Educational Studies in Mathematics

, Volume 96, Issue 2, pp 145–167 | Cite as

Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

  • Jorge Fiallo
  • Angel GutiérrezEmail author


We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.


Ck¢ model Cognitive unity of theorems Conjecture-and-proof problems Learning conjecture and proof Structural and referential analysis Toulmin scheme Trigonometry 



The authors are grateful to the anonymous reviewers of this paper and the editors of the special issue for their thorough revision and many valuable suggestions that helped us to improve earlier versions of the paper. We are also grateful to the teacher of the Floridablanca school and his pupils for agreeing to collaborate in this experience.


  1. Antonini, S. (2003). Non-examples and proof by contradiction. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME Conference (Vol. 2, pp. 49–56). Honolulu, HI: PME.Google Scholar
  2. Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM—International Journal on Mathematics Education, 40(3), 401–412.Google Scholar
  3. Arzarello, F., Micheletti, C., Olivero, F., & Robutti, O. (1998). A model for analysing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME Conference (Vol. 2, pp. 24–31). Stellenbosch, Republic of South Africa: PME.Google Scholar
  4. Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in Cabri and modalities of transition from conjectures to proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME Conference (Vol. 2, pp. 32–39). Stellenbosch, Republic of South Africa: PME.Google Scholar
  5. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Hodder & Stoughton.Google Scholar
  6. Balacheff, N., & Margolinas, C. (2005). cK¢ modèle de connaissances pour le calcul de situations didactiques. In A. Mercier & C. Margolinas (Eds.), Balises pour la didactique des mathématiques (pp. 75–106). Grenoble, France: La Pensée Sauvage.Google Scholar
  7. Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education. Dordrecht, The Netherlands: Springer.Google Scholar
  8. Boero, P., Garuti, R., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME Conference (Vol. 2, pp. 113–120). Valencia, Spain: PME.Google Scholar
  9. Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In M. M. F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th PME Conference (Vol. 1, pp. 179–209). Belo Horizonte, Brazil: PME.Google Scholar
  10. Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22(3), 233–261.CrossRefGoogle Scholar
  11. Fiallo, J. (2011). Estudio del proceso de demostración en el aprendizaje de las razones trigonométricas en un ambiente de geometría dinámica (Unpublished doctoral dissertation). University of Valencia, Valencia, Spain. Retrieved from
  12. Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and proving in mathematics education. Dordrecht, The Netherlands: Springer.Google Scholar
  13. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education (Vol. III, pp. 234–283). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
  14. Laborde, C., Kynigos, C., Hollebrands, K., & Strässer, R. (2006). Teaching and learning geometry with technology. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 275–304). Rotterdam, The Netherlands: Sense.Google Scholar
  15. Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM - International Journal on Mathematics Education, 43(3), 325–336.Google Scholar
  16. Maher, C. A. (2009). Children’s reasoning: Discovering the idea of mathematical proof. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades. A K-16 perspective (pp. 120–132). New York: Routledge.Google Scholar
  17. Mariotti, M. A. (2001). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281.CrossRefGoogle Scholar
  18. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (pp. 173–204). Rotterdam, The Netherlands: Sense.Google Scholar
  19. Marrades, R., & Gutiérrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1/2), 87–125.CrossRefGoogle Scholar
  20. Ministerio de Educación Nacional (MEN). (2006). Estándares básicos de matemáticas. Bogotá, Colombia: Author.Google Scholar
  21. Pedemonte, B. (2002). Etude didactique et cognitive des rapports de l’argumentation et de la démonstration dans le apprentisage des mathématiques. (Doctoral dissertation). Université Joseph Fourier-Grenoble I, Grenoble, France.Google Scholar
  22. Pedemonte, B. (2005). Quelques outils pour l’analyse cognitive du rapport entre argumentation et démonstration. Recherches en Didactique des Mathématiques, 25(3), 313–348.Google Scholar
  23. Pedemonte, B., & Balacheff, N. (2016). Establishing links between conceptions, argumentation and proof through the ck¢-enriched Toulmin model. The Journal of Mathematical Behavior, 41, 104–122.CrossRefGoogle Scholar
  24. Pratt, D., & Noss, R. (2010). Designing for mathematical abstraction. International Journal of Computers for Mathematical Learning, 15(2), 81–97.CrossRefGoogle Scholar
  25. Reid, D. A., & Knipping, C. (2010). Proof in mathematics education. Rotterdam, The Netherlands: Sense.Google Scholar
  26. Stylianides, A. J., & Stylianides, G. J. (2013). Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education. ZDM—International Journal on Mathematics Education, 45(3), 333–341.Google Scholar
  27. Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10(3), 145–166.CrossRefGoogle Scholar
  28. Toulmin, S. E. (2003). The uses of argument (updated edition of the 1958 book). Cambridge, UK: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Escuela de MatemáticasUniversidad Industrial de SantanderBucaramangaColombia
  2. 2.Dpto. de Didáctica de la MatemáticaUniversidad de ValenciaValenciaSpain

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