Validation of Solutions of Construction Problems in Dynamic Geometry Environments

  • Gabriel J. StylianidesEmail author
  • Andreas J. Stylianides


This paper discusses issues concerning the validation of solutions of construction 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 constructions created using measurement tools to be considered valid; however, these constructions 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 necessarily 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 criterion 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.


drag test Dynamic Geometry Environments (DGEs) Euclidean geometry geometrical constructions proof validation of construction problems 


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  1. Balacheff, N., Sutherland, R. 1994Epistemological domain of validity of microworlds:The case of Logo and Cabri-géomètreLewis, R.Mendelson, P. eds. Lessons from Learning.ElsevierAmsterdam137115Google Scholar
  2. Cipra, B. 1993New computer insights from “transparent” proofsWhat’s Happening in the Mathematical Sciences1712Google Scholar
  3. Dubinsky, E., Tall, D. 1991Advanced mathematical thinking and the computerTall, D. eds. Advanced Mathematical Thinking.Kluwer Academic PublishersNetherlands231274Google Scholar
  4. Goldreich, O. 2002Zero-Knlowledge Twenty Years after its InventionDepartment of Computer Science and Applied Mathematics. Weizmann Institute of ScienceRehovot, IsraelGoogle Scholar
  5. Hanna, G. 1995Challenges to the importance of proofFor the Learning of Mathematics154249Google Scholar
  6. Healy, L., Hoelzl, R., Hoyles, C., Noss, R. 1994Messing upMicromath101416Google Scholar
  7. Hersh, R. 1986Some proposals for reviving the philosophy of mathematicsTymoczko, T. eds. New Directions in the Philosophy of Mathematics.BirkhouserBoston928Google Scholar
  8. Hersh, R. 1993Proving is convincing and explainingEducational Studies in Mathematics24389399CrossRefGoogle Scholar
  9. Jones, K. 2000Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanationsEducational Studies in Mathematics445585CrossRefGoogle Scholar
  10. Laborde, J., Bellemain, F. 1998Cabri Geometry IITexas Instruments SoftwareDallas, TexasGoogle Scholar
  11. Mariotti, M.A. 2000Introduction to proof: The mediation of a dynamic software environmentEducational Studies in Mathematics442553CrossRefGoogle Scholar
  12. Mariotti, M.A. 2001Justifying and proving in the Cabri environmentInternational Journal of Computers for Mathematical Learning6257281CrossRefGoogle Scholar
  13. Noss, R., Healy, L., Hoyles, C. and Hoelzl, R. (1994). Constructing meanings for construction. In J.P. da Ponte and J.F. Matos (Eds.), Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education (pp. 360–367). Lisbon, Portugal.Google Scholar
  14. Ott, E. 1993Chaos in Dynamical SystemsCambridge University PressNew YorkGoogle Scholar
  15. Schoenfeld, A.H. 1988When good teaching leads to bad results: The disasters of “well–taught” mathematics coursesEducational Psychologist23145166Google Scholar
  16. Straesser, R. 2001Cabri-Géome‘tre: Does dynamic geometry software (DGS) change geometry and its teaching and learning?. International Journal of Computers for Mathematical Learning6319333Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Gabriel J. Stylianides
    • 1
    Email author
  • Andreas J. Stylianides
    • 1
  1. 1.School of EducationThe University of MichiganArborMIUSA

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