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From Pseudo-Objects in Dynamic Explorations to Proof by Contradiction

  • Anna Baccaglini-Frank
  • Samuele Antonini
  • Allen Leung
  • Maria Alessandra Mariotti
Article

Abstract

Proof by contradiction presents various difficulties for students relating especially to the formulation and interpretation of a negation, the managing of impossible mathematical objects, and the acceptability of the validity of the statement once a contradiction has been reached from its negation. This article discusses how a Dynamic Geometry Environment (DGE) can contribute to students’ argumentation processes when trying to explain contradictions. Four cases are presented and analysed, involving students from high school, as well as undergraduate and graduate students. The approach of the analyses makes use of a symbolic logical chain and the notion of pseudo-object. Such analyses lead to a hypothesis, that experiencing a pseudo-object during an exploration can foster DGE-supported processes of argumentation culminating in geometrical proofs by contradiction, while the lack of experience of a pseudo-object may hinder such processes. If this hypothesis is confirmed by further studies, we foresee important didactical implications since it sheds light on the transition from students’ DGE-based argumentations to proofs by contradiction.

Keywords

Dynamic geometry Indirect argument Proof by contradiction Pseudo-object 

References

  1. Antonini, S. (2004). A statement, the contrapositive and the inverse: Intuition and argumentation. In M. Høines & B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 47–54). Bergen: Bergen University College.Google Scholar
  2. Antonini, S. (2010). A model to analyse argumentations supporting impossibilities in mathematics. In I. M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 153–160). Belo Horizonte: PME.Google Scholar
  3. Antonini, S., & Mariotti, M. (2006). Reasoning in an absurd world: Difficulties with proof by contradiction. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 65–72). Prague: Faculty of Education, Charles University.Google Scholar
  4. Antonini, S., & Mariotti, M. (2007). Indirect proof: An interpreting model. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 541–550). Larnaca: University of Cyprus/CERME.Google Scholar
  5. Antonini, S., & Mariotti, M. (2008). Indirect proof: What is specific to this way of proving? ZDM Mathematics Education, 40(3), 401–412.CrossRefGoogle Scholar
  6. Baccaglini-Frank, A. (2010). Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging. Unpublished doctoral dissertation. Durham: University of New Hampshire.Google Scholar
  7. Baccaglini-Frank, A., & Mariotti, M. (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
  8. Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. (2011). Reasoning by contradiction in dynamic geometry. In B. Ubuz (Ed.), Proceedings of the 35rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 81–88). Ankara: PME.Google Scholar
  9. Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. (2013). Reasoning by contradiction in dynamic geometry. PNA: Revista de Investigación en Didáctica de la Matemática, 7(2), 63–73.Google Scholar
  10. Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. (2017). Designing non-constructability tasks in a dynamic geometry environment. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematics education tasks: Potential and pitfalls (pp. 99–120). Cham: Springer.CrossRefGoogle Scholar
  11. Boero, P. (Ed.). (2007). Theorems in school: From history, epistemology and cognition to classroom practice (pp. 249–264). Rotterdam: Sense Publishers.Google Scholar
  12. Boero, P., Garuti, R., & Mariotti, M. (1996). Some dynamic mental process underlying producing and proving conjectures. In A. Gutiérrez & L. Puig (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 121–128). Valencia: Departamento de Didáctica de la Matemática, Universita de València.Google Scholar
  13. de Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. International Journal of Mathematical Education in Science and Technology, 35(5), 703–724.CrossRefGoogle Scholar
  14. Duval, R. (1993). Argumenter, démontrer, expliquer: Continuité ou rupture cognitive? Petit x, 31, 37–61.Google Scholar
  15. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel.Google Scholar
  16. Garuti, R., Boero, P., Lemut, E., & Mariotti, M. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In A. Gutiérrez & L. Puig (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 113–120). Valencia: Departamento de Didáctica de la Matemática, Universita de València.Google Scholar
  17. Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and proving in mathematics education: The 19th ICMI study. Dordrecht: Springer.Google Scholar
  18. Healy, L. (2000). Identifying and explaining geometric 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. 1, pp. 103–117). Hiroshima: PME.Google Scholar
  19. Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1–3), 151–161.CrossRefGoogle Scholar
  20. Laborde, C., & Laborde, J.-M. (1992). Problem solving in geometry: From microworlds to intelligent computer environments. In J. Ponte, J. Matos, J. Matos, & D. Fernandes (Eds.), Mathematical problem solving and new information technologies (pp. 177–192). Berlin: Springer-Verlag.CrossRefGoogle Scholar
  21. Laborde, J.-M., & Sträßer, R. (1990). Cabri-géomètre: A microworld of geometry for guided discovery learning. ZDM Mathematics Education, 22(5), 171–177.Google Scholar
  22. Leron, U. (1985). A direct approach to indirect proofs. Educational Studies in Mathematics, 16(3), 321–325.CrossRefGoogle Scholar
  23. 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(2), 145–165.CrossRefGoogle Scholar
  24. Leung, A., Baccaglini-Frank, A., & Mariotti, M. (2013). Discernment in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460.CrossRefGoogle Scholar
  25. Mariotti, M. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1–3), 25–53.CrossRefGoogle Scholar
  26. Mariotti, M. (2002). The influence of technological advances on students’ mathematical learning. In L. English (Ed.), Handbook of international research in mathematics education (pp. 695–723). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  27. Mariotti, M. (2014). Transforming images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), Transformation: A fundamental idea of mathematics education (pp. 156–175). New York: Springer.Google Scholar
  28. Mariotti, M., & Antonini, S. (2009). Breakdown and reconstruction of figural concepts in proofs by contradiction in geometry. In F.-L. Lin, F. Hsieh, G. Hanna, & M. de Villers (Eds.), Proof and proving in mathematics education: ICMI Study 19 Conference Proceedings (Vol. 2, pp. 82–87). Taipei: National Taiwan Normal University.Google Scholar
  29. Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom: The case of function. International Journal of Educational Research, 51–52, 10–27.CrossRefGoogle Scholar
  30. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.CrossRefGoogle Scholar
  31. Sinclair, N., & Robutti, O. (2013). Technology and the role of proof: The case of dynamic geometry. In M. Clements, A. Bishop, C. Keitel-Kreidt, J. Kilpatrick, & F. Leung (Eds.), Third international handbook of mathematics education (pp. 571–596). New York: Springer.Google Scholar
  32. Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. Leder, & P. Boero (Eds.), The second handbook on the psychology of mathematics education: The journey continues (pp. 315–351). Rotterdam: Sense Publishers.Google Scholar
  33. Vinner, S. (1999). The possible and the impossible (a review of G. Martin, 1998, Geometric constructions). ZDM Mathematics Education, 31(2), 77.Google Scholar
  34. Wu Yu, J.-Y., Lin, F.-L., & Lee, Y.-S. (2003). Students’ understanding of proof by contradiction. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PME-NA (Vol. 4, pp. 443–449). Honolulu: College of Education, University of Hawaii.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Anna Baccaglini-Frank
    • 1
  • Samuele Antonini
    • 2
  • Allen Leung
    • 3
  • Maria Alessandra Mariotti
    • 4
  1. 1.Universita degli Studi di PisaPisaItaly
  2. 2.Università degli Studi di PaviaPaviaItaly
  3. 3.Hong Kong Baptist UniversityKowloon TongHong Kong
  4. 4.Università degli Studi di SienaSienaItaly

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