Alain Kuzniak

Un Exemple de Cadre Conceptuel pour L'Etude de L'Enseignement de la Geometrie en Formation des Maitres.

We are interested in the teaching of geometry to primary school (children from 3 to 11) teachers. We define a priori a conceptual frame, which organises geometry upon three kinds of knowledge: intuition, experience and deduction. Drawing on Gonseths works, we bring out three syntheses of elementary geometry: natural geometry (geometry I), natural axiomatic geometry (geometry II) and formalist axiomatic geometry (geometry III). Next we illustrate this conceptual frame with examples of teaching geometry. Last we bring out different conceptions of geometry in scholar system which could lead to cross purposes.

Understanding the Nature of the Geometric Work Through Its Development and Its Transformations

The question of the teaching and learning of geometry has been profoundly renewed by the appearance of Dynamic Geometry Software (DGS). These new artefacts and tools have modified the nature of geometry by changing the methods of construction and validation. They also have profoundly altered the cognitive nature of student work, giving new meaning to visualisation and experimentation. In our presentation, we show how the study of some geneses (figural, instrumental and discursive) could clarify the transformation of geometric knowledge in school context. The argumentation is supported on the framework of Geometrical paradigms and Spaces for Geometric Work that articulates two basic views on a geometer’s work: cognitive and epistemological.

Understanding Geometric Work through Its Development and Its Transformations

The question of the teaching and learning of geometry has been profoundly renewed by the appearance of dynamic geometry software (DGS). These new artefacts and tools have modified the nature of geometry by changing the methods of construction and validation. They also have profoundly altered the cognitive nature of student work, giving new meaning to visualization and experimentation. In our paper, we assert how the study of some geneses (figural, instrumental and discursive) could clarify the transformation of geometric knowledge in the school context. The argumentation is supported on the framework of geometric work space that articulates two basic views on a geometer’s work: cognitive and epistemological.

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.

Invited Lectures from the 13th International Congress on Mathematical Education

Practice-based initial teacher education reforms are typically organised around a set of core teaching practices, a set of normative principles to guide teachers’ judgement, and the knowledge needed to teach mathematics. Developing more than understandings, practices, and visions, practice-based pedagogies also need to support prospective teachers’ emergent dispositions for teaching. Based on the premise that an inquiry stance is a key attribute of adaptive expertise and teacher professionalism this paper examines the function and value of inquiry within practice-based learning. Findings from the Learning the Work of Ambitious Mathematics Teaching project are used to illustrate how opportunities to engage in critical and collaborative reflective practices can contribute to prospective teachers’ development of an inquiry-oriented stance. Exemplars of prospective teachers’ inquiry processes in action—both within rehearsal activities and a classroom inquiry—highlight the potential value of practice-based opportunities to learn the work of teaching.

Thinking About the Teaching of Geometry Through the Lens of the Theory of Geometric Working Spaces

In this communication, I argue that shared theoretical frameworks and specific topics need to be developed in international research in geometry education to move forward. My purpose is supported both by my experience as chair and participant in different international conferences (CERME, ICME), and also by a research program on Geometric Working Spaces and geometric paradigms. I show how this framework allows thinking about the nature of geometric work in various educational contexts.

Introduction: Perspectives on Didactics of Mathematics through Michèle Artigue’s Contributions

Since the early 1970s and up to the present day, Michele Artigue has been closely linked to the emergence and the development of the didactics of mathematics. By observing her exemplary professional history, one can witness a new and specific research domain taking form, as well as see the difficulties that accompanied its recognition by both the academic community and, more generally, the whole education community. Following this notion of recognition, we have organised this opening chapter around some of the major issues related to the past, the present, and the future of the didactics of mathematics, and more generally of mathematics education: didactics as a specific research domain, the role of theoretical frameworks, the relationship to connected fields of research, and finally, the way didactics considers its relationship with the outside world of mathematics teaching and learning.