Discernment of invariants in dynamic geometry environments

In this paper, we discuss discernment of invariants in dynamic geometry environments (DGE) based on a combined perspective that puts together the lens of variation and the maintaining dragging strategy developed previously by the authors. We interpret and describe a model of discerning invariants in DGE through types of variation awareness and simultaneity, and sensorimotor perception leading to awareness of dragging control. In this model, level-1 invariants and level-2 invariants are distinguished. We discuss the connection between these two levels of invariants through the concept of path that can play an important role during explorations in DGE, leading from discernment of level-1 invariants to discernment of level-2 invariants. The emergence of a path and the usefulness of the model will be illustrated by analysing two students’ DGE exploration episodes. We end the paper by discussing a possible pathway between the phenomenal world of DGE and the axiomatic world of Euclidean geometry by introducing a dragging exploration principle.     

Introduction to Proof: The Mediation of a Dynamic Software Environment

This paper, which reports on a long-term teaching experiment carried out in the 9^th and 10^th grades of a scientific highschool, is part of a larger co-ordinated research project. The workconstitutes a `research for innovation', in which action in the classroom isboth a means and result of a study aimed at introducing pupils totheoretical thinking and at studying the ways in which this process isrealised.The purpose of the study is to clarify the role of a particular software,Cabri-géomètre, in the teaching/learning process. Assuming aVygotskian perspective, attention is focussed on the social construction ofknowledge and on the semiotic mediation accomplished through culturalartefacts; the functioning of specific elements of the software will bedescribed and analysed as instruments of semiotic mediation used by theteacher in classroom activities.This revised version was published online in September 2005 with corrections to the Cover Date.     

The wood of “C” and “F” Roman ships found in the ancient harbour of Pisa (Tuscany, Italy): the utilisation of different timbers and the probable geographical area which supplied them

During the excavation of the Etruscan and Roman harbour of Pisa, several shipwrecks were found. The wooden timbers constituting the ship C and ship F (which date back to the first and the second century A.D., respectively, as attested by archaeological findings) were selected in order to collect information about the technological knowledge of the time. Pinus pinaster Aiton was essentially utilised for the planking of ship C and Quercus sp. caducifolia for that of ship F. The choice of timber for the other parts of ship C hull was much more differentiated. Ship C seems to be built in a careful way and its characteristics when added to its overall lightness, seem to reflect its use as a higher capacity boat built for sea sailing. Ship F seems more linked to short voyages in inner fresh waters. The utilisation of different woods was linked not only to the technological characteristics of wood, but also to their easy availability. Palynological analysis, carried out on the clay sediments embedding the shipwrecks, has shown that the flora of Pisa area in that time period would have allowed the acquisition of all the timber species used for both the ships’ construction, with the exception of the fig wood. However, a foreign origin of the timber from somewhere else in the Northern Mediterranean area has not been excluded.     

Synergies between theoretical approaches to mathematics education with technology: A case study through a cross-analysis methodology

Mathematics education as a research domain is characterized by a plurality of theoretical approaches. Acknowledging the existence of such diversity and the risks of an excessive theoretical fragmentation does not mean to search for a unifying theory but to urge the community to develop strategies for coping with this diversity. This article is meant to show the potential of a “cross-analysis” methodology for establishing connections between different theoretical approaches to mathematics education with technology. Within the frame of the ReMath European Project, two Teaching Experiments were realized, centred on the use of a same ICT tool—Casyopée. Two distinct theoretical approaches shaped both the Teaching Experiments design and their enactments: the Theory of Didactical Situations and the Theory of Semiotic Mediation. The two Teaching Experiments have then been analysed from both theoretical points of view. In this article we will provide some examples drawn from this cross-analysis that show the synergy which can be established between the aforementioned theoretical approaches. Beyond contributing to a deeper understanding of the observed “didactical phenomena”, that synergy allows establishing connections between the two approaches that lead to their reciprocal enrichment.     

Using Performance-Based Assessments to Prepare Safe Science Teachers

Standard 9 of the National Science Teachers Association Standards for Science Teacher Preparation is designed to ensure that science teacher preparation programs provide preservice science teachers with the knowledge and skills to understand and successfully engage students in a safe and ethical manner. This standard contains four components describing science teachers’ legal and ethical responsibilities, appropriate use of instructional materials (chemicals in particular), emergency procedures and safety equipment, and guidelines for proper use of living organisms in the classroom. In this article, we describe the requirements of Standard 9 and provide guidance on assessments that can be used to present evidence for preservice teachers’ competence in each of the four components.     

Approaching functions: Cabri tools as instruments of semiotic mediation

Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking the Vygotskian perspective of semiotic mediation, a teaching experiment was designed with the aim of introducing students to the idea of function. This paper focuses on the use of the Trace tool and its potentialities for constructing the meaning of function. In particular, starting from a dynamic approach aimed at grounding the meaning of function in the experience of covariation, the Trace tool can be used to introduce the twofold meaning of trajectory, at the same time global and pointwise, and leads students to grasp the notion of function.     

Networking theoretical frames: the ReMath enterprise

In this article of the special issue devoted to the European project ReMath, we present and discuss the affordances of this project in terms of networking between theoretical frameworks. After clarifying the vision of theories and networking adopted in this project, we introduce the methodological constructs that have been developed and used in ReMath for performing the planned networking enterprise. We then present its outcomes, focusing on different facets of this networking activity: the identification of possible connections and complementarities between frameworks, the identification and elaboration of boundary objects between cultures, and the progressive building of a shared theoretical framework regarding semiotic representations. In the last part of this article, we review the whole networking enterprise from the perspective of research praxeology.