ZDM

, Volume 43, Issue 2, pp 205–217 | Cite as

Generating examples: focus on processes

Original Article

Abstract

Constructing an example can be a rich and complex activity, interesting to investigate mathematical thinking and with many potentialities in mathematics education. In this article, I analyse processes involved in example generation, with particular emphasis on production and transformation of signs representing mathematical objects and on generation of inferences. The richness and complexity of these processes will also be shown through the notions of prototypes, concept image and concept definition. This investigation reveals aspects that are significant both in education and for the reflection on cognitive and cultural aspects of mathematical thinking.

Keywords

Example generation Cognitive processes Semiotic set Argumentation and proof 

References

  1. Alcock, L. (2004). Uses of examples objects in proving. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 17–24). Bergen: Bergen University College.Google Scholar
  2. Alcock, L., & Inglis, M. (2009). Representation systems and undergraduate proof production: a comment on Weber. Journal of Mathematical Behavior, 28, 209–211.CrossRefGoogle Scholar
  3. Antonini, S. (2003). Non-examples and proof by contradiction. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (Vol. 2, pp. 49–55). Honolulu, Hawai’i: CRDG College of Education, University of Hawai’i.Google Scholar
  4. Antonini, S. (2006). Graduate students’ processes in generating examples of mathematical objects. In J. Novotnà, H. Moarovà, M. Kràtkà, & N. Stelìchovà (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 57–64). Prague: Charles University.Google Scholar
  5. Antonini, S. (2010). A model to analyse argumentations supporting impossibilities in mathematics. In Proceedings of the 34th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 153–160). Belo Horizonte: PME.Google Scholar
  6. Antonini, S., Furinghetti, F., Morselli, F., & Tosetto, E. (2007). University students generating examples in real analysis: where is the definition? In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the fifth congress of the European Society for research in mathematics education, CERME 5 (pp. 2241–2249). Larnaca, Cyprus. http://ermeweb.free.fr/CERME5b/.
  7. Antonini, S., & Mariotti, M. A. (2008). Indirect proof: what is specific to this way of proving? Zentralblatt für Didaktik der Mathematik, 40(3), 401–412.CrossRefGoogle Scholar
  8. Arzarello, F. (2006). Semiosis as a multimodal process. Relime, Número Especial, 267–299.Google Scholar
  9. Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18(2), 147–176.CrossRefGoogle Scholar
  10. Bills, L., Mason, J., Watson, A., & Zaslavsky, O. (2006). Research Forum 02. Exemplification: the use of examples in teaching and learning mathematics). In J. Novotnà, H. Moarovà, M. Kràtkà, & N. Stelìchovà (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 25–153). Prague: Charles University.Google Scholar
  11. Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 99–119). Dordrecht: Kluwer.Google Scholar
  12. Boero, P., Garuti, R., & Lemut, E. (1999). About the generation of conditionality of statements and its links with proving. In O. Zaslavski (Ed.), Proceedings of the 23rd conference of the international group for the psychology of mathematics education (Vol. 2, pp. 137–144). Haifa: PME.Google Scholar
  13. Bratina, T. A. (1986). Can your students give examples? Mathematics Teacher, 79, 524–526.Google Scholar
  14. Capobianco, M., & Molluzzo, J. C. (1978). Examples and counterexamples in graph theory. New York: North Holland.Google Scholar
  15. Carlson, M. P., & Bloom, I. (2005). The cyclic nature of problem solving: an emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58(1), 45–75.CrossRefGoogle Scholar
  16. Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33, 283–299.CrossRefGoogle Scholar
  17. Duval, R. (1995). Sémiosis et Pensée Humaine. Bern: Peter Lang.Google Scholar
  18. Edwards, A., & Alcock, L. (2009). How do undergraduate students navigate their example spaces? In Proceedings of the 13th annual conference on research in undergraduate mathematics education. http://sigmaa.maa.org/rume/crume2010/Archive/Edwards.pdf.
  19. Furinghetti, F., Morselli, F., & Antonini, S. (2011). To exist or not to exist: example generation in real analysis. Zentralblatt für Didaktik der Mathematik (this issue).Google Scholar
  20. Gelbaum, B. R., & Olmsted, J. M. H. (1964). Counterexamples in analysis. San Francisco: Holden-Day.Google Scholar
  21. Gelbaum, B. R., & Olmsted, J. M. H. (1990). Theorems and counterexamples in Mathematics. New York: Springer.Google Scholar
  22. Harel, G., & Sowder, L. (1998). Students’ proof schemes: results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research on collegiate mathematics education (Vol. 3, pp. 234–283). Providence: American Mathematical Society.Google Scholar
  23. Hazzan, O., & Zazkis, R. (1997). Constructing knowledge by constructing examples for mathematical concepts. In E. Pehkonen (Ed.), Proceedings of the 21st conference of the international group for the psychology of mathematics education (Vol. 4, pp. 299–306). Helsinki: University of Helsinki-Lahti Research and Training Center.Google Scholar
  24. Hintikka, J., & Remes, U. (1974). The method of analysis: Its geometrical origin and its general significance. Dordrecht: Reidel Publishing Company.Google Scholar
  25. Iannone, P., Inglis, M., Mejia-Ramos, J. P., Siemons, J., & Weber, K. (2009). How do undergraduate students generate examples of mathematical concepts? In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd conference of the international group for the psychology of mathematics education (Vol. 3, pp. 217–224). Thessaloniki: PME.Google Scholar
  26. Khaleelulla, S. M. (1982). Counterexamples in topological vector spaces. Berlin: Springer.Google Scholar
  27. Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. Cambridge: Cambridge University Press.Google Scholar
  28. Lakoff, G. (1987). Women, fire, and dangerous things. Chicago: University of Chicago Press.Google Scholar
  29. Mason, J., & Klymchuk, S. (2009). Using Counter-examples in calculus. London: Imperial College Press.Google Scholar
  30. Polya, G. (1945). How to solve it. Princeton: Princeton University Press.Google Scholar
  31. Presmeg, N. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595–610.CrossRefGoogle Scholar
  32. Romano, J. P., & Siegel, A. F. (1986). Counterexamples in probability and statistics. Monterey: Wadsworth & Brooks.Google Scholar
  33. Rosch, E. (1977). Human categorization. In N. Warren (Ed.), Studies in cross-cultural psychology (Vol. 1, pp. 1–49). London: Academic Press.Google Scholar
  34. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In D. A. Grows (Ed.), Handbook of research in mathematics learning and teaching (pp. 334–370). New York: MacMillan.Google Scholar
  35. Simon, A. M. (1996). Beyond inductive and deductive reasoning: the search for a sense of knowing. Educational Studies in Mathematics, 30, 197–210.CrossRefGoogle Scholar
  36. Steen, L. A., & Seebach, J. A, Jr. (1978). Counterexamples in topology. New York: Springer.Google Scholar
  37. Stoyanov, J. M. (1987). Counterexamples in probability. New York: Wiley.Google Scholar
  38. Tall, D. O., & Vinner, S. (1981). Concept image and concept definition with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar
  39. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: learners generating examples. Mahwah: Erlbaum.Google Scholar
  40. Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Journal of Mathematical Behavior, 28, 200–208.CrossRefGoogle Scholar
  41. Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student-teachers: The case of binary operation. Journal for Research in Mathematics Education, 27(1), 67–78.CrossRefGoogle Scholar
  42. Zazkis, R., & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27(2), 15–21.Google Scholar

Copyright information

© FIZ Karlsruhe 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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