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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.

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.

Introducing the slope of a curve as the limit of the slope of secant lines is a well-known challenge in mathematics education. As an alternative, three other approaches can be recognized, based on linear approximation, based on multiplicities, or based on transition points. In this study we investigated which of these approaches fits students most by analyzing students’ inventions during a lesson scenario revolving around a design problem. The problem is set in a context that is meaningful to students and invites them to invent methods to construct a tangent line to a curve: an implementation of the guided reinvention principle from Realistic Mathematics Education (RME). The teaching scenario is based on the phased lesson structure of the Theory of Didactical Situations (TDS). The scenario was tested with 44 groups of three students in six grade 9 or 10 classrooms. We classified the strategies used by students and, using the emergent models-principle from RME, investigated to which of the four approaches the student strategies connect best. The results show that the groups produced a variety of strategies in each classroom and these strategies contributed to a meaningful institutionalization of the notion of slope of a curve.

Este artículo presenta el análisis de la gestión didáctica de dos profesoras de matemáticas en Chile, a partir de la observación directa de sus clases de geometría del teorema de Pitágoras, en el grado 7 y alumnos de 12 a 13 años. El objetivo del estudio consiste en detectar, en la gestión de la clase, los índices de las decisiones didácticas que toman las docentes sobre el tema, y verificar si sus gestiones siguen algún modelo didáctico de la acción pedagógica. Este estudio se centra en el análisis de episodios seleccionados de las clases, el cual se inscribe en una perspectiva constructivista del aprendizaje y de la enseñanza de la matemática, basado en ideas claves de la teoría de situaciones didácticas de Brousseau. La gestión de los profesores conduce a un modelo de transmisión de conocimientos que dificultan el desarrollo de las matemáticas y el protagonismo del estudiante. This paper reports the analysis of the didactical management of two mathematics teachers in Chile, from direct observation of their geometry classes about the Pythagorean Theorem, in grade 7 and students 12 to 13 years old. The aim of the study is to detect in classroom management, indicators of teachers´didactical decisions about the topic and see if their efforts follow a didactic model of pedagogical action. This study focuses on the analysis of selected episodes of the lessons, which is part of a constructivist approach to learning and teaching of mathematics, based on key ideas of the theory of didactic situations of Brousseau. The management of both teachers leads to a transmission model of knowledge that hampers the development of mathematics and student protagonism.

Resumen 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. Abstract 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.

When parents and educators think about numeracy and how to help children’s use of number, they often try to help them to memorise the numbers in order and to ask them to count how many objects are present. This is certainly useful in order to attain the level of numeracy which is important at school, but it is not sufficient. Parents and educators generally pay little attention to the way in which the objects are pointed to in order to give the right number. This chapter will focus on this aspect of numeracy that has been named “enumeration” by researchers in mathematics education (Brousseau in L’enseignement de l’enumeration. International Congress on Mathematical Education, 1984. http://guy-brousseau.com/2297/l%E2%80%99enseignement-de-l%E2%80%99enumeration-1984/; Briand in Rech Didact Math 19:41–76, 1999. https://halshs.archives-ouvertes.fr/halshs-00494924).

International audience; This chapter analyses specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts and following influences on, and interactions with the international communities of research. This complex genesis is traced in four sections: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education. Beyond historical and cultural specificities, the chapter aims to evidence the potential of interactions between teachers and researchers, as well as interactions between researchers in mathematics and mathematics education for improving our understanding of learning and teaching issues in mathematics.

International audience; In what is often called the "French didactical culture", design has always played an essential role in research. This is attested by the introduction and institutionalization of a specific concept, that of didactical engineering, already in the early eighties and by the way didactical engineering has accompanied the development of didactical research, both in its fundamental and applied dimensions. In this chapter, I present this vision of design and its characteristics as a research methodology, coming back to its historical origin in close connection with the development of the theory of didactical situations, tracing its evolution along the last three decades, and illustrating this methodology by some particular examples. I also consider current developments within this design culture, especially those linked to the integration of a design dimension into the anthropological theory of didactics and also to the idea of didactical engineering of second generation introduced for addressing more efficiently the development dimension of didactical engineering.

International audience; The chapter briefly introduces the Theory of Didactical Situations (TDS) by referring to the data from Chap. 2. TDS provides a systemic framework for investigating teaching and learning processes in mathematics, and for supporting didactical design. The theory is structured around the notions of a-didactical and didactical situations and includes a corpus of concepts relevant for teaching and learning in mathematics classrooms.

Le dépôt correspond d'une part à ma note de synthèse d'HDR intitulée "Le couple (contrat didactique, milieu) et les conditions de la rencontre avec le savoir en mathématiques : de l'analyse de séquences ordinaires au développement de situations pour les classes ordinaires" et d'autre part au mémoire complémentaire demandé l'université de Nantes pour les HDR intitulé " Empirisme et rationalité à l'école élémentaire, vers la preuve au cycle 3". Ce mémoire figure en annexe de ce dépôt.La note de synthèse présente mes travaux sur la notion de contrat didactique et la façon dont on peut utiliser les notions de contrat didactique et de milieu pour l'analyse de situations ordinaires. Ce travail s'inscrit dans le cadre de la théorie des situations didactiques de Brousseau. La note de synthèse ouvre sur des perspectives d'ingénieries didactiques pour les classes ordinaires. Le mémoire complémentaire rend compte d'une recherche originale qui correspond à un travail d'ingénierie didactique au cycle 3 (élèves de 9 à 11 ans) à propos de l'impossible en mathématiques. Ce travail est effectué en appui sur le cadre de l'apprentissage par problématisation (Fabre et Orange), en articulation avec celui de la théorie des situations didactique. Cette recherche constitue un travail pionner de l'usage du cadre de l'apprentissage par problématisation en didactique des mathématiques.