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Research on the epistemic, logical, and discursive complexity of proof learning has generated a wealth of literature over the past two decades. The results contribute to a more precise understanding of the difficulties encountered by students and those encountered by teachers. They support the design of situations, especially validation situations in the sense of the Theory of Didactical Situations (TDS--Brousseau, 1998), in which proof is a problem-solving tool. However, there remains the difficulty of grasping proof as an object, recognizing its mathematical specificities and institutionalizing it as such. This is the topic of this text. This text complements those given at the Séminaire national de didactique des mathématiques in 2017 and at CORFEM in 2019. The common theme of these three presentations was the learning and the teaching of proof before the introduction of mathematical proof as the canonical form of proof in mathematics. After an introduction recalling the institutional and scientific context, the first part (sections 2 to 4) is devoted to a review of the state of research, taking into account the reports of outstanding work from different approaches. In the second part (section 5), proposals are made to provide a basis for future research. The conclusion considers the issues raised by the need for situation-specific engineering to encourage and support the genesis and recognition of proof norms in the mathematics classroom, prior to the explicit teaching of the introduction of mathematical proof. [ABSTRACT FROM AUTHOR]

This article reports qualitative research, which had as subjects a group of students from a higher education course in Computer Science, with the proposal of solving an issue related to the nqueens problem, a generalization of the original problem, which consisted of having 8 queens on a chessboard, considering different positions, so that the pieces do not capture each other. The specific didactic sequence consisted of proposing a generalization whose application provides the number of diagonals to be considered for solving the problem on any n-by-n board, with n greater than 3. Based on the assumptions of Didactic Engineering, and having as main theoretical supports the Theory of Didactic Situations (TSD) and the work of Zazkis and Liljedahal on close and distant generalizations, the students developed an autonomous investigative trajectory, based on collaborations, to present acceptable solutions to the proposed problem. The results allow us to infer that the experience around solving mathematical problems is relevant as a learning resource in higher education Computer Science courses, considering a scenario of intensive use of digital technologies. [ABSTRACT FROM AUTHOR]

This article presents the results of a research, based on a didactic experiment, which had as subjects a group of students from the ninth grade of elementary school. The study had as its theme the generalization of standards and foresaw the realization of sessions to solve problems that had as part of the strategy the use of technologies, including digital ones, with an emphasis on the GeoGebra software. Thus, the investigation assumed a qualitative character, with an outline supported by the concepts of didactic engineering. In theoretical terms, the investigation found support in the theory of didactic situations and in authors related to the central theme, such as Dreyfus, Zazkis and Mason. The interactions between the pairs formed to solve the problems showed that the planned didactic strategy allowed the subjects to reflect on the proposals addressed, providing mathematically valid solutions for the activities, using a set of available technologies, digital and non-digital, in convergence. [ABSTRACT FROM AUTHOR]

This article discusses the didactic potential of a teaching intervention supported by games, manipulative materials and collective discussions, to contribute to the construction of the concept of Decimal Numbering System, by elementary school students. The study considered the National Pact for Literacy at the Right Age and the Mathematical Literacy of the Bahia Pact and had the theoretical support of the Theory of Didactic Situations. The study included an experimental group (EG) and a control group (CG). Both underwent a pre and a post test, and a teaching intervention between them. Given the results obtained, we can infer the existence of a positive effect on the teaching intervention proposed for the EG. [ABSTRACT FROM AUTHOR]

As teacher educators involved in the initial education of prospective Early Childhood Education teachers, we observe evident signs of the monumentalistic paradigm (visiting some pre-stablished works in Mathematics as well as in Didactics of Mathematics) when the education of teachers is structured following the traditional scheme lecture-practice. In this paper, we aim at clearly identifying the challenge of a functional education of prospective teachers, formulating it as a research problem within the ATD, and exploring the potential of the lesson study device as tool to develop prospective teachers' praxeological equipment as responses to live and authentic professional questions. We will describe de design of such device, for the case of the initial education of prospective Early Childhood Education teachers around the teaching of numbers and numbering. [ABSTRACT FROM AUTHOR]

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic. [ABSTRACT FROM AUTHOR]

Our research is concerned with teacher’s knowledge, and especially with teacher’s processes of learning, in the classroom, from observing and interacting with students’ work. In the first part of the paper, we outline the theoretical framework of our study and distinguish it from some other perspectives. We argue for the importance of distinguishing a kind of teacher’s knowledge, which we call didactic knowledge. In this paper, we concentrate on a subcategory of this knowledge, namely observational didactic knowledge, which grows from teacher’s observation and reflection upon students’ mathematical activity in the classroom. In modeling the processes of evolution of this particular knowledge in teachers, we are inspired, among others, by some general aspects of the theory of didactic situations. In the second part of the paper, the model is applied in two case studies of teachers conducting ordinary lessons. In conclusion, we will discuss what seems to be taken into account by teachers as they observe students’ activity, and how in-service teacher training can play a role in modifying their knowledge about students’ ways of dealing with mathematical problems. [ABSTRACT FROM AUTHOR]

In this paper, we use the theory of didactic situations to characterize a mathematics teaching practice, currently used in secondary schools in France, which we have called interactive synthesis discussion. We have studied this practice in ordinary classes, i.e. classes where the researcher intervenes neither in the preparation nor in the management of the lessons. We have looked at the didactic situations the teacher chooses, and how he manages his teaching project, the students’ work in the classroom and at home, and classroom interactions. We present two case studies of experienced teachers, one in grade 8, and the other in grade 10. [ABSTRACT FROM AUTHOR]

In this commentary, I will critically elaborate on the potential impact of the coordinated papers of this volume on further development of research in mathematics education. The papers, which share common theoretical frameworks, will be categorized into three different classes: ‘demolishers of illusions’, ‘economizers of thought’ and ‘energizers of practice’. I will analyze the role played by psychology and related sciences as a possible enrichment of the frameworks, especially where technologies are concerned. Finally, I will discuss the possible conflict between the need to consider the phenomena elicited in this kind of studies and the sophistication required by the theoretical constructs, which makes the results of these studies very difficult to communicate to the international community. [ABSTRACT FROM AUTHOR]

This volume gathers contributions that share the same double concern: to focus on teaching situations in classrooms, especially the work of the teacher, and to be strongly anchored in original theoretical frameworks allowing to take the classroom situation as unit of analysis. The contributions are not a representative sample of all research sharing this focus worldwide. The theoretical frameworks are grounded mainly (but not solely) in the theory of didactic situations (Brousseau, 1997) and the anthropological theory of didactics (Chevallard, 1992, 1999). There are 11 articles altogether, 9 of which present research works within the chosen theme and focus. The other two are commentary papers offering a reflection on studies of classroom situations from the point of view of other theoretical viewpoints. [ABSTRACT FROM AUTHOR]

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of each student with the procedures generated by other students during stage 1. The two kinds of processes – adaptation to a milieu and social interactions – play a critical role in the change of “rationality” required for the move from arithmetic to algebra. Both the design of the initial mathematical problem given to the students and the organization of the interactions leading to the debate under study in this article are based on this hypothesis. The research presented in this article is set in a broader work of didactic engineering that aims at studying didactic conditions for making a connection between arithmetic practices and algebraic practices. [ABSTRACT FROM AUTHOR]