Maria Alessandra Mariotti

Defining in Classroom Activities.

This paper discusses some aspects concerning the defining process in geometrical context, in the reference frame of the theory of ‘figural concepts’. The discussion will consider two different, but not antithetical, points of view. On the one hand, the problem of definitions will be considered in the general context of geometrical reasoning; on the other hand, the problem of definition will be considered an educational problem and consequently, analysed in the context of school activities. An introductory discussion focuses on definitions from the point of view of both Mathematics and education. The core of the paper concerns the analysis of some examples taken from a teaching experiment at the 6th grade level. The interaction between figural and conceptual aspects of geometrical reasoning emerges from the dynamic of collective discussions: the contributions of different voices in the discussion allows conflicts to appear and draw toward a harmony between figural and conceptual components. A basic role is played by the intervention of the teacher in guiding the discussion and mediating the defining process.

Justifying and Proving in the Cabri Environment

This paper describes a long term teaching experiment carried out with students from the 9th–10th grades. Geometrical constructions in the Cabri environment were selected as a specific field of experience, within which the sense of theory may emerge. The idea of construction constitutes the key to accessing the idea of theorem, moving from a generic idea of justification towards the idea of validating within a geometrical system. The study aims at clarifying the role of the Cabri environment in this teaching-learning processes: analysis of protocols shows the possible evolution of a justification into a proof but at the same time indicates that this evolution is not expected to be simple and spontaneous.

Generating Conjectures in Dynamic Geometry: The Maintaining Dragging Model

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact students’ approach to investigating open problems in Euclidean geometry. We particularly focus on cognitive processes that might be induced by certain ways of dragging in Cabri. Building on the work of Arzarello, Olivero and other researchers, we have conceived a model describing some cognitive processes that can occur during the production of conjectures in dynamic geometry and that seem to be related to the use of specific dragging modalities. While describing such cognitive processes, our model introduces key elements and describes how these are developed during the exploratory phase and how they evolve into the basic components of the statement of the conjecture (premise, conclusion, and conditional link between them). In this paper we present our model and use it to analyze students’ explorations of open problems. The description of the model and the data presented are part of a more general qualitative study aimed at investigating cognitive processes during conjecture-generation in a DGS, in relation to specific dragging modalities. During the study the participants were introduced to certain ways of dragging and then interviewed while working on open problem activities.

Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

This chapter discusses some strands of experimental mathematics from both an epistemological and a didactical point of view. We introduce some ancient and recent historical examples in Western and Eastern cultures in order to illustrate how the use of mathematical tools has driven the genesis of many abstract mathematical concepts. We show how the interaction between concrete tools and abstract ideas introduces an “experimental” dimension in mathematics and a dynamic tension between the empirical nature of the activities with the tools and the deductive nature of the discipline. We then discuss how the heavy use of the new technology in mathematics teaching gives new dynamism to this dialectic, specifically through students’ proving activities in digital electronic environments. Finally, we introduce some theoretical frameworks to examine and interpret students’ thoughts and actions whilst the students work in such environments to explore problematic situations, formulate conjectures and logically prove them. The chapter is followed by a response by Jonathan Borwein and Judy-anne Osborn.

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.

Transforming Images in a DGS: The Semiotic Potential of the Dragging Tool for Introducing the Notion of Conditional Statement

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact on students’ approach to Euclidean Geometry and specifically on investigating open problems asking for producing conjectures. Building on the work of Arzarello, Olivero, and other researchers, the study addresses the use of specific dragging modalities in the solution of conjecture problems. Within the frame of the theory of semiotic mediation (TSM), the investigation aims at describing the semiotic potential of the dragging tool: how personal meanings emerging from students’ activities in a DGS can potentially be transformed into mathematical meanings. A theoretical discussion is presented, concerning the possible meanings, emerging in respect to the different dragging modalities, their relationship with mathematical meanings concerning conjectures, and conditional statements. Further, it is described how meanings emerge during different exploratory processes and how they may be related to the basic components of a conditional statement: premise, conclusion, and conditional link between them. Some examples discussed are drawn from a teaching experiment where participants were introduced to certain ways of dragging and then interviewed while working on open problem activities.

Resources for the Teacher from a Semiotic Mediation Perspective

The potential of ICT tools for learning have been extensively studied, with a main focus on their possible use by students and the subsequent benefits for them. However, there has been the tendency to underestimate the complexity of the teacher’s role in exploiting this potential. In this chapter, we assume a semiotic mediation perspective and discuss different kinds of resources which are offered to teachers to enhance the teaching–learning activity centred on the use of an ICT tool: by resource for the teacher we mean any artefact which may help the teacher accomplish her educational objectives. Taking a semiotic mediation perspective means to acknowledge the central role of signs in teaching–learning activity: the focus is on semiotic processes, specifically production of signs and their transformation. Fostering or guiding these processes is a crucial issue and a demanding task for the teacher. Consequently, from such a perspective, a resource for the teacher is any artefact that can help her promoting and sustaining the development of those semiotic processes. In this chapter, we discuss the teacher’s use of specific resources – namely written texts – with respect to the semiotic processes which she is expected to orchestrate in the different moments of the teaching sequence.

Semiotic Mediation in the Primary School

“During the seventeenth century geometrical perception became separated, so to say, into two relatively distinct forms of geometry, into two different geometrical styles. One of these is represented by the work of Descartes: the geometry of mechanical-metric activity. The straight line in Cartesian geometry corresponds to an axis of rotation or to the stiffness of a measuring rod. The other geometrical style is represented by the work of Desargues. The straight line of Desarguesian geometry is the ray of light or the line of sight.” These sentences were written by Michael Otte in 1997. Aim of this paper is to present the rationale, design and early findings of a teaching experiment, where the Desarguesian form of geometry was approached at by 5th graders, through the use of a cultural artifact and guidance of the teacher.

Proofs, Semiotics and Artefacts of Information Technologies

This paper discusses some aspects of long-term teaching experiments, carried out with the goal of introducing pupils to proof. I will present two experimental examples of contexts for approaching proof centred on using a computer-based environment, structuring my discussion around the notion of semiotic mediation and its derived didactic model (Mariotti 2002; Bartolini Bussi and Mariotti in press). The experiments are part of a joint research program on semiotic mediation in the mathematics classroom (Bartolini Bussi and Mariotti in press); we adopted the paradigm of research for innovation in the mathematics classroom (Arzarello and Bartolini Bussi 1998). In this paradigm, practice and theory nurture each other in a complex interlaced process. We carried out successive teaching experiments on a single class of students with a single teacher over the ninth and tenth grades. We designed our teaching experiments with strict and continuous collaboration with the teacher, with whom we designed the pedagogical plan and analysed students’ behaviours and productions. Initially our implementation of the innovative didactic strategies was driven by a number of vague pedagogical assumptions. During any teaching experiment, we tried to formulate, cyclically refine and clarify our theoretical hypotheses. Thus, over several years, we developed a theoretical framework that clarifies and formalizes our initial, vague intuition (Mariotti 1996), placing this framework within a Vygotskian approach based on the key notion of semiotic mediation. [AU1]

Designing Non-constructability Tasks in a Dynamic Geometry Environment

This chapter highlights specific design features of tasks proposed in a Dynamic Geometry Environment (DGE) that can foster the production of indirect argumentations and proof by contradiction. We introduce the notion of open construction problem and describe the design of two types of problems, analysing their potential a priori, with the goal of elaborating on the potentials of designing problems in a DGE with respect to fostering processes of indirect argumentation. Specifically, we aim at showing how particular open construction problems, that we refer to as non-constructability problems, are expected to make indirect argumentations emerge.