Philippe R. Richard

• Ph. D.
• Professor at Université de Montréal

This article touches up the complementary presentations provided by the authors during the conference, addressing the design, evolution, and utilization of digital technologies in education. It is composed of two distinct parts. The first one offers a perspective from the field of educational sciences, while the second focuses on the didactics of m...

Il est essentiel de se pencher sur les interactions entre l’intelligence artificielle (IA) et la didactique, encore plus à notre époque où l’impact de l’IA sur la société et l’économie est aussi profond. Tout d’abord, nous remettons en question la notion dite d’intelligence et les préjugés qu’elle peut susciter lorsqu’on réfléchit à l’IA et à ses d...

This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic app...

Preprint Mars 2023. Accepted for publication in “Handbook of Digital Resources in Mathematics Education" edited by Prof. Dr. Birgit Pepin, Prof. Ghislaine Gueudet, Prof. Jeffrey Choppin This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techni...

The purpose of the chapter is to identify and present the future challenges of the theory of MWS, on which we believe future research must make progress. The guiding idea of the chapter is to deepen the study of mathematical work through different perspectives by considering various mathematical domains, other scientific fields and looking beyond t...

The central focus of this chapter is the relationship between the notions introduced by Vérillon and Rabardel (1995), “artifact” and “instrumental genesis,” and the homonym notions in the theory of MWS. Following a brief overview of the Piagetian and Activity Theory perspectives which inspired Vérillon and Rabardel, we explore how didactic research...

This chapter focuses on the particularities of mathematical work in the digital age. It opens with historical considerations that relate to the development of mathematical work to do arithmetic when symbolic or mechanical tools and algorithmic methods were used. He goes on to define the new mathematical work by introducing concepts of reference in...

This chapter contains an overview of the main points of the theory of MWS in relation to frequently asked questions. The aim is to form the basis of an online glossary of the theory, to be completed according to the contributions of researchers.

The central focus of this chapter is the relationship between the notions introduced by Vérillon and Rabardel (1995), ‘artefact’ and ‘instrumental genesis’, and the homonym notions in the theory of MWS. Following a brief overview of the Piagetian and Activity Theory perspectives which inspired Vérillon and Rabardel, we explore how didactic research...

This book highlights the contribution of artificial intelligence for mathematics education. It provides concrete ideas supported by mathematical work obtained through dynamic international collaboration, and discusses the flourishing of new mathematics in the contemporary world from a sustainable development perspective.

When considering the mathematics competences referential (Niss & Højgaard, 2019), whether from Québec Education Program, the Standards of the National Council of Teachers of Mathematics or some large international studies such as the Programme for International StudentStudent Assessment, reasoning, and mathematical proofs are always at the forefron...

This chapter contains an overview of the main points of the theory of MWS in relation to frequently asked questions. The aim is to form the basis of an online glossary of the theory, to be completed according to the contributions of researchers.

The reflective narratives of three pre-service teachers are analysed from the perspective of discourse analysis and reflection on practice, applied to the acquisition of geometric competences and the role of dynamic geometry. In the context of the master's degree for future secondary school teachers in mathematics and a pedagogical innovation proje...

In this study, we explore automated reasoning tools (ART) in geometry education and we argue that these tools are part of a wider, nascent ecosystem for computer-supported geometric reasoning. To provide some context, we set out to summarize the capabilities of ART in GeoGebra (GGb), and we discuss the first research proposals of its use in the cla...

The Symposium on Artificial Intelligence for Mathematics Education (AI4ME 2020) held at the CIEM (International Centre for Mathematical meetings, ICMM) in Castro Urdiales, in the Spanish region of Cantabria, has been a space for research and reflection to better understand the interconnected challenges of instrumental learning of mathematics and in...

In this proposal we deal with a well-known Computational Geometry construction: the diagram of Delaunay. We will show how the GeoGebra implementation of the corresponding command brings the opportunity to explore, through Delaunay eyes, some recent artwork from two Spanish artists. This visual exploration leads, from our perspective, to a conjectur...

A blocking state is a measurable state on an intelligent tutoring systems’ user interface, which mirrors a student’s cognitive state where she/he cannot temporarily make any progress toward finding a solution to a problem. In this paper, we present the development of four probabilistic models to detect a blocking state of students while they are so...

When working on intelligent tutor systems designed for mathematics education and its specificities, an interesting objective is to provide relevant help to the students by anticipating their next steps. This can only be done by knowing, beforehand, the possible ways to solve a problem. Hence the need for an automated theorem prover that provide pro...

Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof...

In this proposal we will briefly introduce a well-known Computational Geometry construction: the diagram of Delaunay, summarily describing some of its basic properties. The GeoGebra implementation of the corresponding command will bring the opportunity to explore, through Delaunay eyes, some recent artwork from two Spanish artists, yielding, once m...

This paper reflects on the design of tasks that could take profit from the new features for Automated Reasoning, recently available in the dynamic geometry program GeoGebra. We report on some ongoing experiment, involving initial mathematics teacher students, aiming to develop and evaluate the characteristics of tasks that, with the concurrence of...

Cet atelier vise à montrer comment les représentations sémiotiques et les outils technologiques influencent le point de vue traditionnel de la notion de preuve pour la réalisation du travail mathématique. À partir de considérations didactiques, épistémologiques et cognitives, nous distinguons la logique du constructeur (constitution des mathématiqu...

Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof...

The software GeoGebra, as any other dynamic geometry system, offers some notorious ways to allow the student to perform his/her work as a mathematician, improving under certain conditions the exploration, the construction and the implementation of Euclidean plane geometry theorems. On the other hand, some specific features allowing the automatic an...

This chapter concludes the collection of chapters, each of which expands on the papers presented during the Topic Study Group on the teaching and learning of secondary school geometry at ICME-13. In articulating a vision for where the field of secondary school geometry could go in the near future, the chapter revisits issues of methodologies for da...

This chapter introduces the book by providing an orientation to the field of research and practice in the teaching and learning of secondary school geometry. The introduction then outlines the chapters in the book, each of which expands on the papers presented during the Topic Study Group on the teaching and learning of secondary school geometry at...

This book presents current perspectives on theoretical and empirical issues related to the teaching and learning of geometry at secondary schools. It contains chapters contributing to three main areas. A first set of chapters examines mathematical, epistemological, and curricular perspectives. A second set of chapters presents studies on geometry i...

Our contribution shows the anticipated effect of what we call connected problems in developing the competencies of students and their acquisition of mathematical knowledge. Whilst our theoretical approach focuses on didactic and cognitive interactions, we give special attention to a model to reason about learners’ conceptions, and the ideas of math...

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the reso...

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the reso...

This article proposes a state of the art of tutorial systems for high school planar geometry proof learning. The chosen approach is part of exploring the research problem that motivates the development, by our research team, of a tutorial system named QED-Tutrix, that we will present in another paper. In the following article, a synthesis and a com...

GeoGebra Automated Reasoning Tools (GGB-ART) are a collection of GeoGebra tools and commands ready to automatically derive, discover and/or prove geometric statements in a dynamic geometric construction. The aim of this workshop is to present, through examples, the use of GGB-ART and to argue about its potential impact in the classroom.

GeoGebra Automated Reasoning Tools (GGB-ART) are a collection of GeoGebra tools and commands ready to automatically derive, discover and/or prove geometric statements in a dynamic geometric construction. The aim of this workshop is to present, through examples, the use of GGBART and to argue about its potential impact in the classroom. Keywords: Au...

Our paper aims at showing how the instrumented learning of the properties in Euclidean geometry can be introduced in continuity with the mathematical competences developed at the elementary school. The principle founder of our research is based on the relation of subordination between the constraints of a property, posed using dynamic geometry soft...

Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and learning of proof, with or without technology. After posing our hypotheses on the relationship of proof with mathematical work, the pedagogical intent of historical elements of geometry and...

Cette contribution montre l’éclairage apporté par le modèle des espaces de travail géométrique (ETG) dans la conception et la validation du système tutoriel intelligent geogebraTUTOR (GGBT). Conçu pour être employé par des élèves de l’école secondaire, ce système se destine au développement de la pensée géométrique dans un contexte de résolution de...

Il arrive souvent que les didacticiels pour l’enseignement des mathématiques se constituent en référence à des modèles informatiques, laissant implicites les modèles d’apprentissage qui découlent de l’usage de l’outil et qui auraient pu contribuer à sa conception. Dans le jeu des allers et retours entre les concepteurs et les usagers, si les nouvel...

Este número especial de la Revista Latinoamericana de Investigación en Matemática Educativa (RELIME) es el resultado de distintos artículos elaborados para el tercer simposio Espacio de Trabajo Matemático (ETM), cuyo objeto de estudio es el desarrollo y los usos posibles de la noción de ETM en la didáctica de las matemáticas. El trabajo matemático...

Our paper presents a project that involves two research questions: does the choice of a related problem by the tutorial system allow the problem solving process which is blocked for the student to be restarted? What information about learning do related problems returned by the system provide us? We answer the first question according to the didact...

Abstract. This paper explores the role of mathematics teachers in dynamic geometry (DG) resource analysis, using a questionnaire as a means for professional development, and the integration of DG in their classes. The questionnaire aims to allow the evaluation of mathematical, technical, educational and ergonomic quality of resources available on t...

In order to support the development of a modelling competency at the beginning of secondary school, we briefly analyse two problem-situations through the use of the GeoGebra3D software. The first problem results in the conics of Apollonius, while the second raises a few considerations on beehives. Our proposal aims to demonstrate the effect of inst...

This paper explores the role of mathematics teachers in dynamic geometry (DG) resource analysis, using a questionnaire as a means for professional development, and the integration of DG in their classes. The questionnaire aims to allow the evaluation of mathematical, technical, educational and ergonomic quality of resources available on the i2geo p...

To help understand the diversity that exists in Canadian mathematics education, this contribution focuses on the reality of English Canada, complementing the separate studies we have completed on Québec and NewBrunswick. As a country that is officially “bilingual” (English/French), Canada has unique constitutional considerations within education: e...

To increase understanding of the diversity that exists in Canadian mathematics education, this contribution focuses on Québec, complementing the separate papers that we have also written on New Brunswick and English Canada. The Canadian educational context involves both linguistic and constitutional factors: each region must provide primary and sec...

To help understand the diversity of Canadian mathematics education, our contribution relates to the New Brunswick reality, having completed separate papers on the Quebec and English Canada contexts. As a country that is officially “bilingual” (English/French), Canada has unique constitutional considerations within education: each region must provid...

This paper aims at showing the didactic and theoretical-based perspectives in the experimental development of the geogebraTUTOR system (GGBT) in interaction with the students. As a research and technological realization developed in a convergent way between mathematical education and computer science, GGBT is an intelligent tutorial system, which s...

Dynamic figures in a mathematical workspace for the learning of geometrical properties. Our paper aims at showing how dynamic figures are useful in the learning of the use of geometrical properties at a high school level, in continuity with the practices inherited during the primary school education. After considering the general contextual rooting...

La reedición de textos matemáticos clásicos, o su reciente digitalización para un gran público, abre al mundo de la educación un universo antes reservado a los historiadores u otros especialistas en epistemología. Con todo, muchos textos del siglo XVIII ya se presentaban como obras en las cuales los autores pretendían explicar los conceptos matemát...

The new edition of classical mathematical texts, or its recent digitization for the general public, represents an approach to the universe that has been traditionally reserved to historians and other experts on epistemology. Many texts from the 18th century were already presented like works aimed at the explanation of certain mathematical concepts,...

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary sc...

Tutoring System for Improvement of Argumentative Competencies in High School Mathematics This article tries to show how secondary school pupils can improve their argument skills with the help of tutorial systems intended for learning geometry. After having established the conceptual framework at the intersection of mathematics teaching and computer...

This paper presents the development of an intelligent tutoring system called TURING (French acronym of «TUtoRiel INtelligent en Géométrie») in a multidisciplinary project, that joint recent research in didactic of mathematics with the possibilities of computer-based learning environments. From the educational point of view, the system is helpful fo...

This article aims at legitimating the use of the graphic registers in the written expression of mathematical reasoning. More than the defence of this legitimacy, the article shows also the risks of its use: the problems of communication which it causes as well as the difficulties, for the teachers, to evaluate the reasoning of the pupils when they...

Résumé Cet article vise à montrer l’utilité d’une approche fonctionnelle-structurelle pour l’étude des manuels scolaires en mathématiques. L’approche s’inspire de trois sources : la théorie des fonctions du langage de Duval, le modèle des fonctions du langage dans la communication de Jakobson et le modèle des structures sémiotiques développé par Ri...

Resumo: O uso limitado de tecnologias nas salas de aula pode ser conseqüência da sua falta nas escolas, da despreocupação dos professores sobre os recursos disponíveis e da necessidade de adaptá-las às situações das aulas; ou da insuficiência teórica e prática para que os professores as integrem ao seu ensino. Neste artigo propomos formas e descrev...

Résumé. Le but de notre exposé est de montrer une nouvelle approche pour la recherche sur l'apprentissage de la géométrie à l'école secondaire à partir d'un environnement informatique d'apprentissage humain conçu par notre équipe de recherche (geogebraTUTOR). Le texte présente les objectifs de notre projet, le contexte de réalisation, dont les axes...

RÉSUMÉ : Notre communication présente le travail de recherche lié au développement d'un système tutoriel intelligent appelé TURING (TUtoRiel INtelligent en Géométrie). TURING est un projet multidisciplinaire qui intègre la recherche récente en didactique des mathématiques aux possibilités des environnements informatiques d'apprentissage humain. Du...

Our aim in this paper is to show how students at secondary school level can improve their argumentative competence by means of an Intelligent Tutorial System (ITS) designed for learning geometry. After establishing the theoretical frame for our research, we compare the heuristic and discursive characteristics of some tutorial systems, including the...

This paper presents work related to the development of an intelligent tutoring system called TURING (French acronym of «TUtoRiel INtelligent en Géométrie»). TURING is a multidisciplinary project, that joints recent research in didactic of mathematics with the possibilities of computer-based learning environments. From the educational point of view,...

In this article, we present some indications for the evaluation of problem solving and reasoning in mathematics by an intelligent tutoring system to be installed at schools. With such indicators we aim to ameliorate the type of messages currently produced in order to help students all along a problem solving process. Hence, making a follow-up repor...

En: Suma Zaragoza 2002, n. 40, junio ; p. 19-24 Se recoge la intervención del propio autor el 11 de abril de 2002 ante la ponencia sobre la situación comparativa de las enseñanzas científicas con los países europeos en la educación secundaria creada en 2001 en la Comisión de Educación, Cultura y Deporte del Senado Español. En ella se trata el estad...