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Compartmentalisation of Mathematical Sectors: The Case of Continuous Probability Distributions and Integrals | SpringerLink

Compartmentalisation of Mathematical Sectors: The Case of Continuous Probability Distributions and Integrals


This paper investigates the phenomenon of compartmentalisation of knowledge in the teaching and learning of continuous probability distributions and integral calculus at the secondary-tertiary transition in France. Using the Anthropological Theory of the Didactic (ATD), and in particular the key notion of praxeology, we investigate in which sense those two sectors may be described as compartmentalised in current textbooks. We then study, by means of a questionnaire, the educational effects of the compartmentalisation: do students’ difficulties in completing “bridging tasks” (tasks that require to relate the two sectors) reflect the partial disconnections revealed by the praxeological analyses? The key notion of ostensive, combined with the role played by the technology in the sense of ATD, is used to interpret the data. Altogether, this study sheds light on the deficit of cognitive flexibility required to change mathematical sectors, which is understood as a result of deficient praxeologies developed within the institutions.

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    This theoretical framework puts into the fore the notion of institution and its consequences on teaching and learning.

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    The syllabus makes the distinction between “fluctuation” and confidence intervals. In the former, the frequency is to be estimated from the probability which is given; in the latter, the situation is reversed.

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    Centre National de la Recherche Scientifique; https://www.insmi.cnrs.fr/

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    Didactique et Épistémologie des Mathématiques, liens avec l’Informatique et la Physique, dans le Supérieur; https://demips.math.cnrs.fr/


  1. Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: the case of limits of functions in spanish hish schools. Educational Studies in Mathematics, 59, 235–268.

    Article  Google Scholar 

  2. Batanero, C., Tauber, L., & Meyer, R. (1999). From data analysis to inference: a research project on the teaching of normal distributions. In Bulletin of the International Statistical Institute: Proceedings of the Fifty-Second Session of the International Statistical Institute (Tome LVIII, Book 1) (pp. 57–58). Helsinki: Internatio.

  3. Batanero, C., Tauber, L.M., & Sánchez, V. (2004). Students’ reasoning about the normal distribution. In Kluwer, B.-Z.D., & Garfield, J. (Eds.) The Challenge of Developing Statistical Literacy, Reasoning and Thinking. http://link.springer.com/content/pdf/10.1007/1-4020-2278-6_11.pdf (pp. 257–276). Dordrecht: Springer.

  4. Bosch, M., & Chevallard, Y. (1999). La sensibilité de l’activité mathématique aux ostensifs. Recherches en Didactique des Mathématiques, 19(1), 77–124.

    Google Scholar 

  5. Bosch, M., & Gascón, J. (2014). Introduction to the Anthropological Theory of the Didactic. In Bikner-Ahsbahs, A., & Prediger, S. (Eds.) Networking Theories as a Research Practice in Mathematics Education. Springer (pp. 67–83).

  6. Bourbaki, N. (2007). Éléments d’histoire des mathématiques. Springer.

  7. Derouet, C., & Parzysz, B. (2016). How can histograms be useful for introducing continuous probability distributions?. ZDM - Mathematics Education, 48(6), 757–773.

    Article  Google Scholar 

  8. Derouet, C., Planchon, P., Hausberger, T., & Hochmuth, R. (2018). Bridging probability and calculus: the case of continuous distributions and integrals at the secondary-tertiary transition. In Durand-Guerrier, V., Hochmuth, R., Goodchild, S., & Hogstad, N.M. (Eds.) Proceedings of the Second Conference of the International Network for Didactic Research in University Mathematics (pp. 497–506). Kristiansand: University of Agder and INDRUM.

  9. Derouet, C. (2019). Introduire la notion de fonction de densité de probabilité : dynamiques entre trois domaines mathématiques. Recherches en Didactique des Mathématiques, 39(2), 213–266.

    Google Scholar 

  10. Dieudonné, J. (1992). Abrégé d’histoire des mathématiques. Hermann.

  11. Eraslan, A. (2007). The notion of compartmentalization: the case of Richard. Journal of Mathematical Education in Science and Technology, 38(8), 1065–1073.

    Article  Google Scholar 

  12. Florensa, I., Bosch, M., & Gascón, J. (2015). The epistemological dimension in didactics: Two problematic issues. In Krainer, K., & Vondrová, N. (Eds.) Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education. Prague: Faculty of Education, Charles University (pp. 2635–2641).

  13. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.

    Google Scholar 

  14. Jahnke, H.N. (2003). A history of analysis. American Mathematical Society.

  15. Kondratieva, M., & Winsløw, C. (2018). Klein’s plan b in the early teaching of analysis: Two theoretical cases of exploring mathematical links. International Journal of research in Undergraduate Mathematics Education, 4(1), 119–138.

    Article  Google Scholar 

  16. Kouropatov, A., & Dreyfus, T. (2013). Constructing the integral concept on the basis of the idea of accumulation: Suggestion for a high school curriculum. International Journal of Mathematical Education in Science and Technology, 44, 641–651.

    Article  Google Scholar 

  17. Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM - Mathematics Education, 46(4), 533–548.

    Article  Google Scholar 

  18. Mandl, H., Gruber, H., & Renkl, A. (1993). Misconsceptions and knowledge compartmentalization. In Wender, G., & Strube, F.K. (Eds.), The cognitive psychology of knowledge (pp. 161–176). Amsterdam: North-Holland.

  19. Michel, A. (1992). Constitution de la théorie moderne de l’intégration. Paris: Vrin.

  20. Pfannkuch, M., & Reading, C. (2006). Reasoning about distribution: A complex process. Statistics Education Research Journal, 5(2), 4–9. http://www.stat.auckland.ac.nz/~iase/serj/SERJ5(2).pdf#page=7.

    Google Scholar 

  21. Rosenthal, B. (1992). Discovering and experiencing the fundamental theorem of calculus. PRIMUS: Problems, Resources, and Issues in Undergraduate Studies, 2(2), 131–154.

    Article  Google Scholar 

  22. Schoenfeld, A.H. (1988). When good teaching leads to bad results: the disaster of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.

    Article  Google Scholar 

  23. Stigler, S.M. (1986). The history of statisitics: the measurement of unvertainty before 1900. Harvard University Press.

  24. Thompson, P.W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26(2-3), 229–274.

    Article  Google Scholar 

  25. Thompson, P.W., & Silverman, J. (2008). The concept of accumulation in calculus. In Carlson, M., & Rasmussen, C. (Eds.) Making the connection: Research and teaching in undergraduate mathematics (pp. 43–52). Washington: MAA.

  26. Verret, M. (1975). Le temps des études I. Paris: Librairie Honoré Champion.

    Google Scholar 

  27. Vinner, S., Hershkowitz, R., & Bruckheimer, M. (1981). Some cognitive factors as causes of mistakes in the addition of fractions. Journal for Research in Mathematics Education, 12, 70–76.

    Google Scholar 

  28. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.

    Article  Google Scholar 

  29. Wagner, J.F. (2018). Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral. International Journal of Research in Undergraduate Mathematics Education, 4(3), 327–356.

    Article  Google Scholar 

  30. Wijayanti, D. (2019). Analysing textbook treatment of similarity in plane geometry. Annales de Didactique et des Sciences Cognitives, 24, 107–132.

    Google Scholar 

  31. Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171–202.

    Article  Google Scholar 

  32. Winsløw, C. (2015). Mathematical Analysis in High School: a Fundamental Dilemma. In Bergsten, C., & Sriraman, B. (Eds.) Refractions of Mathematics Education: Festschrift for Eva Jablonka (pp. 197–213). Charlotte: Information Age Publishing.

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This research grew out of a project initiated within a thematic working group of the CNRSFootnote 4 consortium DEMIPSFootnote 5 that federates university mathematics education research in France.

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Correspondence to Thomas Hausberger.

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Hausberger, T., Derouet, C., Hochmuth, R. et al. Compartmentalisation of Mathematical Sectors: The Case of Continuous Probability Distributions and Integrals. Int. J. Res. Undergrad. Math. Ed. (2021). https://doi.org/10.1007/s40753-021-00143-y

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  • Continuous probability distributions
  • Integral calculus
  • Secondary-tertiary transition
  • Compartmentalisation of knowledge
  • Anthropological theory of the didactic