## Abstract

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a loss of evidence in various ways, owing to its particular logical structure and to the difficulty in managing geometrical figures with contradictory properties. In this paper, we observe that students produce argumentation by starting with the assumption that the claim is false (indirect argumentation), and that they seem to accept this as more evident than the proofs by contradiction. On the basis of the notion of intuitive knowledge developed by Fischbein and through the analysis of task-based interviews, we investigate the intuitive acceptance of proof by contradiction and of indirect argumentation, underlining, in particular, that indirect argumentation can be produced as a compromise between a proof by contradiction and the need for a more evident argument.

## Keywords

Formal and intuitive knowledge Proof by contradiction Indirect argumentation Figural concepts## Notes

### Acknowledgements

I wish to thank Anna Baccaglini-Frank and Maria Alessandra Mariotti for many interesting and helpful discussions. The exchanges with them have led to many ideas in this paper.

## References

- Antonini, S. (2004). A statement, the contrapositive and the inverse: intuition and argumentation. In M. Johnsen Høines, & A. Berit Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 47–54). Norway: Bergen.Google Scholar - Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving?
*ZDM Mathematics Education,**40*(3), 401–412.CrossRefGoogle Scholar - Antonini, S., & Mariotti, M. A. (2010). Abduction and the explanation of anomalies: The case of proof by contradiction. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.),
*Proceedings of the 6th Conference of European Research in Mathematics Education*(pp. 322–331). France: Lyon.Google Scholar - Baccaglini-Frank, A. (2010).
*Conjecturing in dynamic geometry: A model for conjecture*-*generation through maintaining dragging*. Doctoral dissertation, University of New Hampshire, Durham, NH. ProQuest.Google Scholar - Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (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). New York: Springer.CrossRefGoogle Scholar - Baccaglini-Frank, A., Antonini, S., Leung, A. S., & Mariotti, M. A. (2018). From pseudo-objects in dynamic explorations to proof by contradiction.
*Digital Experiences in Mathematics Education,**4*(2–3), 87–109.CrossRefGoogle Scholar - Barbin, E. (1988). La démonstration mathématique: Significations épistémologiques et questions didactiques.
*Bulletin APMEP,**366,*591–620.Google Scholar - Boero, P. (Ed.). (2007).
*Theorems in school: From history, epistemology and cognition to classroom practice*. Rotterdam: Sense Publishers.Google Scholar - Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof.
*Educational Studies in Mathematics,**24*(4), 359–387.CrossRefGoogle Scholar - de Villiers, M. (1990). The role and function of proof in mathematics.
*Pythagoras,**24,*17–24.Google Scholar - Dummett, M. (1977).
*Elements of intuitionism*. New York: Oxford University Press.Google Scholar - Duval, R. (1992–1993). Argumenter, demontrer, expliquer: coninuité ou rupture cognitive?
*Petit x*,*31*, 37–61.Google Scholar - Duval, R. (1995).
*Sémiosis et Pensée Humain*. Bern: Peter Lang.Google Scholar - Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.),
*Perspectives on the teaching and learning of Geometry for the 21st Century*(pp. 37–52). Dordrecht: Kluwer.Google Scholar - Fischbein, E. (1982). Intuition and proof.
*For the Learning of mathematics,**3*(2), 9–24.Google Scholar - Fischbein, E. (1987).
*Intuition in science and mathematics*. Dordrecht: Kluwer.Google Scholar - Fischbein, E. (1993). The theory of figural concepts.
*Educational Studies in Mathematics,**24*(2), 139–162.CrossRefGoogle Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht: Reidel.Google Scholar - Giaquinto, M. (1992). Visualizing as a means of geometrical discovery.
*Mind and Language,**7,*381–401.Google Scholar - Hanna, G., & de Villiers, M. (Eds.). (2012).
*Proof and proving in mathematics education. The 19th ICMI study*. Dordrecht: Springer.Google Scholar - Hanna, G., & Sidoli, N. (2007). Visualisation and proof: A brief survey of philosophical perspectives.
*ZDM Mathematics Education,**39,*73–78.CrossRefGoogle Scholar - Harel, G. (2007). Students’ proof schemes revisited. In P. Boero (Ed.),
*Theorems in school: From history, epistemology and cognition to classroom practice*(pp. 65–78). Rotterdam: Sense Publishers.Google Scholar - Laborde, C. (1998). Relationship between the spatial and the theoretical in geometry: The role of computer dynamic representations in problem solving. In D. Tinsley & D. Johnson (Eds.),
*Information and communication technologies in school mathematics*(pp. 183–194). London: Chapman & Hall.CrossRefGoogle Scholar - Leron, U. (1985). A direct approach to indirect proofs.
*Educational Studies in Mathematics,**16*(3), 321–325.CrossRefGoogle Scholar - 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 - Lolli, G. (2005). A cosa servono le dimostrazioni nella scuola e nella ricerca. http://docplayer.it/70935634-A-cosa-servono-le-dimostrazioni-nella-scuola-e-nella-ricerca.html. Accessed 14 May 2018.
- Magnani, L. (2001).
*Abduction, reason, and science. Processes of discovery and explanation*. New York: Kluwer.CrossRefGoogle Scholar - Mancosu, P. (1996).
*Philosophy of mathematical practice in the 17th century*. New York: Oxford University Press.Google Scholar - Mariotti, M. A. (1995). Images and concepts in Geometrical reasoning. In R. Sutherland & J. Mason (Eds.),
*Exploiting mental imagery with computer in mathematics education*(pp. 97–116). Berlin: Springer.CrossRefGoogle Scholar - Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 173–204). Rotterdam: Sense Publishers.Google Scholar - Mariotti, M. A., & Antonini, S. (2009). Breakdown and reconstruction of figural concepts in proofs by contradiction in geometry. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.),
*Proof and proving in mathematics education, ICMI Study 19 Conference Proceedings*(vol. 2, pp. 82–87). Taiwan: Taipei.Google Scholar - Mariotti, M. A., Bartolini Bussi, M., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: From history and epistemology to cognition. In E. Pehkonen (Ed.),
*Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 180–195). Finland: Lahti.Google Scholar - Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities.
*Educational Studies in Mathematics,**34,*219–224.CrossRefGoogle Scholar - Murray, J. A. H., et al. (Eds.). (1961).
*The Oxford English Dictionary: Being a corrected re-issue with an introduction, supplement, and bibliography of A New English Dictionary on Historical Principles 3: D-E. The Philological Society*. Oxford: The Clarendon Press.Google Scholar - Partridge, E. (1958).
*Origins: A short etymological dictionary of modern English*. London: Routledge & Kegan-Paul.Google Scholar - Reid, D., & Dobbin, J. (1998). Why is proof by contradiction difficult? In A. Olivier & K. Newstead (Eds.),
*Proceedings of the 22th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 41–48). Stellenbosch, South Africa.Google Scholar - Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. Leder, & P. Boero (Eds.),
*2nd handbook on the psychology of mathematics education*(pp. 315–351). Rotterdam: Sense Publishers.Google Scholar - Szabó, A. (1978).
*The beginnings of Greek mathematics*. Dordrecht: Reidel.CrossRefGoogle Scholar - Thompson, D. R. (1996). Learning and teaching indirect proof.
*The Mathematics Teacher,**89*(6), 474–482.Google Scholar - 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). Hawaii: Honolulu.Google Scholar