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Preformal proving: Examples and reflections

Abstract

The starting point of our reflections is a classroom situation in grade 12 in which it was to be proved intuitively that non-trivial solutions of the differential equationf′=f have no zeros. We give a working definition of the concept of preformal proving, as well as three examples of preformal proofs. Then we furnish several such proofs of the aforesaid fact, and we analyse these proofs in detail. Finally, we draw some conclusions for mathematics in school and in teacher training.

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Revised and extended version of a conference proceedings contribution in German: Warum haben nicht-triviale Lösungen vonf′=f keine Nullstellen? Beobachtungen und Bemerkungen zum inhaltlich-anschaulichen Beweisen. In: Kautschitsch, H. and Metzler, W. (eds.): Anschauliches Beweisen. Schriftenreihe Didaktik der Mathematik, Band 18. Wien/Stuttgart 1989, pp. 199–209. We would like to thank Prof. K. Heidenreich (Reutlingen) as well as the editors of ESM for several valuable hints.

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Blum, W., Kirsch, A. Preformal proving: Examples and reflections. Educ Stud Math 22, 183–203 (1991). https://doi.org/10.1007/BF00555722

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Keywords

  • Teacher Training
  • Classroom Situation