Mathematics Education Research Journal

, Volume 20, Issue 3, pp 6–31 | Cite as

Learning from “Didactikids”: An impetus for revisiting the empty number line

  • Marja van den Heuvel-Panhuizen


This article discusses students’ perceptions as a source for understanding education. It addresses what didactically experienced children, called “didactikids”, taught us about the empty number line as a didactical model for teaching whole number calculations. The article mainly reports on a student consultancy study carried out in the Netherlands. The findings are similar to what was revealed in an Australian study. Both studies explain what can go wrong when the number line is applied rigidly and wrongly implemented.


Mathematics Education Number Line Realistic Mathematic Textbook Series Early Childhood Research 
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  1. Barkhuisen, G. P. (1998). Discovering learners’ perceptions of ESL classroom teaching/learning activities in a South African context.TESOL Quarterly, 32(1), 85–107.CrossRefGoogle Scholar
  2. Bergen, D. (Ed.). (1988).Play as a medium for learning and development: A handbook of theory and practice. Portsmouth, NH: Heinemann.Google Scholar
  3. Bobis, J., & Bobis, E. (2005). The empty numberline: Making children’s thinking visible. In M. Coupland, J. Anderson, & T. Spencer (Eds.),Making mathematics vital: Proceedings of the 20th biennial conference of the Australian Association of Mathematics Teachers (pp. 66–72). Sydney: AAMT.Google Scholar
  4. Clarke, D. J. (2005). Personal communication.Google Scholar
  5. Dahlberg, G., Moss, P., & Pence, A. (1999).Beyond quality in early childhood education and care: Postmodern perspectives. London: Falmer.Google Scholar
  6. Desforges, C., & Cockburn, A. (1987).Understanding the mathematics teacher. A study of practice in first schools. London: Falmer.Google Scholar
  7. Freudenthal, H. (1973).Mathematics as an educational task. Dordrecht, The Netherlands: Reidel.Google Scholar
  8. Freudenthal, H. (April, 1979). Learning processes. Lecture at pre-session of the NCTM meeting in Boston, 18 April.Google Scholar
  9. Freudenthal, H. (1978).Weeding and sowing. Preface to a science of mathematical education. Dordrecht, The Netherlands: Reidel.Google Scholar
  10. Freudenthal, H. (1984a).Appels en peren / wiskunde en psychologie [Apples and pears / mathematics and psychology]. Apeldoorn, The Netherlands: Van Walraven.Google Scholar
  11. Freudenthal, H. F. (1984b). Onderzoek van onderwijs: Voorbeelden en voorwaarden [Research of education: Examples and conditions]. In P. G. Vos, K. Koster, & J. Kingma (Eds.),Rekenen. Balans van standpunten in theorievorming en empirisch onderzoek [Arithmetic. Balance of viewpoints in generating theory and empirical research]. Lisse, The Netherlands: Swets & Zeitlinger.Google Scholar
  12. Freudenthal, H. (1988). Ontwikkelingsonderzoek [Developmental research]. In K. Gravemeijer & K. Koster (Eds.),Onderzoek, ontwikkeling en ontwikkelingsonderzoek [Research, development and developmental research]. Utrecht, the Netherlands: OC&OW, University of Utrecht.Google Scholar
  13. Fraser, B. J. (1998). Science learning environments: Assessment, effects and determinants. In B. J. Fraser & K. G. Tobin (Eds.),International handbook of science education — Part one (pp. 527–564). London: Kluwer.Google Scholar
  14. Hagborg, W. J. (1994). Student and teacher perceptions of classroom instructional methods and evaluation procedures.Evaluation and Program Planning, 17(3), 257–260.CrossRefGoogle Scholar
  15. Kumaravadivelu, B. (1991). Language-learning tasks: Teacher intention and learner interpretation.English Language Teaching Journal, 45(2), 98–107.Google Scholar
  16. Keitel, C. (2003).Values in mathematics classroom practice: The students’ perspective. Paper presented at the Conference of the Learner’s Perspective Study, International Research Team, University of Melbourne, December 1–3, 2003.Google Scholar
  17. Kraemer, J-M., Janssen, J., van der Schoot, F., & Hemker, B. (2005).Balans (31) van het reken-wiskundeonderwijs halverwege de basisschool 4 [Account (31) of mathematics education halfway primary school]. Arnhem, The Netherlands: Cito.Google Scholar
  18. La Bastide-Van Gemert, S. (2006). “Elke positieve actie begint met critiek”. Hans Freudenthal en de didactiek van de wiskunde [“Criticism is the start of all positive action”. Hans Freudenthal and the didactics of mathematics]. Hilversum, The Netherlands: Uitgeverij Verloren.Google Scholar
  19. Malmberg B. V. (2000–2003).Pluspunt [Plus point]. Hertogenbosch, The Netherlands: Malmberg B.V.Google Scholar
  20. McRobbie, C. J., Fisher, D. L., & Wong, A. F. L. (1998). Personal and class forms of classroom environment instruments. In B. J. Fraser & K. G. Tobin (Eds.),International handbook of science education — Part one (pp. 581–594). London: Kluwer.Google Scholar
  21. Menne, J. J. M. (2001).Met sprongen vooruit. Een productief oefenprogramma voor zwakke rekenaars in het getallengebied tot 100 — een onderwijsexperiment [Jumping ahead. A productive training program for low achievers in mathematics in the domain of numbers up to 100]. Utrecht, The Netherlands: Freudenthal Institute.Google Scholar
  22. Ministry of Education (2005).Book 1: The number framework. Numeracy professional development projects. Wellington: Ministry of Education.Google Scholar
  23. Owens, K., & Perry, B. (2001).Mathematics K-10 Literature Review for NSW Board of Studies., accessed march 1, 2008.Google Scholar
  24. Shimizu, Y. (April, 2002).Discrepancies in perceptions of lesson structure between the teacher and the students in the mathematics classroom. Paper presented at the symposiumInternational Perspectives on Mathematics Classrooms, at the Annual Meeting of the American Educational Research Association, New Orleans, April 1–5, 2002.Google Scholar
  25. Spratt, M. (1999). How good are we at knowing what learners like?System, 27, 141–155.CrossRefGoogle Scholar
  26. Stacey, K., Helme, S., & Steinle, V. (2001). Confusions between decimals, fraction and negative numbers: A consequence of the mirror as a conceptual metaphor in three different ways. In M. van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Volume 4, pp. 217–224). Utrecht, The Netherlands: Freudenthal Institute, Utrecht University.Google Scholar
  27. Treffers, A. (1991a). Didactical background of a mathematics program for primary education. In L. Streefland (Ed.),Realistic Mathematics Education in Primary School (pp. 21–56). Utrecht, The Netherlands: CD-β Press / Freudenthal Institute, Utrecht University.Google Scholar
  28. Treffers, A. (1991b). Meeting innumeracy at primary school.Educational Studies in Mathematics, 22, 333–352.CrossRefGoogle Scholar
  29. Treffers, A., & De Moor, E. (1990).Proeve van een nationaal programma voor het reken-wiskundeonderwijs op de basisschool. Deel 2 Basisvaardigheden en cijferen [Design of a National Curriculum for mathematics education in primary school. Part 2. Basic skills and algorithms]. Tilburg, The Netherlands: Zwijsen.Google Scholar
  30. Van den Brink, J., & Streefland, L. (1979). Young children (6–8): Ratio and proportion.Educational Studies in Mathematics, 10, 403–420.CrossRefGoogle Scholar
  31. Van den Brink, J. (1980). Onderwijzende kinderen [Children who teach]. InIOWO, Kijk op Hans [View on Hans] (pp. 6–8). Utrecht, the Netherlands: IOWO.Google Scholar
  32. Van den Heuvel-Panhuizen, M. (1986). Het rekenonderwijs op de lom-school opnieuw ter discussie [Mathematics education in special education again under discussion].Tijdschrift voor orthopedagogiek [Journal for Special Education], 25(3), 137–145.Google Scholar
  33. Van den Heuvel-Panhuizen, M. (2001a). Realistic Mathematics Education in the Netherlands. In J. Anghileri (Ed),Principles and practices in arithmetic teaching: Innovative approaches for the primary classroom (pp. 49–63). Buckingham/Philadelphia: Open University Press.Google Scholar
  34. Van den Heuvel-Panhuizen, M. (Ed). (2001b).Children learn mathematics. A learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Utrecht, The Netherlands: Freudenthal Institute, Utrecht University / SLO.Google Scholar
  35. Van den Heuvel-Panhuizen, M. (2002). Realistic Mathematics Education as work in progress. In F-L. Lin (Ed.),Common sense in mathematics education. Proceedings of 2001 The Netherlands and Taiwan Conference on Mathematics Education, Taipei, Taiwan (pp. 1–42). Taipei, Taiwan: National Taiwan Normal University.Google Scholar
  36. Van den Heuvel-Panhuizen, M., & Bodin-Baarends, C. (2004). All or nothing: Problem solving by high achievers in mathematics.Journal of the Korea Society of Mathematical Education, 8(3), 115–121.Google Scholar
  37. Van den Heuvel-Panhuizen, M. (2005). Twee didactikids over de lege getallenlijn — Freudenthals observaties als inspiratiebron [Two didactikids about the empty number line — Freudenthal’s observations as a source of inspiration].Reken-wiskundeonderwijs: onderzoek, ontwikkeling, praktijk [Mathematics education: research, development, practice], 24(3) / Nieuwe Wiskrant [New Journal for Mathematics], 25(1), 82–89; Freudenthal 100 — Speciale editie ter gelegenheid van de honderdste geboortedag van Professor Hans Freudenthal; edited by H. ter Heege, T. Goris, R. Keijzer, & L. Wesker.Google Scholar
  38. Van der Velden, B. (2000). Between “Bastiaan ou de l’éducation” and “Bastiaan und die Detektive”.Zentralblatt für Didaktik der Mathematik [ZDM], 6, 201–202.Google Scholar
  39. Wing, L. A. (1995). Play is not the work of the child: Young children’s perceptions of work and play.Early Childhood Research Quarterly, 10, 223–247.CrossRefGoogle Scholar
  40. Whitney, H. (1985). Taking responsibility in school mathematics education. In L. Streefland (Ed.),Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (Vol. II, pp. 123–141). Utrecht, the Netherlands: OW&OC, Utrecht University.Google Scholar
  41. Wiltz, N. W., & Klein, E. L. (2001). “What do you do in child care?” Children’s perceptions of high and low quality classrooms.Early Childhood Research Quarterly, 16, 209–236CrossRefGoogle Scholar

Copyright information

© Mathematics Education Research Group of Australasia Inc. 2008

Authors and Affiliations

  • Marja van den Heuvel-Panhuizen
    • 1
    • 2
  1. 1.Freudenthal InstituteUtrecht Universitythe Netherlands
  2. 2.Institut zur Qualitätsentwicklung im BildungswesenHumboldt UniversityBerlinGermany

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