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

  • Katharina Loibl
  • Timo Leuders
Originalarbeit/Original Article


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.


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


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.


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

MESC Classification

C33 C73 D53 F43 



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


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Copyright information

© GDM 2018

Authors and Affiliations

  1. 1.University of Education FreiburgFreiburgGermany

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