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Abductive Arguments Supporting Students’ Construction of Proofs

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Handbook of Abductive Cognition

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Abstract

Research studies show that it is not easy for the instructor to modify students’ argumentations based on conceptions that hardly evolve into theorems. Extending these studies, this chapter shows that abductive arguments can be effectively used by the instructor to support students in completing their argumentations, when they are struggling in solving a problem or they are using incorrect rules to solve it. Test results drawn from 60 undergraduate students who solved two algebraic problems, a factorization problem and a system of linear equations, are presented. Students who provided incomplete or incorrect solutions to one of these problems were selected for a one-on-one meeting with the instructor. The analysis of three cases shows how an instructor’s abductive argument can help students recognize their mistakes and modify their argumentations in solving these problems. Toulmin’s model is used to analyze the interaction between a student’s argumentation and an instructor’s intervention to show that instructors’ abductive arguments should be appropriately constructed in ways that support cognitive unity between students’ argumentation and proof.

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References

  • Arzarello, F., Micheletti, C., Olivero, F., & Robutti, O. (1998). A model for analyzing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the twentieth-second annual conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 24–31). Stellenbosch, South Africa.

    Google Scholar 

  • Balacheff, N. (2009). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 115–135). Springer.

    Google Scholar 

  • Boero, P., Garuti, R., & Mariotti M. A. (1996). Some dynamic mental processes underlying producing and proving conjectures. In L. Puig & A. Gutierrez (Eds.), Proceedings of the twentieth conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 121–128). Valencia.

    Google Scholar 

  • Bonfantini, M., & Proni, G. (1983). To guess or not to guess. In U. Eco & T. Sebeok (Eds.), The sign of three: Dupin, Holmes, Peirce (pp. 119–134). Indiana University Press.

    Google Scholar 

  • Cifarelli, V. (1999). Abductive inference: connections between problem posing and solving. In O. Zaslavsky (Ed.), Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 217–224). Haifa, Israel.

    Google Scholar 

  • Cifarelli, V., & Sáenz-Ludlow, A. (1996). Abductive processes and mathematics learning. In E. Jakubowski, D. Watkins, & H. Biske (Eds.), Proceedings of the eighteenth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 161–166). ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

    Google Scholar 

  • Eco, U. (1983). Horns, Hooves, Insteps: Some hypotheses on three types of abduction. In U. Eco & T. Sebeok (Eds.), The sign of three: Dupin, Holmes, Peirce (pp. 198–220). Indiana University Press.

    Google Scholar 

  • Ferrando, E. (2006). The Abductive System. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the thirtieth conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 57–64). PME.

    Google Scholar 

  • Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems. In L. Puig & A. Gutierrez (Eds.), Proceedings of the twentieth conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 113–120). Valencia.

    Google Scholar 

  • Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulty of proof. In A. Olivier & K. Newstead (Eds.), Proceedings of the twentieth-second annual conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 345–352). Stellenbosch, South Africa.

    Google Scholar 

  • Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3–21.

    Article  Google Scholar 

  • Knipping, C. (2003). Argumentation structures in classroom proving situations. In M. A. Mariotti (Ed.), Proceedings of the third conference of the European Society in Mathematics Education (unpaginated). Retrieved from http://ermeweb.free.fr/CERME3/Groups/TG4/TG4_Knipping_cerme3.pdf

  • Knipping, C. (2008). A method for revealing structures of argumentation in classroom proving processes. ZDM – The International Journal on Mathematics Education, 40(3), 427–441.

    Article  Google Scholar 

  • Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. The Journal of Mathematical Behavior, 26(1), 60–82.

    Article  Google Scholar 

  • Lavy, I. (2006). A case study of different types of arguments emerging from explorations in an interactive computerized environment. The Journal of Mathematical Behavior, 25, 153–169.

    Article  Google Scholar 

  • Magnani, L. (2001). Abduction, reason and science: Processes of discovery and explanation. Kluwer Academic Publishers.

    Book  Google Scholar 

  • Magnani, L. (2004). Conjectures and manipulations: Computational modeling and the extra-theoretical dimension of scientific discovery. Minds and Machines, 14, 507–537.

    Article  Google Scholar 

  • Mason, J. (1996). Abduction at the heart of mathematical being. In E. Gray (Ed.), Thinking about mathematics & Music of the spheres: Papers presented for the inaugural lecture of Professor David Tall (pp. 34–40). Mathematics Education Research Centre.

    Google Scholar 

  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structures for all. Mathematics Education Research Journal, 21(2), 10–32.

    Article  Google Scholar 

  • Nardi, E., Biza, I., & Zachariades, T. (2012). Warrant revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. Educational Studies in Mathematics, 79, 157–173.

    Article  Google Scholar 

  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66, 23–41.

    Article  Google Scholar 

  • Pedemonte, B. (2008). Argumentation and algebraic proof. ZDM – The International Journal on Mathematics Education, 40(3), 385–400.

    Article  Google Scholar 

  • Pedemonte, B. (2018). How can a teacher support students in constructing a proof? In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving. An international perspective (pp. 115–130). Springer. ISSN:2520-8322.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Pedemonte, B., & Reid, D. (2010). The role of abduction in proving processes. Educational Studies in Mathematics, 76(3), 281–303.

    Article  Google Scholar 

  • Peirce, C. S. (1878, August 13). Deduction, induction, and hypothesis. Popular Science Monthly, 470–482. (Compiled in Peirce, C. S., 1960, 2.619-644).

    Google Scholar 

  • Peirce, C. S. (1960). Collected papers. Harvard University Press.

    MATH  Google Scholar 

  • Polya, G. (1962). How to solve it? Princeton University Press (French translation Mesnage C. Comment poser et résoudre un problème. Dunod (Ed.), Paris).

    Google Scholar 

  • Rivera, F. (2017). Abduction and the emergence of necessary mathematical knowledge. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 441–457). Springer.

    Google Scholar 

  • Rivera, F., & Becker, J.-R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. The Journal of Mathematical Behavior, 26, 140–155.

    Article  Google Scholar 

  • Rivera, F., & Becker, J. R. (2016). Middle School Student’s patterning performance on semi-free generalization tasks. The Journal of Mathematical Behavior, 43, 53–69.

    Article  Google Scholar 

  • Sáenz-Ludlow, A. (2016). Abduction in proving. In A. Sáenz-Ludlow & G. Kadunz (Eds.), Semiotics as a tool for learning mathematics. Semiotic perspectives in the teaching and learning of mathematics series (pp. 155–179). Sense Publishers. https://doi.org/10.1007/978-94-6300-337-7_8

    Chapter  Google Scholar 

  • Son, J. W. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84, 49–70.

    Article  Google Scholar 

  • Tall, D. (1995). Cognitive development, representations & proof, justifying and proving in school mathematics (pp. 27–38). Institute of Education.

    Google Scholar 

  • Toulmin, S. E. (1958). The uses of argument. Cambridge University Press.

    Google Scholar 

  • Toulmin, S. E. (1993). Les usages de l’argumentation (P. De Brabanter, Trans.). Presse Universitaire de France.

    Google Scholar 

  • Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. The Journal of Mathematical Behavior, 24(3–4), 351–360.

    Article  Google Scholar 

  • Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proof. For the Learning of Mathematics, 25(1), 34–38.

    Google Scholar 

  • Wood, T. (1999). Creating a context for argument in Mathematics Class Young Children’s concepts of shape. Journal for Research in Mathematics Education, 30(2), 171–191.

    Article  Google Scholar 

  • Yackel, E. (2001). Explanation, Justification and argumentation in mathematics classrooms. In M. Van den Heuvel-Panhuizen (Eds.), Proceedings of the 25th conference of the international group for the psychology of mathematics education (vol. 1, pp. 1–9). Utrecht, Olanda.

    Google Scholar 

  • Yackel, E., & Rasmussen, C. (2002). Beliefs and norms in the mathematics classroom. In G. Toerner, E. Pehkonen, & G. Leder (Eds.), Mathematical beliefs and implications for teaching and learning. Kluwer.

    Google Scholar 

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Authors and Affiliations

  1. Department of Neurology, UCSF Dyslexia Center, UCSF Memory and Aging Center, San Francisco, CA, USA

    Bettina Pedemonte

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  1. Bettina Pedemonte
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Correspondence to Bettina Pedemonte .

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Editors and Affiliations

  1. Department of Humanities, University of Pavia, Pavia, Italy

    Lorenzo Magnani

Section Editor information

  1. Department of Mathematics and Statistics, San Jose State University, San Jose, CA, USA

    Ferdinand Rivera

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Pedemonte, B. (2022). Abductive Arguments Supporting Students’ Construction of Proofs. In: Magnani, L. (eds) Handbook of Abductive Cognition. Springer, Cham. https://doi.org/10.1007/978-3-030-68436-5_72-1

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  • DOI: https://doi.org/10.1007/978-3-030-68436-5_72-1

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  • Print ISBN: 978-3-030-68436-5

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