Advertisement

Didactical Issues at the Interface of Mathematics and Computer Science

  • Viviane Durand-GuerrierEmail author
  • Antoine Meyer
  • Simon Modeste
Chapter
  • 99 Downloads
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 14)

Abstract

This contribution takes place in the context of a research project on the epistemological and didactical issues of interactions between mathematics and computer science. We make the hypothesis that, with the introduction of digital tools and computer science content in most curricula, significantly taking into account the epistemology of mathematics, computer science and their interactions is essential in order to tackle the challenges of mathematics and computer science education in the digital era. In view of this, addressing the question of proof in mathematics and computer science is a central didactical issue, which we examine in this contribution. We will elaborate on the links between the concepts of algorithm, proof, and program, and will argue for their significance in a general reflection on didactical issues in mathematics and computer science, in their teaching at high school and undergraduate levels.

Notes

Acknowledgements

Research funded by the french Agence Nationale pour la Recherche, project number <ANR-16-CE38-0006-01>.

References

  1. Aho, A. V., Sethi, R., & Ullman, J. D. (1986). Compilers, principles, techniques. Addison Wesley.Google Scholar
  2. Arora, S., & Barak, B. (2009). Computational complexity: A modern approach. Cambridge University Press.Google Scholar
  3. Artaud, M. (1998). Introduction à l’approche écologique du didactique - L’écologie des organisations mathématiques et didactiques. Actes de la IXème école d’été de didactique des mathématiques (pp. 101–139). Caen: ARDM & IUFM.Google Scholar
  4. Arzarello, F., Bussi, M. G. B., Leung, A. Y. L., Mariotti, M. A., & Stevenson, I. (2012). Experimental approaches to theoretical thinking: Artefacts and proofs. In Proof and proving in mathematics education (pp. 97–143). Springer.Google Scholar
  5. Balacheff, N. (2013). Open image in new window , a model to reason on learners’ conceptions. In PME-NA 2013-psychology of mathematics education (pp. 2–15). North American Chapter.Google Scholar
  6. Baron, G. L., & Bruillard, É. (2011). L’informatique et son enseignement dans l’enseignement scolaire général français: enjeux de pouvoir et de savoirs. In: Recherches et expertises pour l’enseignement scientifique (Vol. 1, pp. 79–90). De Boeck Supérieur.Google Scholar
  7. Basu, S., Pollack, R., & Roy, M. F. (2006). Algorithms in real algebraic geometry (2nd ed.). In Algorithms and computation in mathematics (Vol. 10 ). Berlin, New York: Springer.Google Scholar
  8. Bérard, B. (2001). Systems and software verification: Model checking techniques and tools. Springer.Google Scholar
  9. Borwein, J. M. (2012). Exploratory experimentation: Digitally-assisted discovery and proof. In Proof and proving in mathematics education (pp. 69–96). Springer.Google Scholar
  10. Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer Academic Publishers.Google Scholar
  11. Chabert, J. L. (1999). A history of algorithms from the pebble to the microchip. Springer.Google Scholar
  12. Colton, S. (2007). Computational discovery in pure mathematics. In Computational discovery of scientific knowledge (pp. 175–201). Springer.Google Scholar
  13. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms (3rd ed.). The MIT Press.Google Scholar
  14. Da Costa, N. C. A. (1997). Logiques classiques et non classiques: essai sur les fondements de la logique. Paris: Masson.Google Scholar
  15. Douady, R. (1986). Jeux de cadres et dialectique outil-objet. Recherches en didactique des mathématiques, 7(2), 5–31.Google Scholar
  16. Durand-Guerrier, V. (2008). Truth versus validity in mathematical proof. ZDM, 40(3), 373–384.CrossRefGoogle Scholar
  17. Durand-Guerrier, V., & Arsac, G. (2005). An epistemological and didactic study of a specific calculus reasoning rule. Educational Studies in Mathematics, 60(2), 149–172.CrossRefGoogle Scholar
  18. Frege, G. (1882). Über die wissenschaftliche Berechtigung einer Begriffsschrift. Zeitschrift für Philosophie und philosophische Kritik, 81, 48–56. (English translation in Conceptual notation and related articles. Clarendon Press (1972)).Google Scholar
  19. Godot, K., & Grenier, D. (2004). Research situations for teaching: A modelization proposal and examples. In Proceedings of ICME 10, IMFUFA, Roskilde University.Google Scholar
  20. Gravier, S., Payan, C., & Colliard, M. N. (2008). Maths à modeler: Pavages par des dominos. Grand N, 82, 53–68.Google Scholar
  21. Grenier, D., & Payan, C. (1998). Spécificité de la preuve et de la modélisation en mathématiques discrètes. Recherches en didactique des mathématiques, 18(2), 59–100.Google Scholar
  22. Gribomont, P., Ribbens, D., & Wolper, P. (2000). Logique, automates, informatique. In F. Beets & E. Gillet (Eds.), Logique en perspective: Mélanges offerts à Paul Gochet, Ousia (pp. 545–577).Google Scholar
  23. Gueudet, G., Bueno-Ravel, L., Modeste, S., & Trouche, L. (2017). Curriculum in France. A national frame in transition. In International perspectives on mathematics curriculum, research issues in mathematics education series. IAP.Google Scholar
  24. Hart, E. W. (1998). Algorithmic problem solving in discrete mathematics. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithm in school mathematics, 1998 NCTM Yearbook (pp. 251–267). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  25. Hoare, C. A. R. (1969). An axiomatic basis for computer programming. Communications of the ACM, 12(10), 576–580.CrossRefGoogle Scholar
  26. Hopcroft, J., Motwani, R., & Ullman, J. (2007). Introduction to automata theory, languages, and computation (3rd ed.). Addison-Wesley.Google Scholar
  27. Howson, A. G., & Kahane, J. P. (Eds.). (1986). The influence of computers and informatics on mathematics and its teaching, international commission on mathematical instruction edn. ICMI Study Series. Cambridge University Press.Google Scholar
  28. Kahane, J. P. (2002). Enseignement des sciences mathématiques : Commission de réflexion sur l’enseignement des mathématiques : Rapport au ministre de l’éducation nationale (cndp ed.). Paris: Odile Jacob.Google Scholar
  29. Lovász L (2007) Trends in mathematics: How they could change education? In Conférence européenne “The future of mathematics education in Europe”, Lisbonne.Google Scholar
  30. Meyer, A., & Modeste, S. (2018). Recherche binaire et méthode de dichotomie, comparaison et enjeux didactiques à l’interface mathématiques - informatique. In Proceedings of EMF, Paris, France (to appear).Google Scholar
  31. Modeste, S. (2012). Enseigner l’algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l’apprentissage de la preuve ? Ph.D. thesis, Université de Grenoble.Google Scholar
  32. Modeste, S. (2013). Modelling algorithmic thinking: The fundamental notion of problem. In Proceedings of CERME 8, Antalya (Turkey).Google Scholar
  33. Modeste, S. (2016). Impact of informatics on mathematics and its teaching. In F. Gadducci & M. Tavosanis (Eds.), History and philosophy of computing (pp. 243–255). Cham: Springer International Publishing.Google Scholar
  34. Morris, C. W. (1938). Foundations of the theory of signs. In International encyclopedia of unified science (pp. 1–59), Chicago University Press.Google Scholar
  35. Ouvrier-Buffet, C. (2014). Discrete mathematics teaching and learning. In Encyclopedia of mathematics education (pp. 181–186).Google Scholar
  36. Perrin, D. (2007). L’expérimentation en mathématiques. Petit x, 73, 6–34.Google Scholar
  37. Quine, W. V. (1950). Methods of logic. Harvard University Press.Google Scholar
  38. Reynolds, J. C. (1998). Theories of programming languages. Cambridge University Press.Google Scholar
  39. Sedgewick, R., & Flajolet, P. (2013). An introduction to the analysis of algorithms. Pearson Education.Google Scholar
  40. Sinaceur, H. (1991a). Corps Et Modèles: Essai Sur l’Histoire de l’Algèbre Réelle. Vrin.Google Scholar
  41. Sinaceur, H. (1991b). Logique: mathématique ordinaire ou épistémologie effective? In: Hommage à Jean-Toussaint Desanti, Trans-Europ-Repress.Google Scholar
  42. Straubing, H., & Weil, P. (2012). An introduction to finite automata and their connection to logic. In P. S. Deepak D’Souza (Ed.), Modern applications of automata theory (pp. 3–43). IISc Research Monographs: World Scientific.Google Scholar
  43. Tarski, A. (1933). The concept of truth in the languages of the deductive sciences. Prace Towarzystwa Naukowego Warszawskiego, Wydzial III Nauk Matematyczno-Fizycznych 34(13), 172–198 (English translation in Tarski (1983)).Google Scholar
  44. Tarski, A. (1936). On the concept of logical consequence. Przegla̧d Filozoficzny, 39, 58–68 (English translation in Tarski (1983), pp. 409–420)Google Scholar
  45. Tarski, A. (1941). Introduction to logic and to the methodology of the deductive sciences. Oxford University Press (reedited in Tarski (1995))Google Scholar
  46. Tarski, A. (1943). The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research, 4(3), 341–376.CrossRefGoogle Scholar
  47. Tarski, A. (1954). Contributions to the theory of models, I–II. Indagationes Mathematicae, 16, 572–588.CrossRefGoogle Scholar
  48. Tarski, A. (1955). Contributions to the theory of models, III. Indagationes Mathematicae, 17, 56–64.CrossRefGoogle Scholar
  49. Tarski, A. (1983). Logic, semantics, metamathematics: Papers from 1923 to 1938. Hackett (J. H. Woodger, trans. Introduction: J. Corcoran).Google Scholar
  50. Tarski, A. (1995). Introduction to logic and to the methodology of deductive sciences. New York: Dover Publications, INC. (unabridged Dover republication of the edition published by Oxford University Press, New York, 1946)Google Scholar
  51. Thomas, W. (1997). Languages, automata, and logic. In Handbook of formal languages (pp. 389–455). Springer.Google Scholar
  52. Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94.CrossRefGoogle Scholar
  53. Wittgenstein, L. (1921). Logisch-philosophische abhandlung. Annalen der Naturphilosophie, 14 (English translations in C. K. Ogden trans. Routledge & Kegan Paul (1922) and D. F. Pears and B. F. McGuinnes, trans. Routledge (1961)).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Viviane Durand-Guerrier
    • 1
    Email author
  • Antoine Meyer
    • 2
  • Simon Modeste
    • 1
  1. 1.IMAG, Univ Montpellier, CNRSMontpellierFrance
  2. 2.LIGM (UMR 8049), UPEM, CNRS, ESIEE, ENPC, Université Paris-EstMarne-la-ValléeFrance

Personalised recommendations