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.

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.

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.

Didactic Contract from the Perspective of the Theory of Didactical Situations

The phenomena related to the mathematical teaching and learning process have a significant and direct influence on the construction of knowledge and studying these issues is of interest to the theory developed by Guy Brousseau, called Theory of Didactical Situations. When dealing with the relations established between student, knowledge and teacher, Brousseau presents those that are specific, explicitly and implicitly, of the relationship between teacher and students, as well as their influences on the teaching and learning process, defining, for this, the Didactic Contract, which consists of the set of behaviors that the teacher expects from the student and the student from the teacher. With this, the following question arose for this work: What is the relevance of the Didactic Contract when designing teaching sequences with input in the Theory of Didactical Situations? The objective was, then, to carry out an integrative review in order to systematize information present in scientific articles about the Didactic Contract in the perspective of the Theory of Didactical Situations. A search for works was carried out in the databases Scielo, Google Scholar and Portal de Periódicos CAPES/MEC through the descriptors "Didactic Contract" and "Theory of Didactical Situations", with the following exclusion criteria: works not related to the theme and book, theses, dissertations and monographs. That is, only papers of the scientific article type were chosen. A total of one hundred and seventy-seven works were obtained as an initial result, of which thirteen were repeated. After reading the abstracts and adopting the exclusion criteria, sixteen articles remained, of which nine responded to the elaborated question. With the research, it was found, in the period consulted, a limitation of publications that deal specifically with the Didactic Contract and, in spite of that, it was concluded that the Didactic Contract is a very important element in a teaching and learning situation, because it can favor or cause an obstacle in the acquisition of new knowledge by the student.

Digital technologies as a means of accessing powerful mathematical ideas. A study of adults with low schooling in Mexico

This paper derives from a study which main purpose was to investigate how a group of adults with low schooling can have access to powerful mathematical ideas when working with activities that involve the use of technology resources and that take into account the adults’ previous experience with mathematics. Specifically, adults’ previous experience with area calculation was considered. Principles of the Theory of Didactical Situations (TDS) formulated by Brousseau guided the study design, and Pick’s theorem was recreated in a dynamic digital setting, with which it is possible to calculate the area of regular and irregular polygons. In this approach, intuitive notions of area and perimeter are resorted to, seeking to promote the experience with powerful ideas such as ‘the generality of a method’, ‘realizing the existence of different methods used for one and the same end’ and ‘realizing that each method possesses advantages and limitations’. Analysis of interview protocols from three noteworthy cases (which include both adults’ work in the digital setting and their discussions with the researcher) suggests the presence of powerful underlying mathematical ideas, such as the idea of generality and the power of a method and the features of the constituent elements of a geometric figure that are involved in calculating its attributes, attributes such as area.