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The Contribution of Information and Communication Technology to the Teaching of Proof

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Proof Technology in Mathematics Research and Teaching

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 14))

Abstract

This chapter discusses the use of a Dynamic Geometry Environment for fostering students’ introduction to mathematical proof. Grounded in the theory of semiotic mediation, it explores, on the one hand, the link between computational tools and the personal meanings emerging from their use in classroom activities and, on the other hand, the mathematical notions that are the object of instruction. The discussion uses three interrelated perspectives—the epistemological, the cognitive, and the didactic—to elaborate on findings from a number of longstanding teaching experiments in secondary school classrooms. Some illustrative examples are presented, drawn from research studies carried out in previous years and still in progress.

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Notes

  1. 1.

    A rich source of references can be found at http://lettredelapreuve.org.

  2. 2.

    Following Sinclair and Robutti (2012), I use the term “dynamic geometry environment.” As the authors write, since at least 1996, this term has been used over dynamic geometry software “to underscore the fact that we are dealing with microworlds (including pre-existing sketches and designed tasks) and not just a software program.” (p. 571).

  3. 3.

    The term artefact refers to any generic product of human culture purposefully designed to act or interact in a human setting.

  4. 4.

    An exception is that of mathematical induction. But mathematical induction is very rarely presented in comparison to other modalities of proving, which are commonly considered natural and spontaneous ways of reasoning.

  5. 5.

    Actually a DGE provides a larger set of tools, including for instance “measure of an angle,” “rotation of an angle,” and the like. This implies that the whole set of possible constructions does not coincide with that attainable only with ruler and compass. See Stylianides and Stylianides (2005) for a full discussion.

References

  • Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM: Mathematics Education, 40(3), 401–412.

    Article  Google Scholar 

  • Arsac, G., & Mante, M. (1983). Des “problème ouverts” dans nos classes du premier cycle. Petit x, 2, 5–33.

    Google Scholar 

  • Arsac, G. (1992). Initiation au raisonnement au college. Presse Universitaire de Lyon.

    Google Scholar 

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt für Didaktik der Mathematik, 34(3), 66–72.

    Article  Google Scholar 

  • Arzarello, F. (2007). The proof in the 20th century: From Hilbert to automatic theorem proving. In P. Boero (Ed.), Theorems in school from history and epistemology to cognitive and educational issues (pp. 43–64). Rotterdam: Sense Publishers.

    Google Scholar 

  • Arzarello, F. Bartolini Bussi, M. G., Leung, A. Y. L., Mariotti, M. A., & Stevenson, I. (2012) Experimental approaches to mathematical thinking: Artefacts and proof. In G. Hanna, & M. De Villier, Proof and proving in mathematics education (pp. 1–10). Springer, New ICMI Study Series, Volume 15, 2012.

    Google Scholar 

  • Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253.

    Article  Google Scholar 

  • Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2013). Reasoning by contradiction in dynamic geometry. PNA, 7(2), 63–73.

    Google Scholar 

  • Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2018). From pseudo-objects in dynamic explorations to proof by contradiction. Digital Experiences in Mathematics Education, 1–23. https://doi.org/10.1007/s40751-018-0039-2.

    Article  Google Scholar 

  • Balacheff, N. (2008). The role of the researcher’s epistemology in mathematics education: an essay on the case of proof. ZDM: The International Journal on Mathematics Education, 40, 501–512.

    Article  Google Scholar 

  • Bartolini Bussi, M.G. (1996). Mathematical discussion and perspective drawings in primary school. Education Studies in Mathematics 31, 11– 41.

    Article  Google Scholar 

  • Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education, Second revised edition (pp. 746–805). Lawrence Erlbaum, Mahwah, NJ.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (1999). About the generation of conditionality of statements and its links with proving. Proceedings of the Conference of the International Group for, 2, 137–144.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 247–262). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • de Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 369–394). Erlbaum, Mahwah.

    Google Scholar 

  • De Villiers, M. (2001). Papel e funcoes da demonstracao no trabalho com o Sketchpad. Educacao Matematica, 63, 31–36. (retrived online: https://mzone.mweb.co.za/residents/profmd/proofc.pdf).

  • Dreyfus, T., & Hadas, N. (1996). Proof as answer to the question why. Zentralblatt fur Didaktik der Mathematik/International Reviews on Mathematical Education, 28(1), 1–5.

    Google Scholar 

  • Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 138–162). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.

    Article  Google Scholar 

  • Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1–3), 127–150.

    Article  Google Scholar 

  • Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20–25.

    Google Scholar 

  • Hanna, G., & Jahnke, H. N. (2007). Proving and modelling. In W. Blum, P. L. Galbraith, H.W. Henn, & M. Niss (Eds.), Applications and modelling in mathematics education. The 14th ICMI study (pp. 145–152). Dordrecht: Springer.

    Google Scholar 

  • Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. Berlin: Springer.

    Google Scholar 

  • Hanna, G., & De Villiers, M. (Eds.). (2012). Proof and proving in mathematics education: The 19th ICMI study (Vol. 15). Springer Science & Business Media.

    Google Scholar 

  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schonfeld, J. Kaput, & E. Dubinsky (Eds.) Research in collegiate mathematics education III. (Issues in Mathematics Education, Vol. 7, pp. 234–282). American Mathematical Society.

    Google Scholar 

  • Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283–312.

    Article  Google Scholar 

  • Hölzl, R. (1996). How does “dragging” affect the learning of geometry. International Journal of Computer for Mathematical Learning, 1(2), 169–187.

    Article  Google Scholar 

  • Laborde, C. & Laborde, J. M. (1991). Problem solving in geometry: From microworlds to intelligent computer environments. In J. P. Ponte, J. F. Matos, & D. Fernandes (Eds.), Mathematical problem solving and new information technologies (pp. 177–192). NATO AS1 Series F, New York: Springer.

    Google Scholar 

  • Laborde, J. M., & Strässer, R. (1990). Cabri-géomètre: a microworld of geometry for guided discovery learning. Zentralblatt für Didaktik der Mathematik, 90(5), 171–177.

    Google Scholar 

  • Leron, U. (1985). A Direct approach to indirect proofs. Educational Studies in Mathematics, 16(3), 321–325.

    Article  Google 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, 145–165.

    Article  Google Scholar 

  • Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679.

    Article  Google Scholar 

  • Mariotti, M. A. (2001). Justifying and proving in the Cabri environment. International Journal of Computer for Mathematical Learning, 6(3), 257–281.

    Article  Google Scholar 

  • 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 (pp. 173–204). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Mariotti, M. A. (2007). Geometrical proof: The mediation of a microworld. In P. Boero (Ed.), Theorems in school: From history epistemology and cognition to classroom practice (pp. 285–304). Rotterdam, The Netherlands: Sense Publishers.

    Chapter  Google Scholar 

  • Mariotti, M. A. (2009). Artifacts and signs after a Vygotskian perspective: The role of the teacher. ZDM: The International Journal on Mathematics Education, 41, 427–440.

    Article  Google Scholar 

  • Mariotti, M. A. (2010). Proofs, semiotics and artefacts of information technologies. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 169–190). Springer.

    Google Scholar 

  • Mariotti, M. A. (2012). Proof and proving in the classroom: Dynamic geometry systems as tools of semiotic mediation. Research in Mathematics Education 14(2), 163–185 (2012).

    Article  Google Scholar 

  • Mariotti, M. A. (2014). Transforming images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), Transformation. A fundamental idea of mathematics education. Springer New York.

    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 Villers (Eds.), Proof and proving in mathematics education, ICMI study 19 conference proceedings (Vol. 2, pp. 82–87).

    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 PME-XXI, (Vol. 1, pp. 180–195). Lathi, Finland.

    Google Scholar 

  • Miyazaki, M., Fujita, T., & Jones, K. (2015). Flow-chart proofs with open problems as scaffolds for learning about geometrical proofs. ZDM: International Journal on Mathematics Education, 47(7), 1–14.

    Google Scholar 

  • Olivero, F. (2003). Proving within dynamic geometry environments, Doctoral Dissertation, Graduate School of Education, Bristol. https://telearn.archives-ouvertes.fr/file/index/docid/190412/filename/Olivero-f-2002.pdf.

  • Pedemonte, B. (2002) Etude didactique et cognitive des rapports de l’argumentationet de la demonstration en mathématiques, (Unpublished) Thèse de Doctorat, Université Joseph Fourier, Grenoble. http://tel.archives-ouvertes.fr/tel-00004579/.

  • Reid, D. A., & Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Rotterdam, The Netherlands: Sense Publisher.

    Book  Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic press.

    Google Scholar 

  • Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123–151.

    Article  Google Scholar 

  • Sierpinska, A. (2005). On practical and theoretical thinking. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign—Grounding mathematics education. Festschrift for Michael Otte (pp. 117–135) New York: Springer.

    Google Scholar 

  • Simon, M. A. (1996). Beyond inductive and deductive reasoning: The search for a sense of knowing. Educational Studies in Mathematics, 30(2), 197–209.

    Article  Google Scholar 

  • Sinclair, N., & Robutti, O. (2012). Technology and the role of proof: The case of dynamic geometry. In Third international handbook of mathematics education (pp. 571–596). New York, NY: Springer.

    Google Scholar 

  • Sinclair, N., Bussi, M. G. B., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2017). Geometry education, including the use of new technologies: A survey of recent research. In Proceedings of the 13th international congress on mathematical education (pp. 277–287). Cham: Springer.

    Google Scholar 

  • Stylianides, G. J., & Stylianides, A. J. (2005). Validation of solutions of construction problems in dynamic geometry environments. International Journal of Computers for Mathematical Learning, 10(1), 31–47.

    Article  Google Scholar 

  • Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In The second handbook of research on the psychology of mathematics education (pp. 315–351). Rotterdam: Sense Publishers.

    Google Scholar 

  • Thompson, D. R. (1996). Learning and teaching indirect proof. The Mathematics Teacher, 89(6), 474–482.

    Google Scholar 

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Authors and Affiliations

  1. University of Siena, Siena, Italy

    Maria Alessandra Mariotti

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  1. Maria Alessandra Mariotti
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  1. Ontario Institute for Studies in Education, University of Toronto, Toronto, ON, Canada

    Gila Hanna

  2. AG Didaktik, Fachbereich 3, Universität Bremen, Bremen, Germany

    David A. Reid

  3. Faculty of Education, Stellenbosch University, Stellenbosch, South Africa

    Michael de Villiers

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Mariotti, M.A. (2019). The Contribution of Information and Communication Technology to the Teaching of Proof. In: Hanna, G., Reid, D., de Villiers, M. (eds) Proof Technology in Mathematics Research and Teaching . Mathematics Education in the Digital Era, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-28483-1_8

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