaje de conocimientos científicos. En este trabajo se justifica la importancia de un modelo didáctico dialógicocolaborativo para las situaciones de primer encuentro con los objetos de conocimiento matemáticos en el que el profesor y los estudiantes trabajan juntos en la resolución de las situaciones-problemas. La justificación de este modelo didáctico está basada en los supuestos epistemológicos, ontológicos, semióticos e instruccionales del enfoque ontosemiótico del conocimiento y la instrucción matemáticos., The theory of didactic situations in mathematics postulates that learning is produced by adaptation to an antagonistic milieu, given the central role that this theory assigns to the adidactical situations (moments where the students’ autonomous production of knowledge takes place). However, the relevance of constructivist approaches is questioned by sociocultural theories, when referring to learning scientific contents. In this paper, we justify the importance of a dialogical-cooperative didactic model –in which teacher and students work together in the resolution of problems-situations– during the students’ first encounter with mathematical objects. This justification is based on the epistemological, ontological, semiotic and instructional assumptions of the onto-semiotic approach to mathematical knowledge and instruction.

International audience; This paper presents an analysis of Grade 5 pupils' work with similarity, based on an adaptation of Brousseau's classical puzzle. My main interest is in investigating the pupils' language repertoire and mathematical strategies for expressing enlargement in a situation involving similar shapes. A crucial point is to identify elements of the language use that contribute to seeing enlargement to preserve similarity as a multiplicative situation.

In this study, we aimed at introducing to prospective primary school teachers how to present a pattern of numbers through designing a mobile game according to the Theory of Didactical Situations (TDS). Additionally, the games developed by teacher candidates were examined. In our study, the case study method was used, which is one of the qualitative research methods. The participants were 17 prospective primary school teachers (9-females, 8-males). The data were collected through video cameras, audio recorders, and files that include game designs of prospective teachers. In data analysis, the actions concerning mobile gaming experiences of prospective teachers were conducted according to the TDS stages. The games designed by prospective teachers were examined in terms of basic gamification criteria. Findings of the study; it was demonstrated that mobile games designed according to TDS support the prospective teachers in many aspects, such as developing a milieu for using gamification in teaching math, internalizing processes of reflecting gamification criteria to the game, and positively influencing the affective characteristics. Moreover, concerning gamification criteria, it was determined that some prospective teachers are successful at gamification of mathematical knowledge; however, most of them have difficulties in game designing, particularly in giving feedbacks about games.

El postulado del aprendizaje por adaptación a un medio antagonista asumido por la teoría de situaciones didácticas en matemáticas se corresponde con el papel central que esta teoría atribuye a las situaciones adidácticas (momentos en los que tiene lugar la producción autónoma de conocimientos por parte de los estudiantes). Desde el punto de vista de las teorías socioculturales del aprendizaje se cuestiona la pertinencia de los planteamientos constructivistas cuando se trata del aprendizaje de conocimientos científicos. En este trabajo se justifica la importancia de un modelo didáctico dialógico-colaborativo para las situaciones de primer encuentro con los objetos de conocimiento matemáticos en el que el profesor y los estudiantes trabajan juntos en la resolución de las situaciones-problemas. La justificación de este modelo didáctico está basada en los supuestos epistemológicos, ontológicos, semióticos e instruccionales del enfoque ontosemiótico del conocimiento y la instrucción matemáticos. The theory of didactic situations in mathematics postulates that learning is produced by adaptation to an antagonistic milieu, given the central role that this theory assigns to the adidactical situations (moments where the students' autonomous production of knowledge takes place). However, the relevance of constructivist approaches is questioned by sociocultural theories, when referring to learning scientific contents. In this paper, we justify the importance of a dialogical-cooperative didactic model -in which teacher and students work together in the resolution of problems-situations- during the students' first encounter with mathematical objects. This justification is based on the epistemological, ontological, semiotic and instructional assumptions of the onto-semiotic approach to mathematical knowledge and instruction. Investigación realizada como parte del proyecto de investigación EDU2016-74848-P (FEDER, AEI), con apoyo del Grupo de Investigación FQM-126 (Junta de Andalucía, España).

En este artículo presentamos resultados de un estudio sobre la enseñanza del cálculo estimativo para la multiplicación y la división. Los análisis preliminares permitieron inventariar y analizar la variedad de problemas en diferentes fuentes y a partir de variaciones de los mismos diseñamos una secuencia de 14 clases para 5to año de la escuela primaria. El estudio se centró en los conocimientos puestos en juego por los alumnos, en las intervenciones didácticas y en las interacciones en las clases. En este trabajo se documentan y analizan en particular las dificultades a las que se enfrentaron los alumnos y los procedimientos usados, tanto correctos como incorrectos, cuando resuelven problemas que implican anticipar si un cociente será mayor, menor o igual al cociente de otro cálculo dado. Entre los marcos teóricos de referencia para el diseño y el análisis de las clases se destacan la Teoría de las Situaciones Didácticas (Brousseau, 2007) y la metodología de la Ingeniería Didáctica (Artigue, 1998). La intención de este estudio es aportar conocimientos sobre la enseñanza del cálculo estimativo al sistema educativo y al desarrollo de futuras investigaciones. In this article, we present the results of a study regarding the teaching of estimative calculation for multiplication and division. The preliminary anal- ysis enabled the enumeration and analysis of the variety of the problems in different sources. Using their variations as a starting point, we designed a sequence of 14 lessons for the 5th grade of primary school. The study focused on the knowledge of the students, the didactic interventions and the interaction during class. This work documents and analyzes the difficulties faced by the students and the procedures used - correct or incorrect- when they solve problems that involve the anticipation of whether a quotient will be higher, lower or equal to that one of another calculation given. The theoretical frame- work to design and analyze the sequence was the Theory of Didactic Situations (Brousseau, 2007) and the methodology of Didactic Engineering (Artigue, 1998) The aim of this work is not only contribute to the teaching of estimative calculation but also the developing of new didactic investigations. Fil: Broitman, Claudia. Universidad Nacional de La Plata. Facultad de Humanidades y Ciencias de la Educación. Instituto de Investigaciones en Humanidades y Ciencias Sociales (UNLP-CONICET); Argentina.

Bu çalışmada sınıf öğretmeni adaylarının Aritmetiğin Temel Teoremini Didaktik Durumlar Teorisi (DDT) çerçevesinde tasarlanan bir mobil oyun aracılığıyla keşfetme süreçleri incelenmiştir. Araştırmada nitel araştırma yöntemlerinden durum çalışması kullanılmıştır. Araştırmanın katılımcılarını 2018-2019 öğretim yılında üniversitede öğrenim gören 19 (8 kız ve 11 erkek) sınıf öğretmeni adayı oluşturmaktadır. Veriler video kamera ve ses kayıt cihazı ile kayıt altına alınmıştır. Verilerin analizi betimsel analiz yöntemiyle yapılmıştır. Bu bağlamda analizler DDT’nin aşamalarının özelliklerine göre incelenmiştir. Çalışmada elde edilen sonuçlara göre, sınıf öğretmeni adayları bir mobil oyun aracılığı ile Aritmetiğin Temel Teoremine tasarlanan ortamın parametreleri doğrultusunda ulaşabilmişlerdir. DDT’nin aşamalarının bu bağlamda etkili bir yol sunduğu sonucuna ulaşılmıştır., In this study, the exploration processes of pre-service elementary teachers of the Fundamental Theorem of Arithmetic through a mobile game designed within the framework of the Theory of Didactical Situations (TDS) were examined. The case study of qualitative research methods was used in the research. Research participants included 19 (8 female and 11 male) pre-service elementary teachers who studied at the university in the 2018-2019 academic years. Research data was recorded with a video camera and a voice recorder. The analysis of the data was done by descriptive analysis method. In this context, the analyses were examined according to the characteristics of the stages of TDS. According to the results of the study, pre-service elementary teachers were able to comprehend the Fundamental Theorem of Arithmetic through a mobile game in accordance with the parameters of the intended environment. Therefore, this suggests that the stages of TDS present an effective path in this context.

This article addresses how the concept of finding the area of a quadrilateral has been traditionally taught to student teachers majoring in Middle School Mathematics Education at the Catholic University of Santísima Concepción (UCSC) in Chile. It is important to note that Education Majors already possess a notion as to how to teach the concept of finding the area of a quadrilateral which we refer to here as their didactic cultural heritage. This study observes didactic cultural heritage as a product of the Chilean Educational School System and proposes a baseline to monitor and compare results among these students at UCSC and compare student views with fundamental elements of Didactic Theoretical Situations (DTS) according to Guy Brousseau (1986). Current teaching strategies are based on axiomatic structures and reproductive problem resolution and consider teaching as based on a traditional organizational method. However, students have also proposed more practical teaching methods applicable to real-life scenarios which involve increased teacher-student interaction and additional ways of representing diverse geometric objects, reflexive type exercises and active “devolution” alluding to a more conceptual mastery of area. The results conclude that teaching methods tend to be repeated and Education Majors, in this case, focusing on Middle School Math, will ultimately teach a concept according to how they were taught that same concept in the classroom when they were students in Middle School. En este artículo se presenta un estudio en el cual interesa observar la herencia cultural didáctica respecto del concepto área de un cuadrilátero que posee el estudiantado que ingresa a estudiar la carrera de Pedagogía Media en Matemática en la Universidad Católica de la Santísima Concepción (UCSC). Se desea dar cuenta de lo que el alumnado declara cuando se le solicita planificar una unidad didáctica para enseñar el concepto de área de un cuadrilátero. Con esta información se observa la herencia cultural didáctica otorgada por el proceso de enseñanza en el sistema educativo escolar chileno y propone un punto referencial para monitorear y comparar las modificaciones que se producen al efectuar el proceso de formación en la UCSC. Para desarrollar la investigación se contrastaron las propuestas del estudiantado, con elementos fundamentados en la Teoría de las Situaciones Didácticas (TSD) planteada por Guy Brousseau (1986). La forma de enseñanza más presente implica la presentación de las estructuras axiomáticas para luego resolver problemas de tipo reproductivo y piensan la enseñanza como una forma tradicional de organización. Sin embargo, hay estudiantes que presentan una forma de enseñanza más cercana a aplicaciones del mundo real, conlleva procesos de más diálogo entre docentes y alumnado, formas de representación más diversas de objetos geométricos, ejercicios de tipo reflexivo y presencia de devolución. Además, este alumnado muestra mayor dominio conceptual del concepto de área. Los resultados denotan que repite sus formas de enseñanza de acuerdo con cómo se les enseñó en el aula escolar.

This chapter presents the French didactic tradition. It first describes the emergence and development of this tradition according to four key features (role of mathematics and mathematicians, role of theories, role of design of teaching and learning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this tradition on algebra and on line symmetry-reflection. It then questions the influence of this tradition through the contributions of four researchers from Germany, Italy, Mexico and Tunisia, before ending with a short epilogue.