Advertisement

Educational Studies in Mathematics

, Volume 96, Issue 3, pp 305–325 | Cite as

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

  • Stéphane Clivaz
Article

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.

Keywords

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

References

  1. Ambrose, R., Baek, J.-M., & Carpenter, T. P. (2003). Children’s invention of multidigit multiplication and division algorithms. In A. J. Baroody & A. E. Dowker (Eds.), The development of arithmetic concepts and skills: Constructive adaptive expertise (pp. 305-336). Mahwah, NJ: Laurence Erlbaum.Google Scholar
  2. Artigue, M., & Winsløw, C. (2010). International comparative studies on mathematics education: A viewpoint from the anthropological theory of didactics. Recherches en didactiques des mathématiques, 30(1), 47–82.Google Scholar
  3. Ashlock, R. B. (2010). Error patterns in computation: Using error patterns to help each student learn (9th ed.). Upper Saddle River, NJ: Pearson.Google Scholar
  4. Baek, J. M. (1998). Children’s invented algorithms for multidigit multiplication problems. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 151–160). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  5. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching, who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 4–17, 20–22, 43–46.Google Scholar
  6. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.CrossRefGoogle Scholar
  7. Bikner-Ahsbahs, A., & Prediger, S. (2010). Networking of theories—An approach for exploiting the diversity of theoretical approaches. In S. Lerman (Ed.), Theories of mathematics education (pp. 483–506). Berlin: Springer.CrossRefGoogle Scholar
  8. Bloch, I. (1999). L'articulation du travail mathématique du professeur et de l'élève dans l'enseignement de l'analyse en première scientifique. Détermination d'un milieu – connaissances et savoirs. [The articulation of the mathematical work of the teacher and of the student in the teaching of analysis in “première scientifique”. Determination of a milieu—“Connaissance” and “savoir”]. Recherches en didactique des mathématiques, 19(2), 135–194.Google Scholar
  9. Bloch, I. (2005). Quelques apports de la théorie des situations à la didactique des mathématiques dans l'enseignement secondaire et supérieur: contribution à l'étude et à l'évolution de quelques concepts issus de la théorie des situations didactiques en didactique des mathématiques. [Some contributions of the theory of didactical situations to the didactique of mathematics in secondary and higher education: Contribution to the study and to the evolution of some concepts derived from the theory of didactical situations in didactique of mathematics]. (HDR), Paris 7, Paris.Google Scholar
  10. Bloch, I. (2009). Les interactions mathématiques entre professeurs et élèves. Comment travailler leur pertinence en formation ? [Mathematical interactions between teachers and students. How to work on their pertinence in training?]. Petit X, 81, 25–52.Google Scholar
  11. Bloch, I., & Gibel, P. (2011). Un modèle d'analyse des raisonnements dans les situations didactiques : étude des niveaux de preuves dans une situation d’enseignement de la notion de limite. [A model for analyzing the reasoning produced in didactic situations: A study of different levels of proof in teaching the concept of limit]. Recherches en didactique des mathématiques, 31(2), 191–228.Google Scholar
  12. Brousseau, G. (1997). Theory of didactical situations in mathematics. (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Trans.). Dordrecht, The Netherlands: Kluwer.Google Scholar
  13. Chevallard, Y. (1999). Analyse des pratiques enseignantes et didactique des mathématiques : l’approche anthropologique. [Analysis of the teaching practices and didactique of mathematics: The anthropological approach]. Recherches en didactique des mathématiques, 19(2), 225–265.Google Scholar
  14. Clivaz, S. (2011). Des mathématiques pour enseigner, analyse de l’influence des connaissances mathématiques d’enseignants vaudois sur leur enseignement des mathématiques à l’école primaire [Mathematics for teaching: Analysis of the Influence of Vaud's teachers' mathematical knowledge on their teaching of mathematics in primary schools] (Thèse de doctorat). Université de Genève, Genève.Google Scholar
  15. Clivaz, S. (2014). Des mathématiques pour enseigner? Quelle influence les connaissances mathématiques des enseignants ont-elles sur leur enseignement à l’école primaire? [Mathematics for teaching? What influence have teachers’ mathematical knowledge on their teaching in primary school?]. Grenoble: La Pensée Sauvage.Google Scholar
  16. Clivaz, S., Proulx, J., Sangaré, M., & Kuzniak, A. (2012). Articulation des connaissances mathématiques et didactiques pour l’enseignement : pratiques et formation – Compte-rendu du Groupe de Travail n°1. [Articulation of mathematical knowledge and didactical knowledge for teaching: Practices and training—Report of Working Group 1]. In J.-L. Dorier & S. Coutat (Eds.), Enseignement des mathématiques et contrat social : Enjeux et défis pour le 21e siècle – Actes du colloque EMF2012 (pp. GT1, 155–159). Genève.Google Scholar
  17. Comiti, C., Grenier, D., & Margolinas, C. (1995). Niveaux de connaissances en jeu lors d'interactions en situation de classe et modélisation de phénomènes didactiques. [Levels of knowledge involved in the interactions in classroom situations and modeling of didactical phenomena.]. In G. Arsac, J. Gréa, D. Grenier, & A. Tiberghien (Eds.), Différents types de savoirs et leur articulation (pp. 91–127). Grenoble: La Pensée Sauvage.Google Scholar
  18. Coulange, L. (2001). Enseigner les systèmes d’équations en Troisième. Une étude économique et écologique [Teaching systems of equations in grade 9. An economic and ecological study.] Recherches en didactique des mathématiques, 21(3), 305-353.Google Scholar
  19. Danalet, C., Dumas, J.-P., Studer, C., & Villars-Kneubühler, F. (1999). Mathématiques 4ème année: Livre du maître, livre de l'élève et fichier de l'élève [Mathematics grade 4: Teacher's book, student's book and student's file]. Neuchâtel: COROME.Google Scholar
  20. Davis, B., & Renert, M. (2013). Profound understanding of emergent mathematics: Broadening the construct of teachers’ disciplinary knowledge. Educational Studies in Mathematics, 82(2), 245–265.CrossRefGoogle Scholar
  21. Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293–319.CrossRefGoogle Scholar
  22. Depaepe, F., Verschaffel, L., & Kelchtermans, G. (2013). Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research. Teaching and Teacher Education, 34, 12–25.CrossRefGoogle Scholar
  23. DFJ. (2006). Plan d'édudes vaudois [Course of study, Canton of Vaud]. Lausanne: DFJ/DGEO.Google Scholar
  24. Dorier, J.-L. (2012). La démarche d’investigation en classe de mathématiques : quel renouveau pour le questionnement didactique ? [The inquiry based learning in mathematics class: What revival for didactic questioning?]. In B. Calmettes (Ed.), Démarches d'investigation. Références, représentations, pratiques et formation (pp. 35–56). Paris: L'Harmattan.Google Scholar
  25. Fassnacht, C., & Woods, D. K. (2002–2011). Transana (Version 2.42) [Mac]. Madison, WI: University of Wisconsin. Retrieved from http://www.transana.org/
  26. Hart, L., Oesterle, S., & Swars, S. (2013). The juxtaposition of instructor and student perspectives on mathematics courses for elementary teachers. Educational Studies in Mathematics, 83(3), 1–23.CrossRefGoogle Scholar
  27. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.CrossRefGoogle Scholar
  28. Hill, H. C., Blunk, M., Charalambous, C., Lewis, J., Phelps, G., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511.CrossRefGoogle Scholar
  29. Huillet, D. (2009). Mathematics for teaching: An anthropological approach and its use in teacher training. For the learning of mathematics, 29(3), 4–10.Google Scholar
  30. Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305–342.CrossRefGoogle Scholar
  31. Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66(3), 349–371.CrossRefGoogle Scholar
  32. Margolinas, C. (1994). Jeux de l'élève et du professeur dans une situation complexe. [Games of the student and of the teacher in a complex situation]. In Séminaire de didactique et technologies cognitives en mathématiques (pp. 27–83). Grenoble, France: LSDD-IMAG.Google Scholar
  33. Margolinas, C. (1995). La structuration du milieu et ses apports dans l'analyse a posteriori des situations. [The structuring of the milieu and its contributions in the a posteriori analysis of situations]. In C. Margolinas (Ed.), Les débats de didactique des mathématiques : actes du Séminaire national 1993–1994 (pp. 89–102). Grenoble: La Pensée Sauvage.Google Scholar
  34. Margolinas, C. (1998). Etude de situations didactiques "ordinaires" à l'aide du concept de milieu : détermination d'une situation du professeur. [Study of "ordinary" didactical situations using the concept of milieu: Determination of a teacher's situation]. In M. Bailleul, C. Comiti, J.-L. Dorier, J.-B. Lagrange, B. Parzysz, & M.-H. Salin (Eds.), Actes de la 9e école d'été de didactique des mathématiques (pp. 35–43). Paris: ARDM.Google Scholar
  35. Margolinas, C. (1999). Une étude de la transmission des situations didactiques [A study of the transmission of didactical situations]. Paper presented at the Actes du 2ème colloque international “Recherche(s) et formation des enseignants.”Google Scholar
  36. Margolinas, C. (2002). Situations, milieux, connaissances: Analyse de l'activité du professeur [Situations, milieus, knowledge: Analysis of the activity of the teacher]. In J.-L. Dorier, M. Artaud, M. Artigue, R. Berthelot, & R. Floris (Eds.), Actes de la 11e école d'été de didactique des mathématiques (pp. 141-155). Grenoble: La Pensée Sauvage.Google Scholar
  37. Margolinas, C. (2004a). Modeling the teacher’s situation in the classroom. In H. Fujita, Y. Hashimoto, B. Hodgson, P. Lee, S. Lerman, & T. Sawada (Eds.), Proceedings of the Ninth International Congress on Mathematical Education (pp. 171-173). Dordrecht: Springer Netherlands.Google Scholar
  38. Margolinas, C. (2004b). Points de vue de l'élève et du professeur. Essai de développement de la théorie des situations didactiques [Points of view of the student and of the teacher. Essay on the development of the theory of didactical situations]. (HDR), Université de Provence - Aix-Marseille I.Google Scholar
  39. Margolinas, C., Coulange, L., & Bessot, A. (2005). What can the teacher learn in the classroom? Educational Studies in Mathematics, 59, 205–234.CrossRefGoogle Scholar
  40. National Mathematics Advisory Panel. (2008). Fundation for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.Google Scholar
  41. Perrin-Glorian, M.-J. (1999). Analyse d'un problème de fonctions en termes de milieu. Structuration du milieu pour l'élève et pour le maître [Analysis of a problem of functions in terms of milieu. Structuring the milieu for the student and for the teacher]. Paper presented at the Analyse des pratiques enseignantes et didactique des mathématiques, Actes de l'université d'été de La Rochelle.Google Scholar
  42. Perrin-Glorian, M.-J., & Hersant, M. (2003). Milieu et contrat didactique, outils pour l'analyse de séquences ordinaires [Milieu and didactical contract, tools for the analysis of ordinary sequences]. Recherches en didactique des mathématiques, 23(2), 217-276.Google Scholar
  43. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM, 40(2), 165–178.CrossRefGoogle Scholar
  44. Roditi, E. (2011). Recherches sur les pratiques enseignantes en mathématiques: apports d'une intégration de diverses approches et perspectives [Researches on mathematics teaching practices: contribution of an integration of different approaches and prospects]. Université René Descartes - Paris V.Google Scholar
  45. Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69(2), 149–163.CrossRefGoogle Scholar
  46. Rowland, T., & Ruthven, K. (Eds.). (2011). Mathematical knowledge in teaching. Dordrecht: Springer.Google Scholar
  47. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  48. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Revue, 57(1), 1–22.CrossRefGoogle Scholar
  49. Tchoshanov, M. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141–164.CrossRefGoogle Scholar
  50. Van De Walle, J. A., Karp, K., Bay-Williams, J. M., Wray, J. A., & Rigelman, N. R. (2014). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Harlow: Pearson.Google Scholar
  51. Warfield, V. (2014). Invitation to didactique. New York: Springer.CrossRefGoogle Scholar
  52. Watson, A. (2008, March). Developing and deepening mathematical knowledge in teaching: Being and knowing. MKiT 6, Nuffield Seminar Series, 18th March, at University of Loughborough.Google Scholar
  53. Zazkis, R., & Zazkis, D. (2011). The significance of mathematical knowledge in teaching elementary methods courses: Perspectives of mathematics teacher educators. Educational Studies in Mathematics, 76(3), 247–263.CrossRefGoogle Scholar
  54. Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165–182.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.University of Teacher EducationLausanneSwitzerland

Personalised recommendations