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Visual representations as objects of analysis: the number line as an example

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Abstract

Our paper examines the representational nature of number lines as they are used in instructional tasks. The examination is informed by a so-called mathedidactical analysis of the number line as a tool used in teaching students mathematics. This analysis led to the identification of a family of number line models, based on visual aspects of number lines each reflecting different forms and functions. In the article, number line tasks are unpacked to illustrate the visual representational components of particular number line models. We illuminate how these components of the models provide tools to locate whole numbers and integers, operate with them, and facilitate reasoning and understanding of underlying mathematical concepts.

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References

  • Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.

    Article  Google Scholar 

  • Billstein, R., Libeskind, S., & Lott, J. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed.). Redding: Addison-Wesley.

    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 

  • Çemen, P. B. (1993). Adding and subtracting integers on the number line. Arithmetic Teacher, 40(7), 388–389.

    Google Scholar 

  • Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.

    Article  Google Scholar 

  • Detmer-Kratzin, C. (1997). “You can use the empty number line as a ruler, but it’s not as precise”—analysis of mathematics lessons on the use of the empty number line. Developments in mathematics education in Germany. Selected Papers from the Annual Conference on Didactics of Mathematics, Leipzig, 1997 (pp. 45–51). http://Webdoc.sub.gwdg.de/ebook/e/gdm/1997/index.html. Accessed 1 Oct 2011.

  • Devlin, K. (2008). Multiplication and those pesky British spellings. http://www.maa.org/devlin/devlin_09_08.html. Accessed 30 August 2008.

  • Diezmann, C. N., Lowrie, T., & Sugars, L. A. (2010). Primary students’ success on the structured number line. http://www.csu.edu.au/research/glim/downloads/Lowrie_2010_Structured%20number%20line.pdf. Accessed 8 April 2013.

  • Dr. Math. (2001). Rules for subtracting integers. http://mathforum.org/library/drmath/view/59046/html. Accessed 27 Apr 2009.

  • Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3–26). Columbus, OH: Clearinghouse for Science, Mathematics, and Environmental Education.

  • Earnest, D. (2007). In line with student reasoning: A research methodology with pedagogical potential. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (CD-Rom) (pp. 603–610). Reno, NV: University of Nevada, Reno.

  • Ernest, P. (1985). The number line as a teaching aid. Educational Studies in Mathematics, 16, 411–424.

    Article  Google Scholar 

  • Everyday Mathematics (n.d.). Common core instructional planning and pacing guide: Grade 1. http://www.everydaymath.com/pdf/em_pacing_guide_gr1.pdf. Accessed 8 April 2013.

  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: D. Reidel Publishing Company.

    Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Gray, E., & Doritou, M. (2008). The number line: Ambiguity and interpretation. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, A. Sepúlveda (Eds.), Proceedings of the joint Meeting of PME 32 and PME-NA XXX (Vol. 3, pp. 97–104). Mexico: Cinvestav-UMSNH.

  • Hannula, M. S. (2003). Locating fractions on a number line. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th conference of the International Group for the Psychology of Mathematics Education held jointly with the 25th conference of PME-NA (Vol. 3, pp. 17–24). Honolulu: CRDG, College of Education, University of Hawai’i.

    Google Scholar 

  • Heefer, A. (2011). Historical objections against the number line. Science & Education, 20(9), 863–880.

    Article  Google Scholar 

  • Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: a kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 265–281.

    Article  Google Scholar 

  • Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: helping children learn mathematics. Washington, DC: National Academy Press.

    Google Scholar 

  • Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Maletsky, E. M., & Roby, T. (2004). Harcourt math student edition. Orlando: Harcourt.

    Google Scholar 

  • NCTM (n.d.). NCTM illuminations: Resources for teaching math. http://illuminations.nctm.org/LessonDetail.aspx?ID=L53. Accessed 1 Oct 2012.

  • North Carolina Department of Public Instruction (n.d.). Instructional support tools for achieving new standards. http://ncpublicschools.org/docs/acre/standards/common-core-tools/organizers/math/number-lines.pdf. Accessed 8 April 2013.

  • O’Daffer, P., Charles, R., Cooney, T., Dossey, J., & Schielack, J. (1998). Mathematics for elementary school teachers. Redding: Addiso-Wesley.

    Google Scholar 

  • Peirce, C. S. (1998). The essential Peirce: Selected philosophical writings. In The Peirce Edition Project (Vol. 2, pp. 1893–1913). Bloomington, Indiana: Indiana University Press.

  • Peltenburg, M., Van den Heuvel-Panhuizen, M., Robitzsch, A. (2010). ICT-based dynamic assessment to reveal special education students’ potential in mathematics. Research Papers in Education, 25(3), 319–334.

    Google Scholar 

  • Saxe, G. B. (2004). Practices of quantification from a sociocultural perspective. In A. Demetriou & A. Raftopoulos (Eds.), Cognitive developmental change: Theories, models and measurement (pp. 241–263). New York: Cambridge University Press.

    Google Scholar 

  • Selter, C. (1998). Building on children’s mathematics—A teaching experiment in grade three. Educational Studies in Mathematics, 36, 1–27.

    Article  Google Scholar 

  • Skemp, R. R. (2002; a reprint of 1989). Mathematics in the primary school. London: RoutledgeFalmer.

  • Skoumpourdi, C. (2010). The number line: An auxiliary means or an obstacle? International Journal for Mathematics Learning. (October, 12, 2010). http://www.cimt.plymouth.ac.uk/journal/skoumpourdi.pdf. Accessed 8 April 2013.

  • Sonnabend, T. A. (1993). Mathematics for elementary teachers: An interactive approach. USA: Harcourt College Publishers.

  • Steinbring, H. (1997). Epistemolotical investigation of classroom interaction in elementary mathematics teaching. Educational Studies in Mathematics, 32, 49–92.

    Article  Google Scholar 

  • Streefland, L. (1993). The design of a mathematics course. A theoretical reflection. Educational Studies in Mathematics, 25, 190–135.

    Google Scholar 

  • Strickland, S., Wood, M., & Parks, A. (2005). Number lines: Students across grade levels making meaning through metaphor. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Roanoke, VA: Virginia Polytechnic Institute and State University. http://filebox.vt.edu/users/lloyd/pmena2005/short_orals/so_strickland.pdf. Accessed 1 Oct 2012.

  • Van den Heuvel-Panhuizen, M. (2003a). The learning paradox and the learning miracle: thoughts on primary school mathematics education. Journal für Mathematik-Didaktik, 24(2), 96–121.

    Article  Google Scholar 

  • Van den Heuvel-Panhuizen, M. (2003b). The didactical use of models in Realistic Mathematics Education: an example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.

    Article  Google Scholar 

  • Van den Heuvel-Panhuizen, M. (2008). Learning from “didactikids”: an impetus for revisiting the empty number line. Mathematics Education Research Journal, 20(3), 6–31.

    Article  Google Scholar 

  • Van den Heuvel-Panhuizen, M., & Treffers, A. (2009). Mathe-didactical reflections on young children’s understanding and application of subtraction-related principles. Mathematical Thinking and Learning, 11(1–2), 102–112.

    Article  Google Scholar 

  • Yanik, H. B., Helding, B., & Baek, J. M. (2006). Students’ difficulties in understanding fractions as measures. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the joint meeting of PME 32 and PME-NA XXX (Vol. 2, pp. 323–325). Mexico: Cinvestav-UMSNH.

    Google Scholar 

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Authors and Affiliations

  1. Livingston, MT, USA

    Anne Teppo

  2. Freudenthal Institute for Science and Mathematics Education, Faculty of Science & Department of Special Education for Cognitive and Motor Disabilities, Faculty of Social and Behavioural Science, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Marja van den Heuvel-Panhuizen

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  1. Anne Teppo
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  2. Marja van den Heuvel-Panhuizen
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Correspondence to Marja van den Heuvel-Panhuizen.

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Teppo, A., van den Heuvel-Panhuizen, M. Visual representations as objects of analysis: the number line as an example. ZDM Mathematics Education 46, 45–58 (2014). https://doi.org/10.1007/s11858-013-0518-2

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