Skip to main content
SpringerLink
Log in

Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: from Felix Klein to present applications in mathematics classrooms in different parts of the world

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript

Abstract

Most national curricula for both primary and secondary grades encourage the active involvement of learners through the manipulation of materials (either concrete models or dynamic instruments). This trend is rooted in the emphasis given, at the dawn of ICMI, to what might be called an experimental approach: the links between mathematics, natural sciences and technology were in the foreground in the early documents of ICMI and also in the papers of its first president, Felix Klein. However, the presence of this perspective in teaching practice is uneven. In this paper, we shall reconstruct first an outline of what happened in three different parts of the world (Europe, USA and Japan) under the direct influence of Klein. Then, we shall report classroom activities realized in the same regions in three different research centres: the Laboratory of Mathematical Machines at the University of Modena and Reggio Emilia, Italy (http://www.mmlab.unimore.it); the pedagogical space of Kinematical Model for Design Digital Library at Cornell, USA (http://kmoddl.library.cornell.edu/); and the Centre for Research on International Cooperation in Educational Development at Tsukuba University, Japan (http://math-info.criced.tsukuba.ac.jp/). They have maintained the reference to concrete materials (either models or instruments), with original interpretations that take advantage of the different cultural conditions. Although in all cases the reference to history is deep and systematic, the synergy with mathematical modelling and with information and communication technologies has been exploited, not to substitute but to complement the advantages of the direct manipulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Bartolini Bussi, M. G. (1998). Drawing instruments: Theories and practices from history to didactics. Documenta mathematica, extra volume ICM98, pp. 735–746.

  • Bartolini Bussi, M. G. (2001). The geometry of drawing instruments: Arguments for a didactical use of real and virtual copies. Cubo, 3(2), 27–54.

    Google Scholar 

  • Bartolini Bussi, M. G., & Pergola, M. (1996). History in the mathematics classroom: Linkages and kinematic geometry. In H. N. Jahanke, N. Knoche, & M. Otte (Eds.), Geschichte der Mathematik in der Lehre (pp. 36–67). Goettingen: Vandenhoek & Ruprecht.

    Google Scholar 

  • Campedelli, L. (1958). I Modelli Geometrici. In Commissione internazionale per lo studio e il miglioramento dell'insegnamento della matematica (Ed.), Il Materiale per L’insegnamento della Matematica (pp. 143–172). Firenze: La Nuova Italia.

  • Castelnuovo, G. (1928). La Geometria Algebrica e la Scuola Italiana. Atti del Congresso Internazionale dei Matematici (Vol. I, p. 194). Bologna: Zanichelli.

    Google Scholar 

  • Castelnuovo, E. (2008). L’Officina Matematica. Bari: Edizioni La Meridiana.

    Google Scholar 

  • Cavalieri, B. (1632). Lo specchio ustorio (2001). Firenze: Giunti.

    Google Scholar 

  • Clairaut, A. C. (1741). Élements de Geometrie (2006). Éditions Jacques Gabay.

  • Comenius, I. A. (1657). Didactica Magna (1986). Akal Ediciones.

  • Fischer, G. (Ed.). (1986). Mathematical models: From the collections of Universities and Museums. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn.

    Google Scholar 

  • France National Constitution. (1791). http://sourcebook.fsc.edu/history/constitutionof1791.html. Accessed 25 August 2009.

  • Giacardi, L. (2008). Bibiligograpy of Castelnuovo, G. http://www.icmihistory.unito.it/portrait/castelnuovo.php. Accessed 21 November 2008.

  • Henderson, D., & Taimina, D. (2005a). Mathematical aspects of the Peaucellier-Lipkin linkage. http://kmoddl.library.cornell.edu/tutorials/11/. Accessed 21 November 2008.

  • Henderson, D., & Taimina, D. (2005b). Experiencing geometry: Euclidean and non-euclidean with history. Upper Saddle River: Pearson Prentice Hall.

    Google Scholar 

  • Isoda, M. (2004). The development of mathematics education. In Japan International Cooperation Agency (Ed.), The history of Japan’s educational development (pp. 159–172). Tokyo: Institute for International Cooperation.

    Google Scholar 

  • Isoda, M. (2008). Sample videos developed by Isoda, M. and Baldin, Y. Y. in 2008 at UFRJ on HTEM4. http://math-info.criced.tsukuba.ac.jp/software/dbook/dbook_eng/dbook_eng05.html. Accessed 21 November 2008.

  • Isoda, M., Hara, K., Iijima, Y., Uehara, E., & Watanabe, H. (2006). e-Textbooks: Advancing mathematics teaching and learning into the future mathematics. In Chu, S. C. et al. (Eds.), Proceedings of the eleventh Asian technology conference in mathematics (pp. 54–63). VA: ATCM, Inc.

  • Isoda, M., & Matsuzaki, A. (2003). The roles of mediational means for mathematization: The case of mechanics and graphing tools. Journal of Japan Society of Science Education, 27(4), 245–257.

    Google Scholar 

  • Isoda, M., Suzuki, A., Ohneda, Y., Sakamoto, M., Mizutahi, N., & Kawazaki, N. (2001). LEGO project. Tsukuba Journal of Educational Study in Mathematics, 20, 77–92.

    Google Scholar 

  • Kempe, A. B. (1877). How to draw a straight line. London: Macmillan.

    Google Scholar 

  • Klein, F. (1924–1925). Elementary mathematics from an advanced standpoints. (Hedrick, E. R., & Noble, C. A., translated, 1939). NY: Macmillan.

  • Klein, F., & Riecke, E. (1904). Neue Beiträge zur Frage des Mathematischen und Physikalischen Unterrichts an den Höheren Schulen. Leipzig: Teubner.

    Google Scholar 

  • Koenigs, G. (1897). Leçons De Cinematique Paris A. Hermann. http://www.archive.org/stream/leonsdecinma00koenuoft#page/242/mode/2up. Accessed 25 August 2009.

  • Kuroda, M. (1920). Geometry for secondary school. Tokyo: Baifu-kan.

    Google Scholar 

  • Kuroda, M. (1927). New issues of mathematics teaching (selected publication of his posthumous manuscripts). Tokyo: Baifu-kan.

    Google Scholar 

  • Le Blanc, M. (n.d.). Friedrich Froebel: His life and influence on education. http://www.communityplaythings.co.uk/resources/articles/friedrich-froebel.html.

  • Maclaurin, C. (1720). Geometria Organica, Sive Descriptio Linearum Curvarum Universalis. Londini.

  • Maschietto, M. (2005). The laboratory of mathematical machines of modena. Newsletter of the European Mathematical Society, 57, 34–37.

    Google Scholar 

  • Monbusyo Approved. (1943). Mathematics textbooks for secondary school (II). Tokyo: TyutoGakkoKyokasyo Pub.

    Google Scholar 

  • Moon, F. C. (2002). Franz Reuleaux: Contributions to 19th C. kinematics and theory of machines. http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.htmm/2002-2?abstract=. Accessed 21 November 2008.

  • Mueller, W. (2001). Mathematical Wunderkammern. MAA Monthly, 108, 785–796.

    Article  Google Scholar 

  • Palladino, N. (n.d.). http://www.dma.unina.it/~nicla.palladino/catalogo/. Accessed 25 August 2009.

  • Pan, B., Gay, G. K., Saylor, J., Hembrooke, H. A., & Henderson, D. (2004). Usability, learning, and subjective experience: User evaluation of K-MODDL in an undergraduate class. In H. Chen, M. Christel, & E. Lim (Eds.), Proceedings of the fourth ACM/IEEE joint conference on digital libraries (JCDL’ 04) (pp. 188–189). New York: ACM.

    Chapter  Google Scholar 

  • Parshall, K. H., & Rowe, D. (1991). The emergence of the American mathematical research community 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. Providence: American Mathematical Society, London Mathematical Society (Co-publication).

  • Perry, J. (1913). Elementary practical mathematics. London: Macmillan.

    Google Scholar 

  • Reuleaux, F. (1876). The kinematics of machinery: Outlines of a theory of machines. London: Macmillan (reprinted New York: Dover, 1963).

    Google Scholar 

  • Ruthven, K. (2008). Mathematical technologies as a vehicle for intuition and experiment: a foundational theme of the ICMI, and a continuing preoccupation. Paper prepared for the ICMI centennial symposium, Rome 2008. http://www.unige.ch/math/EnsMath/Rome2008/WG4/Papers/RUTHVEN.pdf. Accessed 21 November 2008.

  • Sanden, H. V. (1914). Praktishe analysis. Leipzig und Berlin: Teubner.

    Google Scholar 

  • Schilling, M. (1911). Catalog matematischer Modelle für den Höheren Mathematischen Unterricht. Leipzig: Martin Schilling.

    Google Scholar 

  • Schooten, F. V. (1646). De organica Conicarum Sectionum Constructione. Leyden.

  • Schubring, G. (1989). Pure and applied mathematics in divergent institutional settings. In D. E. Rowe & J. McCleary (Eds.), Ideas and reception (pp. 171–207). Boston: Academic Press.

    Google Scholar 

  • Shell-Gellasch, A., & Acheson, B. (2007). Geometric string models of descriptive geometry, hands on history. In A. Shell-Gelasch (Ed.), MAA notes #72 (pp. 49–62). Washington, DC: MAA.

    Google Scholar 

  • Smith, D. E. (1913). Intuition and experiment in mathematical teaching in the secondary schools. L’Einseignement Mathématique, 14, 507–534.

  • Taimina, D. (2005a). Linkages. http://kmoddl.library.cornell.edu/linkages/. Accessed 21 November 2008.

  • Taimina, D. (2005b). How to draw a straight line. http://kmoddl.library.cornell.edu/tutorials/04/. Accessed 21 November 2008.

  • Taimina, D., & Henderson, D. (2005). Reuleaux Triangle, learning module. http://kmoddl.library.cornell.edu/tutorials/02/. Accessed 21 November 2008.

Download references

Acknowledgments

We are very grateful to David W. Henderson for his help with the final version of this manuscript. M.G. Bartolini Bussi was funded by the project PRIN 2007B2M4EK on “Instruments and representations in the teaching and learning of mathematics: theory and practice”. D. Taimina was partially funded by National Science Foundation (research grant DUE-0226238) and Institute of Museum and Library Services (research grant LG-30-04-0204-04). M. Isoda was funded by the project JSPS 17300243 on “Developing Mathematics Exhibition for Science Museum”.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, Modena, Italy

    Maria G. Bartolini Bussi

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Daina Taimina

  3. Graduate School of Human Comprehensive Science, University of Tsukuba, 305-8572, Tsukuba, Japan

    Masami Isoda

Authors
  1. Maria G. Bartolini Bussi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daina Taimina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Masami Isoda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria G. Bartolini Bussi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bartolini Bussi, M.G., Taimina, D. & Isoda, M. Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: from Felix Klein to present applications in mathematics classrooms in different parts of the world. ZDM Mathematics Education 42, 19–31 (2010). https://doi.org/10.1007/s11858-009-0220-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-009-0220-6

Keywords

Search

Navigation