Teaching multidigit multiplication: combining multiple frameworks to analyse a class episode

Abstract

This paper provides an analysis of a teaching episode of the multidigit algorithm for multiplication, with a focus on the influence of the teacher’s mathematical knowledge on their teaching. The theoretical framework uses Mathematical Knowledge for Teaching, mathematical pertinence of the teacher and structuration of the milieu in a descending and ascending a priori analysis and an a posteriori analysis. This analysis shows a development of different didactical situations and some links between mathematical knowledge and pertinence. In the conclusion, the contribution of the frameworks from both French and Anglo-American origins is briefly addressed.

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Notes

  1. 1.

    Nine to ten-year-old students

  2. 2.

    A detailed presentation of the three frameworks and of the epistemological analysis can be found in Clivaz (2011, 2014).

  3. 3.

    « Une intervention mathématique est pertinente si elle rend compte dans une certaine mesure de la fonctionnalité de l'objet mathématique visé ; ou, s'agissant d'enseignement, si elle permet au moins de progresser dans l'appréhension de cette fonctionnalité, avec des énoncés de propriétés mathématiques contextualisées ou non, des arguments appropriés sur la validité de procédures ou sur la nature des objets mathématiques. » (Bloch, 2009, p. 32)

  4. 4.

    « […] capacité à interagir avec les élèves sur des éléments mathématiques de la situation et à encourager l'activité des élèves par des interventions et des retours sur leur production mathématique. » (Bloch, 2009, p. 33)

  5. 5.

    Milieu is the usual translation for Brousseau’s French term “milieu”, but, in French, it refers not only to the sociological milieu but it is also used in biology or in Piaget’s work. A more accurate translation would be “environment”.

  6. 6.

    « dans le sens qu’elle ne dépend pas des faits d’expérience ou d’observation », my translation.

  7. 7.

    Literally “in columns” for what is known in English as “long multiplication”.

  8. 8.

    « Mettre les dizaines en dessous. C’est. pas très logique, mais ça permet d’avoir les deux en face. »

  9. 9.

    « Quand on travaille avec les dizaines, on ajoute un zéro. »

  10. 10.

    There is no action and no learning intention at this level.

  11. 11.

    “The didactical contract is the rule of the game and the strategy of the didactical situation. It is the justification that the teacher has for presenting the situation, [...] a relationship [...] which determines—explicitly to some extent, but mainly implicitly—what each partner, the teacher and the student, will [...] be responsible to the other person for. This system of reciprocal obligation [and expectation, we argue] resembles a contract.” (Brousseau, 1997, p. 31)

  12. 12.

    For “unités” and “dizaines” in French, “units” and “tens” in English.

  13. 13.

    A stands for Armand, D for Dominique, the teacher, and E for another student in the class.

  14. 14.

    « c’est 1 × 1 ou 10 × 10 ? »

  15. 15.

    Niveaux de codétermination didactique

  16. 16.

    With the notable exception of Quebec and the presence of a Working Group on the topic in EMF congress (Clivaz, Proulx, Sangaré, & Kuzniak, 2012).

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Clivaz, S. Teaching multidigit multiplication: combining multiple frameworks to analyse a class episode. Educ Stud Math 96, 305–325 (2017). https://doi.org/10.1007/s10649-017-9770-7

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Keywords

  • Mathematical knowledge
  • Teacher
  • Elementary teaching
  • Algorithm for multiplication
  • Pertinence
  • Structuration of the milieu