Advertisement

Proof Technology and Learning in Mathematics: Common Issues and Perspectives

  • Nicolas BalacheffEmail author
  • Thierry Boy de la Tour
Chapter
  • 107 Downloads
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 14)

Abstract

Mathematical proof is undoubtedly the cornerstone of mathematics. Indeed, no mathematical work is definitively complete without the final QED. Mathematics educators know this centrality of proof, the challenges it presents in terms of teaching, and the complexity of its learning.

References

  1. Aleven, V. (2010). Rule-based cognitive modeling for intelligent tutoring systems. In R. Nkambou, J. Bourdeau, & R. Mizoguchi (Eds.), Advances in intelligent tutoring systems (Vol. 308, pp. 33–62). Berlin, Heidelberg: Springer.  https://doi.org/10.1007/978-3-642-14363-2_3.Google Scholar
  2. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA, USA: Harvard University Press.Google Scholar
  3. Anderson, J. R., Boyle, C. F., & Yost, G. (1988). The geometry proof tutor (Advanced Computer Tutoring Project). Carnegie-Mellon University, Pittsburgh, PA 15213. Retrieved from http://act-r.psy.cmu.edu/wordpress/wp-content/uploads/2012/12/124GeoTutor.ABYost.pdf.
  4. Anderson, J. R., Corbett, A. T., Koedinger, K. R., & Pelletier, R. (1995). Cognitive tutors: Lessons learned. Journal of the Learning Sciences, 4(2), 167–207.  https://doi.org/10.1207/s15327809jls0402_2.CrossRefGoogle Scholar
  5. Annals of Mathematics. (n.d.). Statement by the editors on computer-assisted proofs. Retrieved 22 January, 2019, from http://annals.math.princeton.edu/board.
  6. Assaf, A., Burel, G., Cauderlier, R., Delahaye, D., Dowek, G., Dubois, C., … Saillard, R. (2016). Expressing theories in the λΠ-calculus modulo theory and in the Dedukti system. Presented at the 22nd International Conference on Types for Proofs and Programs (TYPES 2016), Novi Sad, Serbia: Springer.Google Scholar
  7. 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.  https://doi.org/10.1007/s10758-010-9169-3.CrossRefGoogle Scholar
  8. Balacheff, N. (2010). Bridging knowing and proving in mathematics: An essay from a didactical perspective. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 115–135). Berlin Heidelberg: Springer.CrossRefGoogle Scholar
  9. Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., et al. (2018). The role of the Mizar mathematical library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1–4), 9–32.  https://doi.org/10.1007/s10817-017-9440-6.CrossRefGoogle Scholar
  10. Boero, P., Dapueto, C., Ferrari, P., Ferrero, E., Garuti, R., Lemut, E., … Scali, E. (1995). Aspects of the mathematics—Culture relationship in mathematics teaching-learning in compulsory school. In L. Meira & D. Carraher (Eds.), Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education (17 pp.). Recife. Retrieved from http://didmat.dima.unige.it/progetti/COFIN/biblio/art_boero/boero%26c_PME_XIX.pdf.
  11. Caferra, R., Peltier, N., & Puitg, F. (2001). Emphasizing human techniques in automated geometry theorem proving: A practical realization. In J. Richter-Gebert & D. Wang (Eds.), Presented at the Workshop on Automated Deduction in Geometry, Zurich, Switzlerland (Vol. LNAI 2061, pp. 268–305). Berlin, Heidelberg: Springer. Retrieved from https://link-springer-com.gaelnomade-1.grenet.fr/content/pdf/10.1007%2F3-540-45410-1.pdf.
  12. Cobo, P., Fortuny, J. M., Puertas, E., & Richard, P. R. (2007). AgentGeom: A multiagent system for pedagogical support in geometric proof problems. International Journal of Computers for Mathematical Learning, 12(1), 57–79.  https://doi.org/10.1007/s10758-007-9111-5.CrossRefGoogle Scholar
  13. Dawkins, P. C., & Weber, K. (2016). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics, 95(2), 123–142.  https://doi.org/10.1007/s10649-016-9740-5.CrossRefGoogle Scholar
  14. Duval, R. (1992). Argumenter, prouver, expliquer: continuité ou rupture cognitive? Petit x, 31, 37–61.Google Scholar
  15. 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. 137–161). Sense Publishers.Google Scholar
  16. Ebner, G., Hetzl, S., Leitsch, A., Reis, G., & Weller, D. (2018). On the generation of quantified lemmas. Journal of Automated Reasoning, 1–32.  https://doi.org/10.1007/s10817-018-9462-8.CrossRefGoogle Scholar
  17. Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulties of proof. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 345–352). Stellenbosch (SA). Retrieved from http://www.mat.ufrgs.br/~portosil/garuti.html.
  18. Gelernter, H. (1959). Realization of a geometry-theorem proving machine. In J. H. Siekmann & G. Wrightson (Eds.), Automation of reasoning (pp. 99–122). Berlin, Heidelberg: Springer.  https://doi.org/10.1007/978-3-642-81952-0_8.CrossRefGoogle Scholar
  19. Goldstein, I. (1973). Elementary geometry theorem proving (AIM No. 280) (p. 46). MIT AI Laboratory. Retrieved from https://dspace.mit.edu/bitstream/handle/1721.1/5798/AIM-280.pdf?sequence=2.
  20. Hähnle, R. (2001). Tableaux and related methods. In A. Robinson & A. Voronkov (Eds.), Handbook of automated reasoning (Vol. 1, pp. 101–178). Elsevier Science B.V.Google Scholar
  21. Hanna, G. (2017). Connecting two different views of mathematical explanation. In Enabling mathematical cultures. Mathematical Institute, University of Oxford. Retrieved from https://enablingmaths.wordpress.com/abstracts/.
  22. Harrison, J., Urban, J., & Wiedijk, F. (2014). History of interactive theorem proving. in handbook of the history of logic (Vol. 9, pp. 135–214). Elsevier.  https://doi.org/10.1016/B978-0-444-51624-4.50004-6.CrossRefGoogle Scholar
  23. Hauer, B., Kovács, Z., Recio, T., & Vélez, P. (2018). Automated reasoning in elementary geometry: Towards inquiry learning. Pädagogische Horizonte, 2(2), 14.Google Scholar
  24. Healy, L., Hoelzl, R., Hoyles, C., & Noss, R. (1994). Messing up. Micromath, 10(1), 14–17.Google Scholar
  25. Herbst, P., & Balacheff, N. (2009). Proving and knowing in public: The nature of proof in a classroom. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 40–63). New York: Routledge.Google Scholar
  26. Heule, M. J. H., Kullmann, O., & Marek., V. W. (2016). Solving and Verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer (Vol. LNCS 9710, pp. 228–245). Presented at the SAT 2016. Springer.  https://doi.org/10.1007/978-3-319-40970-2_15.CrossRefGoogle Scholar
  27. Koedinger, K., & Anderson, J. R. (1990). Theoretical and Empirical Motivations for the Design of ANGLE: A New Geometry Learning Environment. Presented at the Knowledge-Based Environments for Learning and Teaching, Standford University. Retrieved from http://pact.cs.cmu.edu/pubs/Koedinger,%20Anderson%20-90.pdf.
  28. Kortenkamp, U., & Richter-Gebert, J. R. (2004). Using automatic theorem proving to improve the usability of geometry software. In Proceedings of MathUI 2004 (p. 12). Retrieved from https://pdfs.semanticscholar.org/8892/faa455ea7442438d3f126bd05ba4d8c51e81.pdf.
  29. Laborde, J.-M. (1990). Cabri-géomètre - Manuel de l’utilisateur.Google Scholar
  30. Leduc, N. (2016). QED-Tutrix: Système tutoriel intelligent pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane. Montréal: Université de Montréal.Google Scholar
  31. Luengo, V. (1997). Cabri-Euclide : un micromonde de preuve intégrant la réfutation. Université Joseph Fourier (Grenoble 1), Grenoble. Retrieved from https://www.researchgate.net/publication/34765259_Cabri-euclide_un_micromonde_de_preuve_integrant_la_refutation_principes_didactiques_et_informatiques_Realisation.
  32. Luengo, V. (1999). Semi-empirical agent to learn mathematical proof. In Proceedings of Artificial Intelligence in education (AIED 99) (p. 10). Le Mans, France: Amsterdam: IOS.Google Scholar
  33. Matsuda, N., & VanLehn, K. (2004). GRAMY: A geometry theorem prover capable of construction. Journal of Automated Reasoning, 32(1), 3–33.  https://doi.org/10.1023/B:JARS.0000021960.39761.b7.CrossRefGoogle Scholar
  34. Nevins, A. J. (1974). Plane geometry theorem proving using forward chaining (AIM No. 303) (p. 35). MIT AI Laboratory. Retrieved from https://dspace.mit.edu/bitstream/handle/1721.1/6218/AIM-303.pdf?sequence=2.
  35. Newell, A., Shaw, & Simon, H. (1959). Report on a general problem-solving program (p. 27). RAND Corporation. Retrieved from http://bitsavers.trailing-edge.com/pdf/rand/ipl/P-1584_Report_On_A_General_Problem-Solving_Program_Feb59.pdf.
  36. Newell, A., & Simon, H. (1956). The logic theory machine—A complex information processing system (No. P-868) (p. 40). The Rand Corporation. Retrieved from http://shelf1.library.cmu.edu/IMLS/MindModels/logictheorymachine.pdf.
  37. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.  https://doi.org/10.1007/s10649-006-9057-x.CrossRefGoogle Scholar
  38. Polya, G. (1945). How to solve it. Princeton University Press. Retrieved from https://press.princeton.edu/titles/669.html.
  39. Sinclair, N., Bartolini Bussi, M. G., 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 G. Kaiser (Ed.), Proceedings of the 13th International Congress on Mathematical Education (pp. 277–287). Cham: Springer International Publishing.  https://doi.org/10.1007/978-3-319-62597-3_18.CrossRefGoogle Scholar
  40. Soury-Lavergne, S. (Ed.). (2003). Baghera an hybrid and emergent educational society (Cahier du laboratoire Leibniz No. 81). Laboratoire Leibniz - IMAG.Google Scholar
  41. Stefferud, E. (1963). The logic theory machine: A model of heuristic program (Memorandum No. RM-3731-CC) (198 pp.). The Rand Corporation. Retrieved from https://history-computer.com/Library/Logic%20Theorist%20memorandum.pdf.
  42. Takeuti, G. (1975). Proof Theory. Amsterdam: North Holland.Google Scholar
  43. Trilling, L. (1996). Rétrospective sur le projet Mentoniezh. Sciences et Technologies de l’Information et de la Communication pour l’Éducation et la Formation, 3(2), 157–162.  https://doi.org/10.3406/stice.1996.1294.CrossRefGoogle Scholar
  44. Wang, K., & Su, Z. (2017). Interactive, intelligent tutoring for auxiliary constructions in geometry proofs. arXiv:1711.07154 [Cs, Math]. Retrieved from http://arxiv.org/abs/1711.07154.
  45. Webber, C., Pesty, S., & Balacheff, N. (2002). A multi-agent and emergent approach to student modelling. In F. van Harmelen (Ed.), 15th European Conference on Artificial Intelligence (ECAI 2002) (pp. 98–102). IOS Press. Retrieved from https://telearn.archives-ouvertes.fr/hal-00003043.
  46. Weiss, M., Herbst, P., & Chen, C. (2009). Teachers’ perspectives on “authentic mathematics” and the two-column proof form. Educational Studies in Mathematics, 70(3), 275–293.  https://doi.org/10.1007/s10649-008-9144-2.CrossRefGoogle Scholar
  47. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.  https://doi.org/10.2307/749877.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, CNRS, Grenoble INP, LIGGrenobleFrance

Personalised recommendations