, Volume 48, Issue 6, pp 861–874 | Cite as

Understanding the development of mathematical work in the context of the classroom

  • Alain Kuzniak
  • Assia NechacheEmail author
  • J. P. Drouhard
Original Article


According to our approach to mathematics education, the optimal aim of the teaching of mathematics is to assist students in achieving efficient mathematical work. But, what does efficient exactly mean in that case? And how can teachers reach this objective? The model of Mathematical Working Spaces with its three dimensions—semiotic, instrumental, discursive—allows us to address these questions in an original way based on a multidimensional approach to the use of tools and instruments and on the notion of complete mathematical work. The Mathematical work is considered complete when a genuine relationship exists between epistemological and cognitive aspects, and when the three dimensions of the model are appropriately articulated. Two teaching situations in probability for Grades 9 and 10 (age 14 and 15) are used to illustrate how the model can help identify either misunderstandings that are not acknowledged by the teacher, or complete mathematical work despite some differences between intended and actual work.


Mathematical work Tools and instruments Mathematical Working Spaces Probability 


  1. Aimani, S., Bonneval, L.-M., Devynck, J.-B., & Yvonnet, P. (2012). Transmath Terminale S. Paris: Nathan.Google Scholar
  2. Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique Des Mathematiques, 1970–1990. Dordrecht: Kluwer.Google Scholar
  3. Carranza, P. (2011). Dualité dans l’enseignement de la probabilité. Apport pour l’enseignement de la statistique. Recherches en Didactique des Mathématiques, 31(2), 229–259.Google Scholar
  4. Caveing, M. (2004). Le problème des objets dans la pensée mathématique. Paris: Vrin.Google Scholar
  5. Douady, R. (1991). Tool, object, setting, window: elements for analysing and constructing didactical situations in mathematics. In A. J. Bishop & S. Melling Olsen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 100–130). Dordrecht: Kluwer.Google Scholar
  6. Drouhard, J.-P. (2009). Epistemography and algebra. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME6. Lyon, France.
  7. Dupuis, C., & Rousset-Bert, S. (1996). Arbres et tableaux de probabilités: analyse en terme de registre de représentation. Repères Irem, 22, 51–72.Google Scholar
  8. Engeström, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit.Google Scholar
  9. Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435.CrossRefGoogle Scholar
  10. Gomez-Chacon, I., & Kuzniak, A. (2015). Spaces for geometric work: Figural, instrumental, and discursive geneses of reasoning in a technological environment. International Journal of Science and Mathematics Education, 13, 201–226.CrossRefGoogle Scholar
  11. Heitele, D. (1975). An epistemological view: On fundamental stochastic ideas. Educational Studies in Mathematics, 6, 187–205.CrossRefGoogle Scholar
  12. Kuzniak, A. (2011). L’Espace de Travail Mathématique et ses genèses. Annales de Didactique et de Sciences Cognitives, 16, 9–24.Google Scholar
  13. Kuzniak, A. (2013). Teaching and learning of geometry and beyond…Plenary lecture. In B. Ubuz, Ç. Haser, & M. A. Mariotti (Eds.), Proceedings of CERME8 (pp. 33–49). Antalya, Turkey.Google Scholar
  14. Kuzniak, A. & Nechache, A. (2015). Using the geometric working spaces to plan a coherent teaching of geometry. In Proceedings of CERME9. Prague, Czech Republik (Forthcoming publication).Google Scholar
  15. Kuzniak, A., & Richard, P. (2014). Spaces for mathematical work: Viewpoints and perspectives. Relime, 17(4.1), 17–26.Google Scholar
  16. MENSEC (The Grade 10 Mathematics curriculum (2009). Grade 10 mathematics classes). Accessed July 2009.
  17. Montoya-Delgadillo, E., Mena-Lorca, A., & Mena-Lorca, J. (2014). Circulaciones y génesis en el Espacio de Trabajo Matematico. Relime, 17(4.1), 181–197.Google Scholar
  18. Nechache, A. (2015). Comparaison de la démarche de la validation dans les espaces de travail idoines en géométrie et en probabilité. In I. Gomez-Chacon (Ed.), Acte du quatrième symposium ETM (pp. 51–67). Madrid: IMI.Google Scholar
  19. Parzysz, B. (1993). Des statistiques aux probabilités. Exploitons les arbres. Repères Irem, 10, 93–104.Google Scholar
  20. Rabardel, P. (1995). Les hommes et les technologies. Une approche cognitive des instruments contemporains. Paris: Armand Colin.Google Scholar
  21. RESSEC (The official resources for Grade 6 to 9 classes Probabilities). Accessed Mars 2008.
  22. Serfati, M. (2010). Symbolic revolution, scientific revolution: mathematical and philosophical aspects. In A. Heeffer & M. Van Dyck (Eds.), Philosophical aspects of symbolic reasoning in early-modern mathematics (pp. 103–122). London: College Publications.Google Scholar
  23. Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification—The case of algebra. Educational Studies in Mathematics, 26, 191–228.CrossRefGoogle Scholar
  24. Sigward, E., Brisoux, F., Bruckner, C., & Monka, Y. (2014). Manuel de Seconde, Odyssée. Paris: Hatier.Google Scholar

Copyright information

© FIZ Karlsruhe 2016

Authors and Affiliations

  • Alain Kuzniak
    • 1
  • Assia Nechache
    • 1
    Email author
  • J. P. Drouhard
    • 2
  1. 1.Laboratoire de Didactique André RevuzUniversité Paris DiderotParisFrance
  2. 2.Universidad de Buenos AiresBuenos AiresArgentine

Personalised recommendations