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Proofs, Mathematical Practice and Argumentation

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Abstract

In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. I consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity.

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Notes

  1. Toulmin’s model was used to analyze a regular argument in Toulmin et al. (1979), p. 126. I would like to thank an anonymous referee for bringing my attention to this and other references that helped me improve this and others points. Many thanks are also due to the second referee for helping me improve this paper.

  2. Examples from Maxwell can also be found in Aberdein (2010). A very similar example to this one from Wikipedia can be found in Krabbe (2008).

  3. With this statement I am not claiming that mathematical proofs are or should be pure logical or formal proofs, but that their underlying inferential structure is deductive.

  4. The problem need not be formulated exactly like this; it can be presented in a more informal way.

  5. This example is an adaptation of the example provided in Chazan and Sandow (2011).

  6. This definition is cited, for example in (Schwarz et al. 2010) and in (Balacheff 2010).

References

  • Aberdein, Andrew. 2005. The uses of argument in mathematics. Argumentation 19: 287–301.

    Article  Google Scholar 

  • Aberdein, Andrew. 2009. Mathematics and argumentation. Foundations of Science 14(1–2): 1–8.

    Article  Google Scholar 

  • Aberdein, Andrew. 2010. Observations on sick mathematics. In Philosophical perspectives on mathematical practices, ed. Bart Van Kerkhove, Jean Paul Van Bendegem, and Jonas de Vuyst, 269–300. London: College Publications.

    Google Scholar 

  • Aberdein, Andrew. 2011. The dialectical tier of mathematical proof. In Argumentation: Cognition & Community. Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (OSSA), ed. F. Zenker. Windsor, Ontario: OSSA.

  • Aberdein, Andrew. 2013. Mathematical wit and mathematical cognition. Topics in Cognitive Science 5(2): 231–250.

  • Aberdein, Andrew, and Ian J. Dove (eds.). 2009. Mathematical Argumentation. Foundations of Science 14(1–2). Dordrecht: Springer

  • Aberdein, Andrew, and Ian J. Dove (eds.). 2013. The argument of mathematics. Dordrecht: Springer.

    Google Scholar 

  • Alcolea Banegas, Jesús. 1998. L’Argumentació en Matemàtiques. In XIIè Congrés Valencià de Filosofia, ed. Enric Casaban i Moya. Valencià. Updated English translation in The Argument of Mathematics, 2013, eds. Andrew Aberdein and Ian J. Dove, 47–60. Dordrecht: Springer.

  • Andriessen, Jerry, Michael Baker, and Dan D. Suthers (eds.). 2003. Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments. Dordrecht: Kluwer.

    Google Scholar 

  • Andriessen, Jerry. 2009. Argumentation in higher education: Examples of actual practices with argumentation tools. In Argumentation and education: theoretical foundations and practice, eds. Nathalie Muller Mirza and Anne-Nelly Perret-Clermont, 195–213. Dordrecht: Springer.

  • Atzmon, Shirley, Rina Hershkowitz, and Baruch B. Schwarz. 2006. The role of teachers in turning claims to arguments. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, ed. Jamila Novotna, 65–72. Prague: Charles University.

  • Balacheff, Nicolas. 2010. Bridging knowing and proving in mathematics: A didactical perspective. In Explanation and proof in mathematics: philosophical and educational perspectives, ed. Gila Hanna, Hans Niels Jahnke, and Helmut Pulte, 115–135. Dordrecht: Springer.

    Chapter  Google Scholar 

  • Chazan, Daniel. 1990. Quasi-empirical views of mathematics and mathematics teaching. Interchange 21(1): 14–23.

    Article  Google Scholar 

  • Chazan, Daniel, and Dara Sandow. 2011. Why did you do that? Reasoning in algebra classrooms. Mathematics Teacher 104(6): 461–464.

    Google Scholar 

  • Coleman, Edwin. 2009. The surveyability of long proofs. Foundations of Science 14(1–2): 27–43.

    Article  Google Scholar 

  • Douek, Nadia. 2005. Communication in the mathematics classroom. Argumentation and development of mathematical knowledge. In Challenging perspectives on mathematics classroom communication, eds. Anna Chronaki and Iben Maj Christiansen, 145–172. Charlotte, NC: Information Age.

  • Douek, Nadia, and Ezio Scali. 2000. About argumentation and conceptualisation. In Proceedings of PME-XXIV, Vol. 2, eds. Tadao Nakahara and Masataka Koyama, 249–256. Hiroshima: Hiroshima University.

  • Dove, Ian J. 2009. Toward a theory of mathematical argument. Foundations of Science 14(1–2): 137–152.

    Article  Google Scholar 

  • Duval, Raymond. 1991. Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics 22(3): 233–261.

    Article  Google Scholar 

  • Eemeren, Frans H. van, and Peter Houtlosser. 2005. Theoretical construction and argumentative reality: An analytic model of critical discussion and conventionalised types of argumentative activity. In The Uses of Argument. Proceedings of a Conference at McMaster University 18–21 May 2005, eds. David Hitchcock and Daniel Farr, 75–84. Hamilton: Ontario Society for the Study of Argumentation.

  • Fallis, Don. 2003. Intentional gaps in mathematical proofs. Synthese 143: 45–69.

    Article  Google Scholar 

  • Ferreirós, José. 2010. Mathematical knowledge and the interplay of practices. In EPSA philosophical issues in science, ed. Mauricio Suárez, Mauro Dorato, and Miklós Rédei, 55–64. Dordrecht: Springer.

    Chapter  Google Scholar 

  • Gabbay, Dov, and John Woods. 2005. The practical turn in logic. In Handbook of philosophical logic, vol. 13, ed. Dov Gabbay, and Frans Guenthner, 15–122. The Netherlands: Springer.

    Chapter  Google Scholar 

  • Hanna, Gila. 2000. Proof, explanation and exploration: An overview. Educational Studies in Mathematics 44: 5–23.

    Article  Google Scholar 

  • Healy, Lulu, and Celia Hoyles. 2000. A study of proof conceptions in algebra. Journal for Research in Mathematics Education 31(4): 396–428.

    Article  Google Scholar 

  • Hitchcock, David. 2002. The practice of argumentative discussion. Argumentation 16(3): 287–298.

    Article  Google Scholar 

  • Johnson, Ralph H. 2000. Manifest rationality. A pragmatic theory of argument. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Johnson, Ralph H. 2005. Theory and practice again: Challenges from Pinto and Toulmin. In The uses of argument: Proceedings of a conference at McMaster University, 18–21 May 2005, ed. David Hitchcock, and Daniel Farr, 222–231. Hamilton: Ontario Society for the Study of Argumentation.

    Google Scholar 

  • Kerber, Manfred, and Martin Pollet. 2007. Informal and formal representations in mathematics. Studies in Logic, Grammar and Rhetoric 10(23): 75–94.

    Google Scholar 

  • Krabbe, Erik C.W. 2008. Strategic maneuvering in mathematical proofs. Argumentation 22: 453–468.

    Article  Google Scholar 

  • Kuhn, Deanna. 1992. Thinking as argument. Harvard Educational Review 62(2): 155–178.

    Article  Google Scholar 

  • Kvernbekk, Tone. 2012. Argumentation in theory and practice: Gap or equilibrium? Informal Logic 32(3): 288–305.

    Google Scholar 

  • Lakatos, Imre. 1976. Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Larvor, Brendan. 2013. What philosophy of mathematical practice can teach argumentation theory about diagrams and pictures. In The argument of mathematics, ed. Andrew Aberdein, and Ian J. Dove, 239–253. Dordrecht: Springer.

    Chapter  Google Scholar 

  • Mancosu, Paolo (ed.). 2008. The philosophy of mathematical practice. Oxford: Oxford University Press.

    Google Scholar 

  • Mancosu, Paolo. 2011. Explanation in Mathematics. In The stanford encyclopedia of philosophy (Summer 2011 Edition), ed. Edward N. Zalta. http://plato.stanford.edu/archives/sum2011/entries/mathematics-explanation/

  • Maxwell, Edwin A. 1959. Fallacies in mathematics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Muller Mirza, Nathalie, and Anne-Nelly Perret-Clermont (eds.). 2009. Argumentation and education: Theoretical foundations and practices. Dordrecht: Springer.

    Google Scholar 

  • Pease, Alison, and Ursula Martin. 2012. Seventy four minutes of mathematics: An analysis of the third mini-polymath project. In Symposium on mathematical practice and cognition II: A symposium at the AISB/IACAP World Congress 2012, ed. Alison Pease, and Larvor Brendan, 19–29. Birmingham: Society for the Study of Artificial Intelligence and the Simulation of Behaviour.

    Google Scholar 

  • Pinto, Robert C. 2001. Argument, inference and dialectic. Dordrecht: Kluwer Academic Publishers.

    Book  Google Scholar 

  • Pólya, George. 1945. How to solve it. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Pólya, George. 1954. Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Prusak, Naomi. 2007. The analysis of dialogues of dyads of pre-service teachers operating in an activity designed to foster argumentation: The case of dynamic geometry. Unpublished Master thesis, Jerusalem: The Hebrew University.

  • Schwarz, Baruch B. 2009. Argumentation and learning. In Argumentation and education: Theoretical foundations and practice, eds. Nathalie Muller Mirza and Anne-Nelly Perret-Clermont, 91–126. Dordrecht: Springer.

  • Schwarz, Baruch B., Naomi Prusak, and Rina Hershkowitz. 2010. Argumentation and mathematics. In Educational dialogues: Understanding and promoting productive interaction, ed. Karen Littleton, and Christine Howe, 103–127. London: Routledge.

    Google Scholar 

  • Tao, Terry. 2011. The Mini-polymath3 Project: 2011 IMO. http://polymathprojects.org/2011/07/19/minipolymath3-project-2011-imo/. Accessed 24 Feb 2013.

  • Thurston, William P. 1994. On proof and progress in mathematics. Bulletin of the American Mathematical Society 30: 161–177.

    Article  Google Scholar 

  • Tindale, Christopher W. 2002. A concept divided: Ralph Johnson’s definition of argument. Argumentation 16: 209–309.

    Article  Google Scholar 

  • Toulmin, Stephen E. 1958. The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  • Toulmin, Stephen, Richard Rieke, and Allan Janik. 1979. An introduction to reasoning. London: Macmillan.

    Google Scholar 

  • Van Bendegem, Jean Paul, and Bart Van Kerkhove. 2009. Mathematical arguments in context. Foundations of Science 14(1–2): 45–57.

  • van Eemeren, Frans H., et al. 1996. Fundamentals of argumentation theory. A handbook of historical backgrounds and contemporary developments. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Vega Reñón, Luis. 1999. Hacer ver, hacer saber (El rigor informal de las pruebas matemáticas clásicas). In X Congreso nacional de filosofía de la Asociación filosófica argentina. Universidad nacional de Córdoba.

  • Walton, Douglas N., Christopher Reed, and Fabrizio Macagno. 2008. Argumentation schemes. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Walton, Douglas N., and Erik C.W. Krabbe. 1995. Commitment in dialogue: Basic concepts of interpersonal reasoning. Albany, NY: State University of New York Press.

    Google Scholar 

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Acknowledgments

This work was partially supported by the FFI 2010-20118 Research Project of the Spanish Ministry of Economy and Competitiveness.

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  1. Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, Avda. de Tolosa 70, 20018, Donostia-San Sebastián, Spain

    Begoña Carrascal

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Carrascal, B. Proofs, Mathematical Practice and Argumentation. Argumentation 29, 305–324 (2015). https://doi.org/10.1007/s10503-014-9344-0

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