Special Issue Title: Proof in Dynamic Geometry Environments

Special Issue Guest Editors: Keith Jones, Ángel Gutiérrez and Maria Alessandra Mariotti

Outline of guest editorial:
Since its creation in 1968, Educational Studies in Mathematics (ESM) is being a leading magazine in the field of Mathematics Education, the most relevant research questions having been raised and discussed on its pages. On the other hand, the International Group for the Psychology of Mathematics Education (PME) became since its first meetings in late 70s the main forum for researchers in Mathematics Education. Every year, researchers all over the world discuss the newest research questions and results, and define future research directions. As a result of a joint effort of ESM and PME, ESM will periodically publish PME special issues aimed to showcase important and substantial aspects of topics worked out by the PME community.

We are very happy to present the first outcome of the PME Special Issue series. It is devoted to analyze the influence of dynamic geometry software (DGS) on students' conceptions of proof while solving geometry problems involving proofs in an environment mediated by such kind of software.

The introductory paper by G. Hanna deals with some theoretical aspects of questions considered in the four research papers. She first address the questions of the role of proofs in secondary school mathematics, and the contrast between abstract proofs and heuristics, explorations, and visual proofs. Her remarks on epistemology towards the end of the paper constitute an important element of the background to the special issue.

The four research papers explore the central question of whether the opportunity offered by DGS environments to "see" mathematical properties so easily might reduce or even replace any need for proof or, on the contrary, it might open new ways of meaningful approach to promote students' understanding of need for and roles of proofs. Each of the papers addresses this issue within a different theoretical framework:

M.A. Mariotti presents an analysis, from a vygotskian viewpoint, of the relevant mediation of several components and commands of the DGS on the interactions in groups of students. She also highlights correspondences among the commands used by students and subjacent axioms and theorems used in their justifications.

K. Jones presents an environment where students are faced to the question of classification of quadrilaterals by means of a set of problems of increasing difficulty. The analysis of students' answers shows that DGS helps students to progress in their understanding of the dependence relationships among components of a figure and among families, and to advance towards a progressive abstraction in their justifications.

R. Marrades and A. Gutiérrez report on a teaching experiment where students are induced to produce deductive justifications of the correctness of their constructions. A framework integrating and expanding previous ones is used to analyze students answers and to show how the way they use the DGS is determinant of their solutions and justifications, and how quality of students' justifications improve with the time.

N. Hadas, R. Hershkowitz, and B.B. Schwarz present an original approach to induce students to produce deductive justifications by proposing them problems where students have to deal with surprising, contradictory or uncertain results while trying to solve the problems. Authors present different categories of students' answers to this kind of problems when solved in a DGS environment.

The final paper, by C. Laborde describes to what extent the four papers complement each other, by offering a global integrating overview of them. It is possible to build many different DGS teaching environments, adapted to specific necessities of students, where students can gain a better understanding of the deductive structure of mathematics, and the need for justifications/proofs in mathematics. Furthermore, C. Laborde highlights the usefulness of DGS to break the traditional separation between action (as manipulation associated to observation and description) and deduction (as intellectual activity detached from specific objects).

The papers in this special issue are the results of their authors' personal research activity. However, they would not have been possible out of the environment of PME annual conferences, where very valuable discussions took place during past years.