Journal für Mathematik-Didaktik

, Volume 39, Issue 1, pp 69–96 | Cite as

Errors During Exploration and Consolidation—The Effectiveness of Productive Failure as Sequentially Guided Discovery Learning

Originalarbeit/Original Article
  • 54 Downloads

Abstract

Discovery learning has a long tradition in educational psychology and in mathematics education. While these two research strands are often only weakly connected, we demonstrate that it is fruitful to draw on the theories and findings from both perspectives. We base the design of our study on the instructional model “productive failure”, which is investigated in educational psychology, and for which one can find analogous models in mathematics education. Productive failure combines a phase of divergent discovery (exploration phase) with a phase of convergent organization encompassing structured instructional elements (instruction phase). Our research questions target both phases: Regarding the exploration phase, we investigated students’ strategies when they attempt to solve fraction problems prior to instruction. Regarding the instruction phase, we tested whether a guided elaboration on typical errors had an impact on students’ understanding. The study included two learning units. In both units, students of all experimental conditions first engaged in an identical exploration activity and then worked on elaboration tasks that introduced: a) only correct solutions; b) correct and erroneous solution attempts; or c) correct and erroneous solution attempts with prompts to compare these solutions. Our qualitative categorization of students’ solutions showed that the identified errors match the solution attempts used for the instruction phase. Posttest results indicate that including erroneous solution attempts in the instruction phase can be beneficial for learning units in which most students fail to come up with a correct solution themselves, but only if students are prompted to compare the erroneous solution attempts with correct solutions.

Keywords

Discovery learning Learning form errors Productive failure Exploration and consolidation Fractions Comparisons prompts 

Fehler beim Erkunden und Ordnen – die Wirksamkeit von Productive Failure als mehrphasiges Entdeckendes Lernen mit Unterstützung

Zusammenfassung

Entdeckendes Lernen hat eine lange Tradition in der pädagogisch-psychologischen und in der mathematikdidaktischen Forschung. Während diese beiden Forschungsstränge oft wenig verknüpft sind, zeigen wir in diesem Artikel, dass es nutzbringend ist, die beiden Perspektiven in einem Forschungsdesign zu verbinden. Unser Studiendesign basiert auf dem Instruktionsmodell „Productive Failure“, das in der pädagogischen Psychologie untersucht wird und ein Pendant in der Mathematikdidaktik hat. Productive Failure kombiniert eine Phase der divergenten Entdeckung (Explorationsphase) mit einer Phase konvergenter Organisation mit strukturierten Instruktionselementen (Instruktionsphase). In unserer Studie untersuchten wir beide Phasen: 1) Was für Lösungen erstellen Schülerinnen und Schüler in der Explorationsphase, wenn sie ohne vorherige inhaltliche Instruktion Bruchaufgaben lösen? 2) Fördert eine Fehlerverarbeitung in der Instruktionsphase, bei der zum Vergleichen angeregt wird, den Wissenserwerb? Unsere Studie beinhaltete zwei Lerneinheiten. In beiden Lerneinheiten bearbeiteten die Lernenden aller Bedingungen zunächst explorativ das gleiche Problem. Danach arbeiteten die Lernenden an Elaborationsaufgaben, die die richtigen Lösungen einführten. In dieser Instruktionsphase arbeiteten die Lernenden mit a) ausschließlich richtigen Lösungen, b) richtigen Lösungen und Lösungsansätzen oder c) richtigen Lösungen und fehlerhaften Lösungsansätzen mit Prompts. Die Prompts forderten sie auf diese fehlerhaften Lösungsansätze und die richtigen Lösungen zu vergleichen. Unsere qualitative Kategorisierung der Schülerlösungen aus der Explorationsphase zeigte, dass die identifizierten Fehler sich überwiegend mit den Fehlern decken, die in der Instruktionsphase eingeführt wurden. Die Posttest-Ergebnisse zeigen, dass das Präsentieren fehlerhafter Lösungsansätze in Lerneinheiten bei denen die meisten Lernenden nicht eigenständig die richtige Lösung entdecken lernförderlich sein kann, wenn die Lernenden dazu aufgefordert werden, diese mit den richtigen Lösungen zu vergleichen.

Schlüsselwörter

Entdeckendes Lernen Fehlerverarbeitung Productive Failure Erkunden und Ordnen Bruchrechnen Vergleichs-Prompts 

MESC Classification

C33 C73 D53 F43 

Notes

Acknowledgements

This research was supported by the Deutsche Forschungsgemeinschaft, DFG (LO 2196/1-1). We used concepts and material from the KOSIMA-project (www.ko-si-ma.de). We would like to thank the participating schools and students as well as our student assistant Simon Schoch.

References

  1. Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103, 1–18.CrossRefGoogle Scholar
  2. Ausubel, D. P. (1964). Some psychological and educational limitations of learning by discovery. The Arithmetic Teacher, 11, 290–302.Google Scholar
  3. Barzel, B., Leuders, T., Prediger, S., & Hußmann, S. (2013). Designing tasks for engaging students in active knowledge organization. In Watson, et al. (Ed.), ICMI Study 22 on Task Design—Proceedings of Study Conference. Oxford. (pp. 285–294).Google Scholar
  4. Berthold, K., Nückles, M., & Renkl, A. (2007). Do learning protocols support learning strategies and outcomes? The role of cognitive and metacognitive prompts. Learning and Instruction, 17(5), 564–577.CrossRefGoogle Scholar
  5. Bruner, J. S. (1961). The act of discovery. Harvard Educational Review, 31, 21–32.Google Scholar
  6. Büchter, A., & Leuders, T. (2005). Mathematikaufgaben selbst entwickeln. Lernen fördern—Leistung überprüfen. Berlin: Cornelsen Scriptor.Google Scholar
  7. Burkhardt, H., & Schoenfeld, A. H. (2003). Improving educational research: toward a more useful, more influential, and better-funded enterprise. Educational Researcher, 32, 3–14.CrossRefGoogle Scholar
  8. Chi, M. T. H., de Leeuw, N., Chiu, M. H., & Lavancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439–477.Google Scholar
  9. Devolder, A., van Braak, J., & Tondeur, J. (2012). Supporting self-regulated learning in computer-based learning environments: systematic review of effects of scaffolding in the domain of science education. Journal of Computer Assisted Learning, 28(6), 557–573.CrossRefGoogle Scholar
  10. Dillenbourg, P., Baker, M., Blaye, A., & O’Malley, C. (1996). The evolution of research on collaborative learning. In H. Spada & P. Reimann (Eds.), Learning in humans and machine: towards an interdisciplinary learning science (pp. 189–211). Oxford: Elsevier.Google Scholar
  11. Duit, R., & Treagust, D. F. (2003). Conceptual change: a powerful framework for improving science teaching and learning. International Journal of Science Education, 25, 671–688.CrossRefGoogle Scholar
  12. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214.CrossRefGoogle Scholar
  13. Eichelmann, A., Narciss, S., Schnaubert, L., & Melis, E. (2012). Typische Fehler bei der Addition und Subtraktion von Brüchen—Ein Review zu empirischen Fehleranalysen. Journal für Mathematik-Didaktik, 33(1), 29–57.CrossRefGoogle Scholar
  14. Ernest, P. (2014). Policy debates in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 480–484). Dordrecht: Springer.Google Scholar
  15. Flewelling, G., & Higginson, W. (2003). Teaching with rich learning tasks: a handbook. Adelaide: AAMT Inc.Google Scholar
  16. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Springer.Google Scholar
  17. Haverty, L., Koedinger, K., Klahr, D., & Alibali, M. (2000). Solving inductive reasoning problems in mathematics: not-so-trivial pursuit. Cognitive Science, 2, 249–298.CrossRefGoogle Scholar
  18. Heemsoth, T., & Heinze, A. (2014). The impact of incorrect examples on learning fractions: a field experiment with 6th grade students. Instructional Science, 42(4), 639–657.CrossRefGoogle Scholar
  19. Hußmann, S., Leuders, T., Barzel, B., & Prediger, S. (2011). Kontexte für sinnstiftendes Mathematiklernen (KOSIMA)—ein fachdidaktisches Forschungs- und Entwicklungsprojekt. In Beiträge zum Mathematikunterricht (pp. 419–422). Münster: wtm.Google Scholar
  20. Johnson, D. W., & Johnson, R. T. (2002). Learning together and alnone. Overview and meta-analysis. Asia Pacific Journal of Education, 22(1), 95–105.CrossRefGoogle Scholar
  21. Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19, 509–539.CrossRefGoogle Scholar
  22. Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). Expertise reversal effect. Educational Psychologist, 38, 23–31.CrossRefGoogle Scholar
  23. Kapur, M. (2012). Productive failure in learning the concept of variance. Instructional Science, 40, 651–672.CrossRefGoogle Scholar
  24. Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45–83.CrossRefGoogle Scholar
  25. Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41, 75–86.CrossRefGoogle Scholar
  26. Klahr, D., & Dunbar, K. (1988). Dual space search during scientific reasoning. Cognitive Science, 12, 1–48.CrossRefGoogle Scholar
  27. Klein, F. (1908). Elementarmathematik vom höheren Standpunkte aus. Teil I: Arithmetik, Algebra, Analysis. Leipzig: Teubner.Google Scholar
  28. Koedinger, K. R., Corbett, A. T., & Perfetti, C. (2012). The knowledge—learning—instruction framework: bridging the science-practice chasm to enhance robust student learning. Cognitive Science, 36(5), 757–798.CrossRefGoogle Scholar
  29. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge Press.CrossRefGoogle Scholar
  30. Lazonder, A. W. (2014). Inquiry learning. In J. M. Spector, M. D. Merrill, J. Elen & M. J. Bishop (Eds.), Handbook of research on educational communications and technology (pp. 453–464). New York: Springer.CrossRefGoogle Scholar
  31. Lazonder, A. W., & Harmsen, R. (2016). Meta-analysis of inquiry-based learning: effects of guidance. Review of Educational Research, 86(3), 681–718.CrossRefGoogle Scholar
  32. Leuders, T. (2014). Entdeckendes Lernen—Produktives Üben. In H. Linneweber-Lamerskitten (Ed.), Fachdidaktik Mathematik. Grundbildung und Kompetenzaufbau im Unterricht der Sek. I und II (pp. 236–263). Seelze: Klett/Kallmeyer.Google Scholar
  33. Leuders, T., Naccarella, D., & Philipp, K. (2011). Experimentelles Denken—Vorgehensweisen beim innermathematischen Experimentieren. Journal für Mathematik-Didaktik, 32(2), 205–231.CrossRefGoogle Scholar
  34. Loibl, K., & Rummel, N. (2014). Knowing what you don’t know makes failure productive. Learning and Instruction, 34, 74–85.CrossRefGoogle Scholar
  35. Loibl, K., Roll, I., & Rummel, N. (2017). Towards a theory of when and how problem solving followed by instruction supports learning. Educational Psychology Review, 29(4), 693–715.CrossRefGoogle Scholar
  36. McNamara, J., & Shaughnessy, M. M. (2011). Student errors: what can they tell us about what students do understand? http://akrti2015.pbworks.com/f/StudentErrors_JM_MS_Article.pdf. Accessed 02.03.2018.Google Scholar
  37. Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. American Psychologist, 59, 14–19.CrossRefGoogle Scholar
  38. Mazziotti, C., Loibl, K., & Rummel, N. (2015). Collaborative or individual learning within productive failure. Does the social form of learning make a difference? In O. Lindwall, P. Häkkinen, T. Koschman, P. Tchounikine & S. Ludvigsen (Eds.), Exploring the material conditions of learning: the computer supported collaborative learning (CSCL) conference 2015 (Vol. 2, pp. 570–575). Gothenburg: ISLS.Google Scholar
  39. Oser, F., Hascher, T., & Spychiger, M. (1999). Lernen aus Fehlern. Zur Psychologie des “negativen” Wissens. In W. Althof (Ed.), Fehlerwelten. Vom Fehlermachen und Lernen aus Fehlern (pp. 11–41). Opladen: Leske + Budrich.Google Scholar
  40. Renkl, A. (2005). The worked-out example principle in multimedia learning. In R. E. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 229–247). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  41. Padberg, F. (2009). Didaktik der Bruchrechnung für Lehrerausbildung und Lehrerfortbildung (4th edn.). Heidelberg: Spektrum Akademischer Verlag.CrossRefGoogle Scholar
  42. Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30(1), 5–38.CrossRefGoogle Scholar
  43. Pólya, G. (1954). Induction and analogy in mathematics. Vol. 1. Oxford: Oxford University Press.Google Scholar
  44. Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3–17.CrossRefGoogle Scholar
  45. Prediger, S. (2011). Anknüpfen, Konfrontieren, Gegenüberstellen. Strategien zur Weiterarbeit mit individuellen Vorstellungen am Beispiel relativer Häufigkeiten. Praxis der Mathematik in der Schule, 53(40), 8–13.Google Scholar
  46. Prediger, S. (2014). Focussing structural relations in the bar board—a design research study for fostering all students’ conceptual understanding of fractions. In B. Ubuz, C. Haser & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education. Ankara. (pp. 343–352).Google Scholar
  47. Prediger, S., & Schink, A. (2009). Three eights of which whole?—Dealing with changing referent wholes as a key to the part-of-part-model for the multiplication of fractions. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 409–416). Thessaloniki: PME.Google Scholar
  48. Prediger, S., Glade, M., & Schmidt, U. (2011). Wozu rechnen wir mit Anteilen? Herausforderung der Sinnstiftung am schwierigen Beispiel der Bruchoperationen. Praxis der Mathematik in der Schule, 53(37), 28–35.Google Scholar
  49. Prediger, S., Hußmann, S., Leuders, T., & Barzel, B. (2014). Kernprozesse—Ein Modell zur Strukturierung von Unterrichtsdesign und Unterrichtshandeln. In I. Bausch, G. Pinkernell & O. Schmitt (Eds.), Unterrichtsentwicklung und Kompetenzorientierung. Festschrift für Regina Bruder (pp. 81–92). Münster: WTM.Google Scholar
  50. Prediger, S., Barzel, B., Hußmann, S., & Leuders, T. (2013). mathewerkstatt 6. Berlin: Cornelsen.Google Scholar
  51. Pressley, M., Wood, E., Woloshyn, V. E., Martin, V., King, A., & Menke, D. (1992). Encouraging mindful use of prior knowledge: attempting to construct explanatory answers facilitates learning. Educational Psychologist, 27, 91–109.CrossRefGoogle Scholar
  52. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: does one lead to the other? Journal of Educational Psychology, 91, 175–189.CrossRefGoogle Scholar
  53. Schworm, S., & Renkl, A. (2007). Learning argumentation skills through the use of prompts for self-explaining examples. Journal of Educational Psychology, 99(2), 285–296.CrossRefGoogle Scholar
  54. Shulman, L. S., & Keislar, E. R. (1966). Learning by discovery: a critical appraisal. Chicago: Rand McNally.Google Scholar
  55. Sierpinska, A. (1994). Understanding in mathematics. London, Washington: The Falmer Press.Google Scholar
  56. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518.CrossRefGoogle Scholar
  57. Strauss, A. L., & Corbin, J. M. (1996). Grounded theory: Grundlagen qualitativer Sozialforschung. Weinheim: Beltz.Google Scholar
  58. Tobias, S. (1989). Another look at research on the adaptation of instruction to student characteristics. Educational Psychologist, 24, 213–227.CrossRefGoogle Scholar
  59. Tobias, S., & Duffy, T. (2009). Constructivist instruction: success or failure? New York: Routledge.Google Scholar
  60. 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.CrossRefGoogle Scholar
  61. Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. Learning and Instruction, 14(5), 445–451.CrossRefGoogle Scholar
  62. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: using variation to structure sense-making. Mathematical thinking and learning, 8(2), 91–111.CrossRefGoogle Scholar
  63. Winter, H. (1989). Entdeckendes Lernen im Mathematikunterricht: Einblicke in die Ideengeschichte und ihre Bedeutung für die Pädagogik. Braunschweig: Vieweg.CrossRefGoogle Scholar
  64. Wittmann, E. C. (1992). Wider die Flut der bunten Hunde und der grauen Päckchen: Die Konzeption des aktiv entdeckenden Lernens und produktiven Übens. In G. N. Müller & E. C. Wittmann (Eds.), Handbuch produktiver Rechenübungen (pp. 152–166). Stuttgart: Klett.Google Scholar

Copyright information

© GDM 2018

Authors and Affiliations

  1. 1.University of Education FreiburgFreiburgGermany

Personalised recommendations