Abstract
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.
Similar content being viewed by others
References
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.
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.
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.
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.
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.
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.
Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education. Dordrecht, The Netherlands: Springer.
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.
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.
Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22(3), 233–261.
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 https://www.educacion.gob.es/teseo/mostrarRef.do?ref=936657
Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and proving in mathematics education. Dordrecht, The Netherlands: Springer.
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.
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.
Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM - International Journal on Mathematics Education, 43(3), 325–336.
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.
Mariotti, M. A. (2001). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281.
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.
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.
Ministerio de Educación Nacional (MEN). (2006). Estándares básicos de matemáticas. Bogotá, Colombia: Author.
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.
Pedemonte, B. (2005). Quelques outils pour l’analyse cognitive du rapport entre argumentation et démonstration. Recherches en Didactique des Mathématiques, 25(3), 313–348.
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.
Pratt, D., & Noss, R. (2010). Designing for mathematical abstraction. International Journal of Computers for Mathematical Learning, 15(2), 81–97.
Reid, D. A., & Knipping, C. (2010). Proof in mathematics education. Rotterdam, The Netherlands: Sense.
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.
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.
Toulmin, S. E. (2003). The uses of argument (updated edition of the 1958 book). Cambridge, UK: Cambridge University Press.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fiallo, J., Gutiérrez, A. Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course. Educ Stud Math 96, 145–167 (2017). https://doi.org/10.1007/s10649-017-9755-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-017-9755-6