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

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Abstract

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.

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References

  • Barkhuisen, G. P. (1998). Discovering learners’ perceptions of ESL classroom teaching/learning activities in a South African context.TESOL Quarterly, 32(1), 85–107.

    Article  Google Scholar 

  • Bergen, D. (Ed.). (1988).Play as a medium for learning and development: A handbook of theory and practice. Portsmouth, NH: Heinemann.

    Google Scholar 

  • 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 

  • Clarke, D. J. (2005). Personal communication.

  • Dahlberg, G., Moss, P., & Pence, A. (1999).Beyond quality in early childhood education and care: Postmodern perspectives. London: Falmer.

    Google Scholar 

  • Desforges, C., & Cockburn, A. (1987).Understanding the mathematics teacher. A study of practice in first schools. London: Falmer.

    Google Scholar 

  • Freudenthal, H. (1973).Mathematics as an educational task. Dordrecht, The Netherlands: Reidel.

    Google Scholar 

  • Freudenthal, H. (April, 1979). Learning processes. Lecture at pre-session of the NCTM meeting in Boston, 18 April.

  • Freudenthal, H. (1978).Weeding and sowing. Preface to a science of mathematical education. Dordrecht, The Netherlands: Reidel.

    Google Scholar 

  • Freudenthal, H. (1984a).Appels en peren / wiskunde en psychologie [Apples and pears / mathematics and psychology]. Apeldoorn, The Netherlands: Van Walraven.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Hagborg, W. J. (1994). Student and teacher perceptions of classroom instructional methods and evaluation procedures.Evaluation and Program Planning, 17(3), 257–260.

    Article  Google Scholar 

  • Kumaravadivelu, B. (1991). Language-learning tasks: Teacher intention and learner interpretation.English Language Teaching Journal, 45(2), 98–107.

    Google Scholar 

  • 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.

  • 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 

  • 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 

  • Malmberg B. V. (2000–2003).Pluspunt [Plus point]. Hertogenbosch, The Netherlands: Malmberg B.V.

    Google Scholar 

  • 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 

  • 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 

  • Ministry of Education (2005).Book 1: The number framework. Numeracy professional development projects. Wellington: Ministry of Education.

    Google Scholar 

  • Owens, K., & Perry, B. (2001).Mathematics K-10 Literature Review for NSW Board of Studies. http://www.boardofstudies.nsw.edu.au/manuals/pdf_doc/maths_k10_lit_review.pdf, accessed march 1, 2008.

  • 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.

  • Spratt, M. (1999). How good are we at knowing what learners like?System, 27, 141–155.

    Article  Google Scholar 

  • 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 

  • 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 

  • Treffers, A. (1991b). Meeting innumeracy at primary school.Educational Studies in Mathematics, 22, 333–352.

    Article  Google Scholar 

  • 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 

  • Van den Brink, J., & Streefland, L. (1979). Young children (6–8): Ratio and proportion.Educational Studies in Mathematics, 10, 403–420.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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, 209–236

    Article  Google Scholar 

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Authors and Affiliations

  1. Freudenthal Institute, Utrecht University, the Netherlands

    Marja van den Heuvel-Panhuizen

  2. Institut zur Qualitätsentwicklung im Bildungswesen, Humboldt University, Berlin, Germany

    Marja van den Heuvel-Panhuizen

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  1. Marja van den Heuvel-Panhuizen
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This paper is an extended version of a paper written in Dutch and published in a joint special issue of two Dutch journals (see Van den Heuvel-Panhuizen, 2005) produced to celebrate the centenary of Freudenthal’s birth.

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van den Heuvel-Panhuizen, M. Learning from “Didactikids”: An impetus for revisiting the empty number line. Math Ed Res J 20, 6–31 (2008). https://doi.org/10.1007/BF03217528

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