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Students constructing meaning for the number line in small-group discussions: negotiation of essential epistemological issues of visual representations

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Abstract

In mathematics classroom interaction, the multiple meanings of mathematical visual diagrams are often ignored; instead, depending on the given situation, they are read in a well-defined and unitary way. Mathematical visual representations are thus used even less in their epistemological function for learning, but more as pre-given subject matter. The purpose of this paper is to elaborate opportunities for negotiating and clarifying differences, dealing with a great variety of ways of interpreting visual diagrams that are brought into focus in interaction. Theory-based qualitative analyses of two exemplifying video episodes of small-group discussions negotiating their ideas on the topic “number line” show differences of meaning and the importance of conventions followed by mathematical deductions. Two mutually exclusive teacher behaviors within the communicative acts, reconstructed as a dominating way of instruction and a moderating way of focusing, are identified.

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Notes

  1. The research reported in this paper is part of the project “Probing and Evaluating Focusing Interaction and Teaching Strategies (ProFIT)”.

  2. We decided to conduct the small-group discussions outside from the class in an experimental setting. The rules for transcription used in this paper can be found in the addendum (Sect. 7).

  3. Such a teaching behavior Bauersfeld (1978) called the “funneling pattern.” For the interested reader also cf. Wood (1994).

References

  • Baraldi, C., Corsi, G., & Esposito, E. (1997). GLU. Glossar zu Niklas Luhmanns Theorie sozialer Systeme. Frankfurt am Main: Suhrkamp.

    Google Scholar 

  • Bauersfeld, H. (1978). Kommunikationsmuster im Mathematikunterricht—Eine Analyse am Beispiel der Handlungsverengung durch Antworterwartung. In H. Bauersfeld (Ed.), Fallstudien und Analysen zum Mathematikunterricht (pp. 158–170). Hannover: Schroedel.

    Google Scholar 

  • Blumer, H. (1969). Symbolic interactionism. Perspective and method. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Brendefur, J., & Frykholm, J. (2000). Promoting mathematical communication in the classroom: Two preservice teachers’ conceptions and practices. Journal of Mathematics Teacher Education, 3, 125–153.

    Article  Google Scholar 

  • Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education XXI, 1999 (pp. 2–25), Cuernavaca, Morelos, Mexico.

  • Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th international conference for the psychology of mathematics education (Vol. I, pp. 55–69). Hiroshima: Nishiki Print Co.

    Google Scholar 

  • Freudenthal, H. (1977). Mathematik als pädagogische Aufgabe (Vol. 1). Stuttgart: Klett.

    Google Scholar 

  • Jahnke, H. N. (1984). Anschauung und Begründung in der Schulmathematik. In Beiträge zum Mathematikunterricht (pp. 32–41). Bad Salzdetfurth: Franzbecker.

  • Krummheuer, G. (1992). Lernen mit „Format“. Elemente einer interaktionistischen Lerntheorie. Diskutiert an Beispielen mathematischen Unterrichts. Weinheim: Deutscher Studien Verlag.

    Google Scholar 

  • Krummheuer, G. (1997). Zum Begriff der “Argumentation” im Rahmen einer Interaktionstheorie des Lernens und Lehrens von Mathematik. ZDM—The International Journal on Mathematics Education, 29(1), 1–11.

    Article  Google Scholar 

  • Lampert, M. (1990). When the problem is not the question and the solutionn is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63.

    Article  Google Scholar 

  • Lorenz, J. H. (2007). Anschauungsmittel als Kommunikationsmittel. Die Grundschulzeitschrift, 201, 14–16.

    Google Scholar 

  • Luhmann, N. (1987). Soziale Systeme. Grundriss einer allgemeinen Theorie. Frankfurt am Main: Suhrkamp.

    Google Scholar 

  • Luhmann, N. (1996). Takt und Zensur im Erziehungssystem. In N. Luhmann & K.-E. Schorr (Eds.), Zwischen System und Umwelt. Fragen an die Pädagogik (pp. 279–294). Frankfurt am Main: Suhrkamp.

  • Maturana, H. R., & Varela, F. J. (1987). Der Baum der Erkenntnis. Die biologischen Grundlagen des menschlichen Erkennens. Bern: Scherz.

    Google Scholar 

  • Miller, M. (1986). Kollektive Lernprozesse. Frankfurt am Main: Suhrkamp.

    Google Scholar 

  • Miller, M. (2006). Dissens. Zur Theorie diskursiven und systemischen Lernens. Bielefeld: Transcript.

    Google Scholar 

  • Nöth, W. (2000). Handbuch der Semiotik. Stuttgart: J.B. Metzler.

    Google Scholar 

  • Söbbeke, E., & Steenpaß, A. (2010). Mathematische Deutungsprozesse zu Anschauungsmitteln unterstützen. In C. Böttinger, et al. (Eds.), Mathematik im Denken der Kinder (pp. 216–244). Seelze: Friedrich Verlag.

    Google Scholar 

  • Steenpaß, A., & Steinbring, H. (2014). Young students’ subjective interpretations of mathematical diagrams—elements of the theoretical construct “frame based interpreting competence”. ZDMThe International Journal on Mathematics Education, 46(1) (this issue). doi:10.1007/s11858-013-0544-0.

  • Steinbring, H. (1994). Die Verwendung strukturierter Diagramme im Arithmetikunterricht der Grundschule. Zum Unterschied zwischen empirischer und theoretischer Mehrdeutigkeit mathematischer Zeichen. Mathematische Unterrichtspraxis, IV, 7–19.

  • Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction. An epistemological perspective. New York: Springer.

    Google Scholar 

  • Steinbring, H. (2006). What makes a sign a mathematical sign? An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61, 133–162.

    Article  Google Scholar 

  • Voigt, J. (1990). Mehrdeutigkeit als ein wesentliches Moment der Unterrichtskultur. In Beiträge zum Mathematikunterricht (pp. 305–308). Bad Salzdetfurth: Franzbecker.

  • Voigt, J. (1994). Negotiation of mathematical meaning and learning mathematics. Educational Studies in Mathematics, 26, 275–298.

    Article  Google Scholar 

  • Wood, T. (1994). Patterns of interaction and the culture of mathematics classrooms. In S. Lerman (Ed.), Cultural perspectives on the mathematics classroom (pp. 149–168). Dordrecht: Kluwer.

    Chapter  Google Scholar 

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Acknowledgments

Thank you to three anonymous reviewers for their ideas and helpful feedback and to a colleague and friend from Oxford University, UK, for editing the English version.

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Correspondence to Andrea Gellert.

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Gellert, A., Steinbring, H. Students constructing meaning for the number line in small-group discussions: negotiation of essential epistemological issues of visual representations. ZDM Mathematics Education 46, 15–27 (2014). https://doi.org/10.1007/s11858-013-0548-9

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