Educational Studies in Mathematics

, Volume 94, Issue 2, pp 185–203 | Cite as

Students’ individual schematization pathways - empirical reconstructions for the case of part-of-part determination for fractions

Article

Abstract

According to the design principle of progressive schematization, learning trajectories towards procedural rules can be organized as independent discoveries when the learning arrangement invites the students first to develop models for mathematical concepts and model-based informal strategies; then to explore the strategies and to discover pattern for progressively developing procedural rules. This article contributes to the theoretical and empirical foundation of the design principle of progressive schematization by empirically investigating students’ individual schematization pathways on the micro-level for the specific case of part-of-part determination of fractions. In design experiments series in laboratory settings, nine pairs of sixth graders explored the part-of-part determination and progressively schematized their graphical strategies before discovering the procedural rule. The qualitative in-depth analysis of 760 min of video shows that progressive schematization is a multi-facetted process that cannot be described by internalization of graphical procedures alone. Instead, the compaction of concepts- and theorems-in-action is crucial, especially for the goal of justifiable procedural rules.

Keywords

Progressive schematization Students’ learning pathways Design research Multiplication of Fractions 

References

  1. Aebli, H. (1981). Denken: das Ordnen des Tuns. Band II: Denkprozesse. Stuttgart: Klett.Google Scholar
  2. Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing tasks for engaging students in active knowledge organization. In A. Watson, M. Ohtani, et al. (Eds.), ICMI study 22 on task design—Proceedings of the study conference (pp. 285–294). Oxford: ICMI.Google Scholar
  3. Behr, M., Cramer, K., Post, T., & Lesh, R. (2009). Rational number project: Initial fraction ideas. Online-Book. Minneapolis. University of Minnesota. Retrieved from  http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html.
  4. Buijs, K. (2008). Leren vermenigvuldigen met meercijferige getallen. Utrecht: Freudenthal Institute.Google Scholar
  5. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13. doi:10.3102/0013189X032001009.CrossRefGoogle Scholar
  6. Cramer, K., & Bezuk, N. (1991). Multiplication of fractions: Teaching for understanding. Arith Teach, 39(3), 34–37.Google Scholar
  7. Freudenthal, H. (1981). Major problems of mathematical education. Educational Studies in Mathematics, 12(2), 133–150. doi:10.1007/BF00305618.CrossRefGoogle Scholar
  8. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer.Google Scholar
  9. Glade, M. (2016). Individuelle Prozesse der fortschreitenden Schematisierung—Empirische Rekonstruktionen zum Anteil vom Anteil. Wiesbaden: Springer. doi:10.1007/978-3-658-11254-7.CrossRefGoogle Scholar
  10. Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. Van der Akker, K. Gravemeijer, S. McKenny, & N. Nieveen (Eds.), Educational design research (pp. 17–51). London: Routledge.Google Scholar
  11. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1–3), 111–129. doi:10.1023/A:1003749919816.
  12. Harun, H. Z. (2011). Evaluating the teaching and learning of fractions through modelling in Brunei (Doctoral dissertation). University of Manchester, UK. Retrieved from  https://www.escholar.manchester.ac.uk/api/datastream?publicationPid=uk-ac-man-scw:119340&datastreamId=FULL-TEXT.PDF.
  13. Krämer, S. (2003). “Schriftbildlichkeit” oder: Über eine (fast) vergessene Dimension der Schrift. In H. Bredekamp & S. Krämer (Eds.), Bild, Schrift, Zahl (pp. 157–176). München: Fink.Google Scholar
  14. Lorange, C. A., & Rinvold, R. A. (2013). Levels of objectification in student’s strategies. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 323–332). Ankara: METU University / ERME.Google Scholar
  15. Lorange, C. A., & Rinvold, R. A. (2014). Students’ strategies of expanding fractions to a common denominator—a semiotic perspective. Nord Math, 19(2), 57–75.Google Scholar
  16. Prediger, S. (2013). Focussing structural relations in the bar board—a design research study for fostering all students’ conceptual understanding of fractions. In B. Ubuz, Ç. Haser, & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 343–352). Ankara: METU University / ERME.Google Scholar
  17. Prediger, S., & Link, M. (2012). Fachdidaktische Entwicklungsforschung - Ein lernprozessfokussierendes Forschungsprogramm mit Verschränkung fachdidaktischer Arbeitsbereiche. In H. Bayrhuber, U. Harms, B. Muszynski, B. Ralle, M. Rothgangel, L.-H. Schön, H. Vollmer, & H.-G. Weigand (Eds.), Formate Fachdidaktischer Forschung (pp. 29–46). Münster: Waxmann.Google Scholar
  18. Prediger, S. & Schnell, S. (2014). Investigating the dynamics of stochastic learning processes: A didactical research perspective. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking (pp. 533–558). Dordrecht: Springer. doi:10.1007/978-94-007-7155-0_29.CrossRefGoogle Scholar
  19. Prediger, S., Schink, A., Schneider, C., & Verschraegen, J. (2013). Kinder weltweit—Anteile in Statistiken. In S. Prediger, B. Barzel, S. Hußmann, & T. Leuders (Eds.), Mathewerkstatt 6 [Textbook for Grade 6] (pp. 143–164). Berlin: Cornelsen.Google Scholar
  20. Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes—an overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877–891. doi:10.1007/s11858-015-0722-3.CrossRefGoogle Scholar
  21. Radford, L. (2012). Early algebraic thinking: Epistemological, semiotic and developmental issues. Paper presented at ICME, Seoul, South Korea. Retrieved from http://www.icme12.org/upload/submission/1942_F.pdf
  22. Stegmaier, W., & Herrmann, T. (1992). Schema, Schematismus. In J. Ritter & K. Gründer (Eds.), Historisches Wörterbuch der Philosophie (pp. 1245–1263). Basel: Schwabe.Google Scholar
  23. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht: Kluwer.CrossRefGoogle Scholar
  24. Treffers, A. (1979). Cijferend vermenigvuldigen en delen: (1) overzicht en achtergronden. Utrecht: Instituut Ontwikkeling Wiskunde Onderwijs.Google Scholar
  25. Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction. Dordrecht: Reidel.Google Scholar
  26. UNESCO (2009). Education for all in least developed countries. Montreal: UNESCO Institute for Statistics. Retrieved from http://www.uis.unesco.org/Education/Pages/gender-education.aspx → survival rate
  27. van den Heuvel-Panhuizen, M. (2001). Realistic mathematics education in the Netherlands. In J. Anghileri (Ed.), Principles and practices in arithmetic teaching (pp. 49–63). Buckingham: Open University Press.Google Scholar
  28. van den Heuvel-Panhuizen, M. (2003). 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. doi:10.1023/B:EDUC.0000005212.03219.dc.
  29. van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen, E., & Keijzer, R. (2008). Fractions, percentages, decimals and proportions. A learning-teaching trajectory for grade 4, 5 and 6. Rotterdam: Sense Publishers.Google Scholar
  30. Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe & P. Nesher (Eds.), Theories of mathematical learning (pp. 219–239). Mahwah: Lawrence Erlbaum.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University Duisburg-EssenFaculty for MathematicsEssenGermany
  2. 2.Institute for Development and Research in Mathematics EducationTU Dortmund UniversityDortmundGermany

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