Philippe R. Richard

AgentGeom: a multiagent system for pedagogical support in geometric proof problems

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 school student can appropriate these abilities through interactions with the system. Our system uses strategic messages of the agent tutor in an argumentative process that collaborates with a student in the construction of a proof.

CERME7 Working Group 4: Geometry teaching and learning

In the WG4 sessions, a consensus was agreed favouring a common approach and discussions on the following specific topics: educational goals and curriculum in geometry; use of geometrical figures and diagrams; and understanding and use of concepts and proof in geometry. Readers were invited to look at past reports to get to know more about these agreements. The participants paid great attention to linking theoretical and empirical aspects of research in geometry education. Two approaches for using theory in research can be distinguished: first, theory can serve as a starting point for initiating a research study; secondly, theory can act as a lens to look into the data. There were a number of theories which were used by the group when analysing the teaching and learning of geometry. For a cognitive and semiotic approach, the Van Hiele (1986) levels, the notion of figural concept, and Duval’s (1995) registers were used. For an epistemological approach, researchers used the geometrical paradigms described by Kuzniak and Rauscher (2011). Braconne-Michoux discussed the link between the paradigms and the van Hiele levels, and Kuzniak referred to a new trend of research that uses the integrative model of Geometrical Work Spaces, which articulates both approaches through a didactical viewpoint. One of the themes discussed concerned manipulation, approximation and proof. Reasoning is expressed by manipulating objects or by means of linguistic tools. Bulf et al. referred to the relationships between the ways that students perceive, act and speak about objects in geometry classes. The mutual relationships between proof and approximation were highlighted in solving real-life problems, from the process of modelling to the interpretation of a geometrical solution in terms of the original problem. Girnat discussed teachers’ beliefs about applying geometry, setting application-oriented beliefs in the context of the whole geometry curriculum. Approximation raises questions about the limits of perceptual information, the reliability of the figural register, and the use of discrete models to represent continuous phenomena during some instrumented approaches with software. Fujita et al. reported on students’ tackling of 3D geometry problems in which primitive conjectures were produced by relying on visual images rather than on geometrical reasoning. A possible route for the apprehension of the geometrical figure was sketched by Deliyianni et al., following Duval’s 1995 contributions. From a different point of view, Gagatsis et al. expounded the use of figure as illustration.

International Perspectives on Secondary Geometry Education: An Introduction

This chapter introduces the book by providing an orientation to the field of research and practice in the teaching and learning of secondary geometry. The editors describe the chapters in the book in terms of how they contribute to address questions asked in the field, outlining different reasons why prospective readers might want to look into specific chapters.

Conclusion: Prospects for Developments and Research in Secondary Geometry Education

This chapter concludes the collection of reports that expanded on the papers presented at ICME 13, in the context of the Topic Study Group on the teaching and learning of secondary geometry. In an effort to articulate a vision for where the field could go in the near future, the editors take this opportunity to revisit issues of methodologies for data collection and data analysis. They propose how new technologies could be integrated into research and practice in secondary geometry and ask questions that the field might expect to address with the aid of such technologies.

International Perspectives on the Teaching and Learning of Geometry in Secondary Schools

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 instruction and teacher knowledge, and a third set of chapters offers studies on geometry thinking and learning. Specific research topics addressed also include teaching practice, learning trajectories, learning difficulties, technological resources, instructional design, assessments, textbook analyses, and teacher education in geometry.

Connectedness of Problems and Impasse Resolution in the Solving Process in Geometry: A Major Educational Challenge

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 mathematical working space and zone of proximal development, in order to explore how connected problems can help to resolve moments of impasse of a student when solving a proof problem in geometry. In particular, we discuss how the notion of interaction moves our theoretical framework closer to the methodological challenges raised in the QED-Tutrix research project jointly being realized in didactics of mathematics and computer engineering.