Educational Studies in Mathematics

, Volume 22, Issue 2, pp 183–203 | Cite as

Preformal proving: Examples and reflections

  • Werner Blum
  • Arnold Kirsch


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.


Teacher Training Classroom Situation 
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  1. BalacheffN.: 1987, ‘Processus de preuve et situations de validation’,Educational Studies in Mathematics 18, 147–176.Google Scholar
  2. BlumW. and KirschA.: 1979, ‘Zur Konzeption des Analysisunterrichts in Grundkursen’,Der Mathematikunterricht 25(3), 6–24.Google Scholar
  3. BranfordB.: 1908,A Study of Mathematical Education, Clarendon, Oxford.Google Scholar
  4. FischbeinE.: 1987, ‘Intuition and analytical thinking in mathematics education’,Zentralblatt für Didaktik der Mathematik 15, 68–74.Google Scholar
  5. FischerR. and MalleG.: 1985,Mensch und Mathematik, Bibliogr. Institut, Mannheim.Google Scholar
  6. HannaG.: 1983,Rigorous Proof in Mathematics Education, OISE, Toronto.Google Scholar
  7. HannaG.: 1989, ‘More than formal proof’,For the Learning of Mathematics 9(1), 20–23.Google Scholar
  8. HeidenreichK.: 1987, ‘Der Rotwein-Beweis’,Praxis der Mathematik 29, 136–138.Google Scholar
  9. Jahnke, H.-N.: 1978,Zum Verhältnis von Wissensentwicklung und Begründung in der Mathematik—Beweisen als didaktisches Problem, IDM Materialien und Studien, vol. 10, Bielefeld.Google Scholar
  10. Kirsch, A.: 1977, ‘Aspects of simplification in mathematics teaching’, in: Athen, H. and Kunle, H. (eds.),Proceedings of the Third International Congress on Mathematical Education ICME-3, Karlsruhe, pp. 98–120.Google Scholar
  11. KirschA.: 1979, ‘Beispiele für “prämathematische” Beweise’, in: DörflerW. and FischerR. (eds.),Beweisen im Mathematikunterricht, Schriftenreihe Didaktik der Mathematik, vol. 2, Teubner, Stuttgart, pp. 261–274.Google Scholar
  12. Lakatos, I.: 1976,Proofs and Refutations, Cambridge Univ. Press.Google Scholar
  13. Lakatos, I.: 1978,Mathematics, Science and Epistemology (Philos. Papers, vol. 2), Cambridge Univ. Press.Google Scholar
  14. NeubrandM.: 1989, ‘Remarks on the acceptance of proofs: The case of some recently tackled major theorems’,For the Learning of Mathematics 9(3), 2–6.Google Scholar
  15. Polya, G.: 1968,Mathematics and Plausible Reasoning, Vol. II, Princeton Univ. Press.Google Scholar
  16. SemadeniZ.: 1984, ‘Action proofs in primary mathematics teaching and in teacher training’,For the Learning of Mathematics 4(1), 32–34.Google Scholar
  17. StahelE.: 1985, ‘Falsche Elementarisierung’,Praxis der Mathematik 27, 72–79.Google Scholar
  18. Thom, R.: 1973, ‘Modern mathematics: Does it exist?’, in: Howson, A. G. (ed.),Developments in Mathematical Education, Cambridge Univ. Press, pp. 194–209.Google Scholar
  19. WittmannE. C.: 1989, ‘The mathematical training of teachers from the point of view of education’,Journal für Mathematik-Didaktik 10, 291–308.Google Scholar
  20. WittmannE. C. and MüllerG.: 1988, ‘Wann ist ein Beweis ein Beweis?’, in: BenderP. (ed.),Mathematikdidaktik: Theorie und Praxis, Cornelsen, Berlin, pp. 237–257.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Werner Blum
    • 1
  • Arnold Kirsch
    • 1
  1. 1.Department of MathematicsKassel UniversityKasselGermany

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