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Proving is convincing and explaining

Abstract

In mathematical research, the purpose of proof is to convince. The test of whether something is a proof is whether it convinces qualified judges. In the classroom, on the other hand, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding, not to meet abstract standards of “rigor” or “honesty.”

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References

  1. Bishop, E.: 1972,Aspects of Constructivism, New Mexico State University, Mathematical Sciences, Las Cruces, N.M.

    Google Scholar 

  2. Davis, P. J.: 1977, ‘Proof, completeness, transcendentals, and sampling’,Journal of the Association for Computing Machinery 24, 298–310.

    Google Scholar 

  3. Davis, P. J. and Hersh, R.: 1981,The Mathematical Experience, Boston, Birkhauser.

    Google Scholar 

  4. de Villiers, M.: 1990, ‘The role and function of proof in mathematics’,Pythagoras 24, 17–24.

    Google Scholar 

  5. de Villiers, M.: 1991, ‘Pupils' needs for conviction and explanation within the context of geometry’,Pythagoras 26, 18–27.

    Google Scholar 

  6. Gale, D.: 1991, ‘Proof as explanation’,The Mathematical Intelligencer 12, 4.

    Google Scholar 

  7. Gleick, J.: 1987,Chaos, Penguin Books.

  8. Halmos, P.: 1990, Address to 75th annual summer meeting of the Mathematical Association of America, Columbus, Ohio. (Tape recording).

  9. Hanna, G.: 1983,Rigorous Proof in Mathematics Education, Toronto, OISE Press.

    Google Scholar 

  10. Hanna, G.: 1990, ‘Some pedagogical aspects of proof’,Interchange 21, 6–13.

    Google Scholar 

  11. Hardy, G. H.: 1929, ‘Mathematical proof’,Mind, XXXVIII,149, 1–25.

    Google Scholar 

  12. Hardy, G. H.: 1967,A Mathematician's Apology, Cambridge University Press.

  13. Hungerford, T. W.: 1990,Abstract Algebra: An Introduction, Saunders College Publishing.

  14. Knuth, D. E.: 1976, ‘Mathematics and computer science: Coping with finiteness’,Science 194, 1235–1242.

    Google Scholar 

  15. Lakatos, I.: 1976,Proofs and Refutations, Cambridge University Press.

  16. Leron, U.: 1983, ‘Structuring mathematical proofs’,American Mathematical Monthly 90, 174–185.

    Google Scholar 

  17. Meyer, A. R.: 1974, ‘The inherent computational complexity of theories of ordered sets’,Proceedings of the International Congress of Mathematicians 1972 2, 481.

    Google Scholar 

  18. Miller, G. L.: 1976, ‘Riemann's hypothesis and tests for primality’,J. Comput. Syst. Sci. 13, 300–317.

    Google Scholar 

  19. Rabin, M. O.: 1976, ‘Probabilistic algorithms’, in J. F. Traub (ed.),Algorithms and Complexity: New Directions and Recent Results, New York, Academic Press.

    Google Scholar 

  20. Renz, P.: 1982, ‘Mathematical proof: What it is and what it ought to be’,The Two-Year College Mathematics Journal 12, 83–103.

    Google Scholar 

  21. Schwartz, J. T.: 1980, ‘Fast probabilistic algorithms for verification of polynomial identities’,Journal of the Association for Computing Machinery 27, 701–717.

    Google Scholar 

  22. Swart, E. R.: 1980, ‘The philosophical implications of the four-color theorem’,The American Mathematical Monthly 87, 697–707.

    Google Scholar 

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Hersh, R. Proving is convincing and explaining. Educ Stud Math 24, 389–399 (1993). https://doi.org/10.1007/BF01273372

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Keywords

  • Mathematics Classroom
  • Mathematical Research
  • Abstract Standard
  • Qualified Judge