GABRIEL J. STYLIANIDES and ANDREAS J. STYLIANIDES VALIDATION OF SOLUTIONS OF CONSTRUCTION PROBLEMS IN DYNAMIC GEOMETRY ENVIRONMENTS 1 ABSTRACT. This paper discusses issues concerning the validation of solutions of con- struction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit con- structions created using measurement tools to be considered valid; however, these con- structions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is neces- sarily problematic, we examine what would constitute the analogue of the straightedge- and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test cri- terion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software. KEY WORDS: drag test, Dynamic Geometry Environments (DGEs), Euclidean geometry, geometrical constructions, proof, validation of construction problems INTRODUCTION Despite their fundamental theoretical value, geometrical construc- tions seem to have lost their centrality in the school geometry cur- riculum (Mariotti, 2001). At the same time, the learning and teaching of these constructions has often been dissociated from meaningful mathematical activity (see, e.g., Schoenfeld, 1988). However, the appearance of computer geometry software packages seems to be spurring a new interest in geometrical constructions and supports the possibility of using construction tasks in Dynamic Geometry International Journal of Computers for Mathematical Learning (2005) 10: 31–47 DOI 10.1007/s10758-004-6999-x Ó Springer 2005