Advertisement

ZDM

, Volume 51, Issue 1, pp 25–37 | Cite as

Understanding place value with numeration units

  • Catherine HoudementEmail author
  • Frédérick Tempier
Original Article

Abstract

Numeration units (ones, tens, hundreds, etc.) are an epistemological foundation for place value. Here, we assume that they are a didactic support in countries where spoken numbers are incongruent with written numbers. In France, teaching practices neglect the relations between units, and numeration units are typically used only to designate the positions of digits. Our research, which adopts the didactical engineering framework, aims to develop a set of reference tasks for grades 1–5 to support a consistent learning of place value with numeration units. This article reports the theoretical framework on which our research is based, and illustrates it with some examples of tasks designed for grade 3 students and implemented in classrooms.

Keywords

Didactical engineering Numeration units Place value Primary school Task design Unitizing 

References

  1. Artigue, M. (2009). Didactical design in mathematics education. In C. Winslow (Ed.), Nordic Research in Mathematics Education, Proceedings from NORMA08 (pp. 7–16). Rotterdam: Sense Publishers.Google Scholar
  2. Barr, D. C. (1978). A comparison of three methods of introducing two-digit numeration. Journal for Research in Mathematics Education, 9(1), 33–43.CrossRefGoogle Scholar
  3. Bartolini Bussi, M. G., & Mariotti, M. A. (2002). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. G. Bartolini, G. Bussi, R. Jones, Lesh & D. Tirosh (Eds.), Handbook of international research in mathematics education, second revised edition (pp. 746–783). Mahwah: Lawrence Erlbaum.Google Scholar
  4. Baturo, A. (2000). Construction of a numeration model: A theoretical analysis. In J. Bana & A. Chapman (Eds.), In Proceedings 23rd Annual Conference of the Mathematics Education Research Group of Australasia (pp. 95–103).Google Scholar
  5. Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13(1), 33–57.CrossRefGoogle Scholar
  6. Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17(1), 25–64.CrossRefGoogle Scholar
  7. Brissiaud, R. (2005). Comprendre la numération décimale: Les deux formes de verbalisme qui donnent l’illusion de cette compréhension. Rééducation Orthophonique, 223, 225–238.Google Scholar
  8. Brousseau, G. (1995). Les mathématiques à l’école. Bulletin de l’APMEP, 400, 831–850.Google Scholar
  9. Brousseau, G.. Balacheff, N., Cooper, M., Sutherland, R., & Warfield, V. (1997). Theory of didactical situations in mathematics (Edited and translated by. Dordrecht: Kluwer.Google Scholar
  10. Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20.CrossRefGoogle Scholar
  11. Chambris, C. (2008). Relations entre les grandeurs et les nombres dans les mathématiques de lécole primaire. Évolution de lenseignement au cours du 20e siècle. Connaissances des élèves actuels. Université Paris Diderot, Thesis.Google Scholar
  12. Chambris, C., & Tempier, F. (2017). Dealing with large numbers: What is important for students and teachers to know? In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (pp. 245–252). Dublin, Ireland: DCU Institute of Education and ERME.Google Scholar
  13. Chandler, C. C., & Kamii, C. (2009). Giving change when payment is made with a dime: The difficulty of tens and ones. Journal for Research in Mathematics Education, 40(2), 97–118.Google Scholar
  14. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1), 1–42.CrossRefGoogle Scholar
  15. Dorier, J. L. (2015). Key issues for teaching numbers within Brousseau’s theory of didactical situations, In X. H. Sun, B. Kaur, & J. Novotna (Eds.), Proceedings of the Twenty-third ICMI Study: Primary mathematics study on whole numbers. (pp. 76–83). Macao, China.Google Scholar
  16. Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work. Constructing number sense, addition, and subtraction. Portsmouth: Heinemann.Google Scholar
  17. 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
  18. Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first and second grade place-value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21(3), 180–226.CrossRefGoogle Scholar
  19. Fuson, K. C., Wearne, D., Hiebert, J., Human, P., Murray, H., Olivier, A., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130–162.CrossRefGoogle Scholar
  20. Godino, J., Batanero, C., Contreras, A., Estepa, A., Lacasta, E., & Wilhemi, M. (2013). Didactic engineering as design-based research in mathematics education. In B. Ubuz, Ç. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eight Congress of the European Society for Research in Mathematics Education (pp. 2810–2819). Antalya, Turkey.Google Scholar
  21. Guitel, G. (1975). Histoire comparée des numérations écrites. Paris: Flammarion.Google Scholar
  22. Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for research in mathematics education, 23(2), 98–122.CrossRefGoogle Scholar
  23. Houdement, C., & Chambris, C. (2013). Why and how to introduce numbers units in 1st and 2nd grades. In B. Ubuz, Ç. Haser, M. A. Mariotti (Eds.), Proceedings of the Eight Congress of the European Mathematical Society for Research in Mathematics Education (pp. 313–322). Antalya, Turkey.Google Scholar
  24. Houdement, C., & Tempier, F. (2015) Teaching numeration units: Why, how and limits. In X. H. Sun, B. Kaur, & J. Novotna (Eds.), Proceedings of the Twenty-third ICMI Study: Primary mathematics study on whole numbers. (pp. 99–106). Macao, China.Google Scholar
  25. Ifrah, G. (1981). Histoire universelle des chiffres. Paris: Editions Seghers.Google Scholar
  26. Kamii, C., & Joseph, L. (2004). Young children continue to reinvent arithmetic, 2nd grade: Implications of Piaget’s theory. New York: Teachers College Press.Google Scholar
  27. Ko, P. Y., & Marton, F. (2004). Variation and the secret of the virtuoso. In F. Marton, A. B. Tsui, P. Chik, P. Y. Ko & M. L. Lo (Eds.), Classroom discourse and the space of learning (pp. 43–62). Mahwah: Lawrence Erlbaum.Google Scholar
  28. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Hillsdale: Lawrence Erlbaum.Google Scholar
  29. Ma, L., & Kessel, C. (2018). The theory of school arithmetic: Whole numbers. In M. G. Bartolini Bussi & X. H. Sun (Eds.), Building the foundation: Whole numbers in the primary grades (pp. 439–464). Cham: New ICMI Study Series, Springer.CrossRefGoogle Scholar
  30. Margolinas, C., & Drijvers, P. (2015). Didactical engineering in France: An insider’s and an outsider’s view on its foundations, its practice and its impact. ZDM, 47(6), 893–903.CrossRefGoogle Scholar
  31. McClain, K., Cobb, P., & Bowers, J. (1998). A contextual investigation of three-digit addition and subtraction. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 141–150). Reston: National Council of Teachers of Mathematics.Google Scholar
  32. Menninger, K. (1969). Number words and number symbols. A cultural history of numbers. Translated from the revised German edition (Göttingen, 1958). Cambridge: M.I.T. Press.Google Scholar
  33. Nunes, T., Dorneles, B. V., Lin, P. J., & Rathgeb-Schnierer, E. (2016). Teaching and learning about whole numbers in primary school. In Teaching and learning about whole numbers in primary school. ICME-13 Topical Surveys (pp. 1–50). Cham: Springer International Publishing.CrossRefGoogle Scholar
  34. Perrin-Glorian, M. J. (2011). L’ingénierie didactique à l’interface de la recherche avec l’enseignement. Développement de ressources et formation des enseignants. In C. Margolinas et al. (Eds.), En amont et en aval des ingénieries didactiques (pp. 57–78). Grenoble: La Pensée Sauvage.Google Scholar
  35. Proust, C. (2000). La multiplication babylonienne: La part non écrite du calcul. Revue d’Histoire des Mathématiques, 6, 293–303.Google Scholar
  36. Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. The Arithmetic Teacher, 36(6), 47–51.Google Scholar
  37. Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York: Springer Verlag.CrossRefGoogle Scholar
  38. Stevin, S. (1585). Premierement descripte en Flameng, & maintenant convertie en Francois. In La Disme [De Thiende] (pp. 206–213). Leyde: Plantin. L’ arithmétique (1958).Google Scholar
  39. Sun, X. H., et al. (2018). The what and why of whole number arithmetic: Foundational ideas from history, language and societal changes. In M. G. Bartolini, Bussi & X. H. Sun (Eds.), Building the foundation: Whole numbers in the primary grades (pp. 91–124). Cham: New ICMI Study Series, Springer.CrossRefGoogle Scholar
  40. Sun, X. H., Kaur, B., & Novotná, J. (Eds.). (2015). Primary mathematics study on whole numbers. Proceedings of the Twenty-third ICMI Study. Macao, China.Google Scholar
  41. Tempier, F. (2013). La numération décimale de position à lécole primaire. Une ingénierie didactique pour le développement dune ressource. Université Paris Diderot, Thesis.Google Scholar
  42. Tempier, F. (2016). New perspectives for didactical engineering. An example for the development of a resource for teaching decimal number system. Journal of Mathematics Teacher Education, 19(2–3), 261–276.CrossRefGoogle Scholar
  43. Thanheiser, E. (2012). Understanding multidigit whole numbers: The role of knowledge components, connections, and context in understanding regrouping 3+- digit numbers. The Journal of Mathematical Behavior, 31(2), 220–234.CrossRefGoogle Scholar
  44. Thomas, N. (2004). The development of structure in the number system. In M. J. Hoines & A. B. Fuglestad (Eds.), 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 305–312). Bergen: Bergen University College Press.Google Scholar
  45. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (Seventh edition). Boston: Allyn & Bacon.Google Scholar
  46. Watson, A., Ohtani, M., Ainley, J., Bolite Frant, J., Doorman, M., Kieran, C., Leung, A., Margolinas, C., Sullivan, P., Thompson, D., & Yang, Y. (2013). Introduction. In C. Margolinas (Ed.) Task design in mathematics education (pp. 9–15). Proceedings of ICMI Study 22.Google Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.Laboratoire de Didactique André Revuz (EA-4434, France)UA-UCP-UPD-UPEC-URN, Université Rouen NormandieRouenFrance
  2. 2.Laboratoire de Didactique André Revuz (EA-4434, France)UA-UCP-UPD-UPEC-URN, Université de Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations