Advertisement

Educational Studies in Mathematics

, Volume 84, Issue 3, pp 439–460 | Cite as

Discernment of invariants in dynamic geometry environments

  • Allen LeungEmail author
  • Anna Baccaglini-Frank
  • Maria Alessandra Mariotti
Article

Abstract

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.

Keywords

Dynamic geometry Discernment Variation Perception Dragging control 

References

  1. Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM, 34(3), 66–72.CrossRefGoogle Scholar
  2. Baccaglini-Frank, A. (2010). Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging. Doctoral dissertation, University of New Hampshire, Durham, NH, USA.Google Scholar
  3. Baccaglini-Frank, A., (2011). Abduction in generating conjectures in dynamic Geometry through maintaining dragging. In M. Pytlak, T. Rowland, E. Swoboda (Eds.), Proceedings the 7th Conference on European Research in Mathematics Education, pp. 110-119. Rzeszow, Poland.Google Scholar
  4. Baccaglini-Frank, A. (2012). Dragging and making sense of invariants in dynamic geometry. Mathematics Teacher, 105(8), 616–620.CrossRefGoogle Scholar
  5. Baccaglini-Frank, A., & Mariotti, M.A. (2009). Conjecturing and proving in dynamic geometry: The elaboration of some research hypotheses. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (eds.), Proceedings of the 6th Conference on European Research in Mathematics Education (pp. 231–240). Lyon, France.Google Scholar
  6. Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures through dragging in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253.CrossRefGoogle Scholar
  7. Baccaglini-Frank, A., & Mariotti, M. A. (2011). Conjecture-generation through dragging and abduction in dynamic geometry. In A. Méndez-Vilas (Ed.), Education in a Technological World: Communicating Current and Emerging Research and Technological Efforts (pp. 100–107). Spain: Formatex.Google Scholar
  8. Boero, P., Garuti, R. & Lemut, E. (1999). About the generation of conditionality of statements and its links with proving. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 137–144). Haifa, Isreal.Google Scholar
  9. Boero P., Garuti R. & Lemut, E. (2007). Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures and condition suitable to enhance them. In: P. Boero (Ed.), Theorems in School (pp. 249–264). Sensepublisher.Google Scholar
  10. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.CrossRefGoogle Scholar
  11. Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 103–117). Hiroshima: Hiroshima University.Google Scholar
  12. Healy, L., & Hoyles, C. (2001). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6, 235–256.Google Scholar
  13. Hölzl, R. (1996). How does “dragging” affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1(2), 169–187.CrossRefGoogle Scholar
  14. Hölzl, R. (2001). Using dynamic geometry software to add contrast to geometric situations–A case study. International Journal of Computers for Mathematical Learning, 6(1), 63–86.CrossRefGoogle Scholar
  15. Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1), 151–161.CrossRefGoogle Scholar
  16. Laborde, C., & Laborde, J. M. (1995). What about a learning environment where Euclidean concepts are manipulated with a mouse? In A. di Sessa, C. Hoyles, R. Noss (Eds.), Computers and exploratory learning (pp. 241-262). NATO ASI Series, Subseries F(146).Google Scholar
  17. Laborde, J. M., & Strässer, R. (1990). Cabri-Géomètre: A microworld of geometry for guided discovery learning. Zentralblatt für Didaktik der Mathematik, 22(5), 171–177.Google Scholar
  18. Leung, A. (2003). Dynamic geometry and the theory of variation. In N. A. Pateman, B. J. Dougherty & J. Zillox (eds.). Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 197–204). Honolulu, USA.Google Scholar
  19. Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13, 135–157.CrossRefGoogle Scholar
  20. Leung, A., & Lopez-Real, F. (2002). Theorem justification and acquisition in dynamic geometry: A case of proof by contradiction. International Journal of Computers for Mathematical Learning, 7, 145–165.CrossRefGoogle Scholar
  21. Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679.CrossRefGoogle Scholar
  22. Mariotti, M. A. (2010). Riflessioni sulla dinamicità delle figure [Thoughts upon the dynamism of figures]. In G. Accascina & E. Rogora (Eds.), Seminari di Geometria Dinamica [Dynamic Geometry Seminars] (pp. 271–296). Roma: Edizioni Nuova Cultura.Google Scholar
  23. Mariotti, M. A., & Baccaglini-Frank, A. (2011). Making conjectures in dynamic geometry: The potential of a particular way of dragging. New England Mathematics Journal, XLIII, 22–33.Google Scholar
  24. Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). New Jersey: Lawrence Erlbaum Associates, INC Publishers.Google Scholar
  25. Neisser, U. (Ed.). (1989). Concepts and conceptual development: Ecological and intellectual factors in categorization. Cambridge, UK: Cambridge University Press.Google Scholar
  26. Olivero, F. (2002). The proving process within a dynamic geometry environment. (Doctoral thesis). Bristol: University of Bristol. 0-86292-535-5.Google Scholar
  27. Strässer, R. (2001). Cabri-Géomètre: Does dynamic geometry software (DGS) change geometry and its teaching and learning? International Journal of Computers for Mathematical Learning, 6, 319–333.CrossRefGoogle Scholar
  28. Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Allen Leung
    • 1
    Email author
  • Anna Baccaglini-Frank
    • 2
  • Maria Alessandra Mariotti
    • 3
  1. 1.Hong Kong Baptist UniversityKowloonHong Kong
  2. 2.Università di Modena e Reggio EmiliaModenaItaly
  3. 3.Università degli Studi di SienaSienaItaly

Personalised recommendations