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Zentralblatt für Didaktik der Mathematik

, Volume 34, Issue 1, pp 29–35 | Cite as

Learning to prove: The idea of heuristic examples

  • Kristina Reiss
  • Alexander Renkl
Analyses

Abstract

Proof is an important topic in the area of mathematics curriculum and an essential aspect of mathematical competence. However, recent studies have revealed wide gaps in student's understanding of proof. Furthermore, effective teaching to prove, for example, by Schoenfeld's approach, is a real challenge for teachers. A very powerful and empirically well founded method of learning mathematics, which is also relatively easy to implement in the classroom, is learning through worked-out examples. It is, however, primarily suited for algorithmic content areas. We propose the concept of using heuristic worked-out examples, which do not provide an algorithmic problem solution but offer instead heuristic steps that lead towards finding a proof. We rely on Boero's model of proving in designing the single sub-steps of a heuristic example. We illustrate our instructional idea by presenting an heuristic example for proving that the interior angles in any triangle add up to 180°.

ZDM-Classifikation

C30 C40 C70 D50 E50 G40 

Kurzreferat

Es ist ein wichtiges Ziel des Mathematikunterrichts in der Sekundarstufe, das die Schülerinnen und Schülern ein Verständrus für mathematisches Argumentieren und Beweisen entwickeln. Doch verschiedene neuere Studien belegen, dass das Erreichen dieses Ziels mit erheblichen Schwierigkeiten für Schüler und Lehrer verbunden ist. Nun ist empirisch gut belegt, dass das Lernen mithilfe ansgearbeiteter Lösungsbeispiele in der Mathematik zu guten Ergebnissen führen kann und darüber hinaus auch leicht in den konkreten Unterricht integriert werden kann Diese Methode ist allerdings im Wesentlichen für algorithmische Inhalte geeignet. Als eine didaktisch sinnvolle Erweiterung wird im folgenden das Konzept heuristischer ausgearbeiteter Lösungsbeispiele betrachtet. Dabei steht nicht ein Lösungsalgorithmus im Vordergrund, sondern die Aufeinanderfolge geeigneter heuristischer Schritte. Wir verwenden das Modell des Beweisens von Boero zum Aufbaus des Konzepts. Am Beispiel des Satzes von der Winkelsumme im Dreieck werden die grundlegenden Ideen konkretisiert.

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Copyright information

© ZDM 2002

Authors and Affiliations

  • Kristina Reiss
    • 1
  • Alexander Renkl
    • 2
  1. 1.Fachbereich MathematikCarl von Ossietzky-Universität OldenburgOldenburgGermany
  2. 2.Psychologisches InstitutUniversität FreiburgFreiburgGermany

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