Didactical situations to treat Lorentz and Galileo transformations in theoretical physics

Notions of Electromagnetism and Special Theory of Relativity (STR) require important mathematical knowledge applied to theoretical physics. Recognizing pedagogical difficulties in the teaching of theoretical physics, the Theory of Didactical Situations (TDS), which consists of a set of practices that aim to contribute to the improvement of mathematics teaching. In this context, the present work is motivated to present a set of practices based on TDS with a focus on teaching Electromagnetism and STR, where problems that require an understanding of the transformations of Galileo and Lorentz. Specifically, the didactic situation is constructed by means of four problem proposals, while in the adidatic situation, the student is invited to understand the roles of these transformations in the study of these problems. Ultimately, the relevance of the educator in the institutionalization situation is reinforced, a moment when it must be clarified how all mathematical relations are strongly related to physical principles.

Prospective Teachers’ Perspective on the Role of Instructor in Fostering Their Mathematical Disposition

Continuous treatment is needed in developing mathematical dispositions. Even at the higher education level, some prospective teachers still need proper treatment in developing their mathematical dispositions. This study tries to reveal instructors' role in fostering their students' mathematical disposition to better didactical situations. This study is qualitative research with a content analysis approach. The results obtained in this study indicate that the instructor's role is still needed in the classroom regardless of the students' mathematical disposition. Instructors ' expected learning role becomes diverse based on the mathematical disposition grouping between good, moderate, and less mathematical dispositions. For prospective teachers who have an excellent mathematical disposition, the instructor's role is to confirm a theorem, confirm answers to questions, and the benefits of contextual topics. For prospective teachers who have a satisfactory mathematical disposition, the instructor's role is to motivate, direct the learning, and keep the classroom conductivity. Whereas for prospective teachers deficient in mathematical disposition, the instructor's role is needed as a direct learning source, structured learning director, and an enjoyable learning rhythm.

Concept Construction on the Area of Oblique Triangles: A Lesson Study

This paper aimed to explain how students construct the concept of the area of oblique triangles. Through Lesson Study using the Theory of Didactical Situations, the researchers designed a research lesson with the raison d'être of improving the teaching of trigonometry. The research lesson was implemented to junior high school student-participants in a national high school in the Philippines, and was attended by a pool of observers. A priori and a posteriori analyses were conducted to validate the teaching and learning situations. Observations, notes, students’ outputs, and post-lesson discussions and reflections were used in the analysis of didactical situations. After the conduct of the Lesson Study, the following themes emerged: (1) explicit mathematics didactical intentions are essential in analyzing students' constructed mathematical knowledge and learning, and (2) balancing adidactical situation and teacher facilitation allows independent learning for students.

Formação para a pesquisa científica: trajetórias híbridas, ativas e colaborativas em um Programa de Pós-Graduação em Educação MatemáticaEducation for scientific research: hybrid, active and collaborative trajectories in a Postgraduate Program in Mathematics Education

ResumoO presente artigo descreve uma investigação que ocorreu no âmbito de um grupo de mestrandos em Educação Matemática que cursavam a disciplina “Metodologia da Pesquisa” do Programa de Estudos Pós-Graduados em Educação Matemática da PUC-SP. A estratégia aplicada no desenvolvimento das aulas foi criada a partir de um planejamento que previa o uso de metodologias ativas, ensino híbrido e que tinha, como referência para as atividades propostas, a Teoria das Situação Didáticas. O principal objetivo era o de criar um contexto de discussões e um espaço de trabalho com interações presenciais e virtuais de modo a fornecer elementos para que os alunos iniciassem suas trajetórias como pesquisadores e começassem a escrever as respectivas dissertações, movimento classificado por eles como de grande dificuldade. O texto detalha as intervenções, em regime colaborativo, que os estudantes construíram, as discussões realizadas, os progressos alcançados, além de detalhes relativos à estrutura da disciplina que foram criados para incentivar o processo de formação pretendido.Palavras-chave: Colaboração, metodologias ativas, ensino híbrido, formação de pesquisadores, teoria das situações didáticas. AbstractThis article describes an investigation that occurred within a group of masters´ degree students in Mathematics Education that studied the discipline "Methodology of Research" of the Program of Postgraduate Studies in Mathematical Education of PUC-SP. The strategy applied in the development of classes was created from a planning that predicted the use of active learning methodologies, hybrid teaching and that had, as a reference for the proposed activities, the Theory of Didactic Situations. The main objective was to create a context of discussions and a workspace with face-to-face and virtual interactions in order to provide elements for the students to begin their trajectories as researchers as well as to begin to write the respective dissertations, a movement classified by them as very difficult. The text details the collaborative interventions the students have built, the discussions held, the progress made, as well as details of the discipline structure that were created to encourage the intended education process.Keywords: Collaboration, active learning methodologies, hybrid teaching, researchers’ education, didactical situations theory. ResumenEste artículo describe una investigación que se desarrolló en el ámbito de un grupo de estudiantes de maestría en Educación Matemática que cursaban la asignatura "Metodología de la Investigación" del Programa de Estudios de Postgrado en Educación Matemática de la PUC-SP. La estrategia aplicada en el desarrollo de las clases se creó a partir de una planificación que preveía el uso de metodologías activas, docencia híbrida y que tenía como referencia para las actividades propuestas, la Teoría de Situaciones Didácticas. El objetivo principal fue crear un contexto de discusiones y un espacio de trabajo con interacciones presenciales y virtuales con el fin de brindar elementos para que los estudiantes iniciasen sus trayectorias como investigadores y empezasen a escribir sus respectivas disertaciones, movimiento que calificaron como de gran dificultad. El texto detalla las intervenciones colaborativas que construyeron los estudiantes, las discusiones realizadas, los avances logrados, así como detalles relacionados con la estructura de la disciplina que se crearon para incentivar el proceso de formación pretendido.Palabras clave: Colaboración, metodologías activas, docencia híbrida, formación de investigadores, teoría de situaciones didácticas.

Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers

This paper seeks to establish what kind of arguments pupils (aged 12–13) use and how they make their assumptions and generalizations. Our research also explored the same phenomenon in the case of graduate mathematics teachers studying for their masters’ degrees in our faculty at that time. The main focus was on algebraic reasoning, in particular pattern exploring and expressing regularities in numbers. In this paper, we introduce the necessary concepts and notations used in the study, briefly characterize the theoretical levels of cognitive development and terms from the Theory of Didactical Situations. We set out to answer three research questions. To collect the research data, we worked with a group of 32 pupils aged 12–13 and 19 university students (all prospective mathematics teachers in the first year of their master’s). We assigned them two flexible tasks to and asked them to explain their findings/formulas. Besides that, we collected additional (supportive) data using a short questionnaire. The supporting data concerned their opinions on the tasks and the explanations. The results and limited scope of the research indicated what should be changed in preparing future mathematics teachers. These changes could positively influence the pupils’ strategies of solving not only flexible tasks but also their ability to generalize.

Unfolding the Educational and Practical Resources Inherent in Mathematics for Teaching Mathematics

The objective of this introductory chapter is to explain the common rationale behind the papers of this volume. The structure is as follows. The first section shows that learning environments are a natural way to address teachers in their main role, teaching, and that therefore this approach is promising for improving mathematics teaching in an effective way. The section ends with a teaching model based on Guy Brousseau’s theory of didactical situations.

Curriculum design through lesson study

Lesson study is increasingly prevalent as a collaborative activity in which teachers take part to explore their practice. There are many variations in how lesson study manifests itself, even in Japan, where it originated. However, in Japan, fundamental to lesson study is a focus on collaboration in researching teachers’ professional practice. In this article, we draw on experiences of our collaborative research with colleagues in Japan to provide theoretical insights into how we might conceptualize and inform future developments of lesson study as action research that informs curriculum design and implementation. The approach taken develops ideas of the theory of didactical situations, and draws on the construct of boundary objects to understand Japanese lesson study. We identify a class of artefacts, didactical devices , that may provide a useful form of boundary object that supports the collaborative action research of lesson study. Although the particular focus of the work presented here is mathematics, the lessons that we draw should have applicability across the curriculum more widely.

Opportunities for language enhancement in a learning environment designed on the basis of the theory of didactical situations

This paper is based on data from two teaching sequences in primary school that are designed using principles from the theory of didactical situations (TDS). The following research question is addressed: “What opportunities can a teaching design based on TDS give a teacher to gain insight into pupils’ language use, and to use this insight to establish shared, and mathematically acceptable, knowledge in a group of primary school pupils?” Empirical data from one teaching sequence on geometrical shapes and another teaching sequence on combinatorial problems are used to answer this question. The research shows that a sharp focus on well-defined learning goals does not limit the pupils’ possibilities in expressing their thoughts and ideas in their own language. The research also shows that despite clear learning goals, the teacher has rich opportunities to build on pupils’ language to connect everyday and scientific language for the purpose of developing a mathematically accepted discourse.