Educational Studies in Mathematics

, Volume 97, Issue 2, pp 185–207 | Cite as

The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school

  • Christine ChambrisEmail author


This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.


New math reform Relation between school and academic mathematics Didactic transposition Place value Numeration unit Unit 



This article is based on a paper that was presented at the pre-conference for the 23rd ICMI study on Whole Number Arithmetic in Macau (China), in June 2015.


  1. Artigue, M., & Robinet, J. (1982). Numération à l'école élémentaire [Numeration in elementary school]. Educational Studies in Mathematics, 13(2), 155–175.Google Scholar
  2. Arzarello, F., & Bartolini-Bussi, M. G. (1998). Italian trends in research in mathematics education: A national case study in the international perspective. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity (Vol. 2, pp. 243–262). Dordrecht: Kluwer.Google Scholar
  3. Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish high schools. Educational Studies in Mathematics, 59, 235–268.CrossRefGoogle Scholar
  4. Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13(1), 33–57.CrossRefGoogle Scholar
  5. Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67(3), 217–235.CrossRefGoogle Scholar
  6. Bezout, E., & Reynaud, A. A. L. (1821). Traité d'arithmétique à l'usage de la marine et de l'artillerie [Treatise of arithmetic for marine and artillery] 9th edition.Google Scholar
  7. Blanc, J.-P., Bramand, P., Debû, P., Gély, J., Peynichou, D., & Vargas, A. (2002). Pour comprendre les mathématiques. CE2 [To understanding mathematics. Grade 3]. Paris: Hachette.Google Scholar
  8. Bosch, M., & Chevallard, Y. (1999). La sensibilité de l'activité mathématique aux ostensifs: Objet d'étude et problématique. [Sensitivity to ostensive objects in doing mathematics: Object of study and research problem]. Recherches en didactique des mathématiques, 19(1), 77–123.Google Scholar
  9. Bosch, M., Gascón, J., & Trigueros, M. (2017). Dialogue between theories interpreted as research praxeologies: The case of APOS and the ATD. Educational Studies in Mathematics, 95(1), 39–52.CrossRefGoogle Scholar
  10. Boucheny, G., & Guérinet, A. (1931), L’arithmétique au cours élémentaire [Arithmetic in the 2nd and 3rd grades]. Paris: Larousse.Google Scholar
  11. Bronner, A. (2008). La question du numérique dans l’enseignement du secondaire [The numeric issue in secondary teaching]. In A. Rouchier & I. Bloch (Eds.), Perspectives en didactique des mathématiques (pp. 17–45). Grenoble: La pensée sauvage.Google Scholar
  12. Bruner, J. S. (1966). Toward a theory of instruction. Harvard: Harvard University Press.Google Scholar
  13. Castela, C., & Romo Vázquez, A. (2011). Des mathématiques à l'automatique: Étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs [From mathematics to automation: Study of effects of transposition on Laplace transform in engineering education]. Recherches en didactique des mathématiques, 31(1), 79–130.Google Scholar
  14. Chambris, C. (2008). Relations entre les grandeurs et les nombres dans les mathématiques de l'école primaire [Relations between quantities and numbers in mathematics for elementary school]. (Unpushed doctoral dissertation). Université Paris–Diderot, Paris.Google Scholar
  15. Chambris, C. (2010). Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: Théories et écologie [Relations between quantities, numbers and operations in mathematics for elementary school in the 20th century: Theories and ecology]. Recherches en didactique des mathématiques, 30(3), 317–366.Google Scholar
  16. Champeyrache, G., & Fatta, J.-C. (2002). Le nouveau Math élem. CE2 [New elementary math grade 3]. Paris: BelinGoogle Scholar
  17. Chevallard, Y. (1985). La transposition didactique [Didactic transposition]. Grenoble: La pensée sauvage.Google Scholar
  18. Chevallard, Y. (1997). Familière et problématique, la figure du professeur [The teacher as a colloquial and problematic figure]. Recherches en didactique des mathématiques, 17(3), 17–54.Google Scholar
  19. Deblois, L. (1996). Une analyse conceptuelle de la numération de position au primaire [A conceptual analysis of numeration in elementary school]. Recherches en didactique des mathématiques, 16(1), 71–127.Google Scholar
  20. Eiller, R., Brini, R., Martineu, M., Ravenel, S., & Ravenel, R. (1979). Math et calcul. CE2 [Math and computation. Grade 3]. Paris: Hachette.Google Scholar
  21. Eiller, R., & Martineu, M. (1972). Math et calcul. CE2 [Math and computation. Grade 3]. Paris: Hachette.Google Scholar
  22. Eiller, R., Martineu, M., Brini, R., Cornibé, R., & Pradillon, F. (1971). Math et calcul. CE1 [Math and computation. Grade 2]. Paris: Hachette.Google Scholar
  23. ERMEL. (1978). Apprentissages mathématiques à l'école élémentaire. Cycle élémentaire. (Vol. 2) [Mathematical learning in elementary school. Second and third grades. Vol. 2]. Paris: SERMAP-Hatier.Google Scholar
  24. Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number. Educational Studies in Mathematics, 61(1–2), 67–101.CrossRefGoogle Scholar
  25. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  26. Furinghetti, F., Menghini, M., Arzarello, F., & Giacardi, L. (2008). ICMI renaissance: The emergence of new issues in mathematics education. In M. Menghini, F. Furinghetti, L. Giacardi, & F. Arzarello (Eds.), The first century of the international commission on mathematical instruction (1908–2008). Reflecting and shaping the world of mathematics education (pp. 131–147). Rome: Istituto della Enciclopedia Italiana.Google Scholar
  27. Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343–403.CrossRefGoogle Scholar
  28. Gispert, H. (2010). Rénover l'enseignement des mathématiques, la dynamique internationale des années 1950 [Renewing mathematics teaching, the international dynamic of the 1950s]. In R. d'Enfert & P. Kahn (Eds.), En attendant la réforme. Politiques éducatives et disciplines scolaires sous la Quatrième République (pp. 131–143). Grenoble: PUG.Google Scholar
  29. Griesel, H. (2007). Reform of the construction of the number system with reference to Gottlob Frege. ZDM, 39(1–2), 31–38.CrossRefGoogle Scholar
  30. Harlé, A. (1987). L'image du nombre dans les manuels d'arithmétique de l'enseignement primaire au début du XXème siècle [Image of numbers in arithmetic textbooks from early 20th century]. In Groupe d’Histoire des Mathématiques (Ed.), Fragments d'histoire des mathématiques II (pp. 22–84). Paris: APMEP.Google Scholar
  31. Howe, R. (2015). The most important thing for your child to learn about arithmetic. In X. Sun, B. Kaur, & J. Novotná (Eds.), Proceedings of the twenty-third ICMI study: Primary mathematics study on whole numbers (pp. 107–114). China: University of Macao.Google Scholar
  32. ICMI. (2014). Discussion document of the twenty-third ICMI study: Primary mathematics study on whole numbers. Retrieved from
  33. Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM, 44(4), 563–571.CrossRefGoogle Scholar
  34. Kline, M. (1973). Why Johnny can't add: The failure of the new math. New York: St. Martin's Press.Google Scholar
  35. Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170–193.CrossRefGoogle Scholar
  36. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  37. Ma, L. (2013). A critique of the structure of U.S. elementary school mathematics. Notices of the AMs, 60(10), 1282–1296.CrossRefGoogle Scholar
  38. Menotti, G., & Ricco, G. (2007). Didactic practice and the construction of the personal relation of six-year-old pupils to an object of knowledge: Numeration. European Journal of Psychology of Education, 22(4), 477–495.CrossRefGoogle Scholar
  39. Moreira, P. C., & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: Some conflicting elements. Journal for Mathematics Teacher Education, 11(1), 23–40.CrossRefGoogle Scholar
  40. Mounier, E. (2013). Y a-t-il des marges de manœuvres pour piloter la classe durant une phase de bouclage? [Is there a degree of latitude for the teacher when he ends the lesson?]. Recherches en didactique des mathématiques, 33(1), 79–113.Google Scholar
  41. Neyret, R. (1995). Contraintes et détermination des processus de formation des enseignants: Nombres décimaux, rationnels et réels dans les Instituts Universitaires de Formation des Maîtres [Constraints and determinations of teacher education: Decimals, rational and real numbers in training institutes] Thèse. Grenoble: Université Joseph Fourier Grenoble.Google Scholar
  42. Otte, M. F. (2007). Mathematical history, philosophy and education. Educational Studies in Mathematics, 66(2), 243–255.CrossRefGoogle Scholar
  43. Otte, M. F. (2011). Evolution, learning, and semiotics from a Peircean point of view. Educational Studies in Mathematics, 77(2–3), 313–329.CrossRefGoogle Scholar
  44. Perret, J. F. (1985). Comprendre l'écriture des nombres [Understanding written numerals]. Bern: P. Lang.Google Scholar
  45. Ross, S. H. (1989). Parts, wholes and place value: A developmental view. Arithmetic Teacher, 36(6), 47–51.Google Scholar
  46. Thanheiser, E. (2009). Preservice elementary school teachers' conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40(3), 251–281.Google Scholar
  47. Thompson, I. (1999). Implications of research on mental calculations for the teaching of place value. Curriculum, 20(3), 185–191.Google Scholar
  48. Wittmann, E. (1975). Natural numbers and groupings. Educational Studies in Mathematics, 6(1), 53–75.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire de didactique André Revuz (EA 4434), UA, UCP, UPD, UPEC, URNUniversité de Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations