Advertisement

The Meaning and Understanding of Mathematics

The Case of Probability
  • Carmen Batanero
  • Carmen Díaz
Part of the Mathematics Education Library book series (MELI, volume 42)

Abstract

We summarize a model with which to analyze the meaning of mathematical concepts, distinguishing five interrelated components. We also distinguish between the personal and the institutional meaning to differentiate between the meaning that has been proposed for a given concept in a specific institution, and the meaning given to the concept by a particular person in the institution. We use these ideas to analyze the historical emergence of probability and its different current meanings (intuitive, classical, frequentist, propensity, logical, subjective and axiomatic). We furthermore describe mathematical activity as a chain of semiotic functions and introduce the idea of semiotic conflict that can be used to give an alternative explanation to some widespread probabilistic misconceptions.

Keywords

Probability history of probability proof semiotics misconceptions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Batanero, C., Henry, M., and Parzysz, B. (2005). The nature of chance and probability. In G. A. Jones (Ed.) Exploring probability in school: Challenges for teaching and learning (15-37). New York: Springer.Google Scholar
  2. Bellhouse, D. R. (2000). De Vetula: a medieval manuscript containing probability calculations. International Statistical Review, 68 (2), 123-136.CrossRefGoogle Scholar
  3. Bernoulli, Jacques (1987). Ars Conjectandi- 4ème partie (N. Meunier, Trans.) Rouen: IREM. (Original work published in 1713).Google Scholar
  4. Biehler, R. (1997). Software for learning and doing statistics. International Statistics Review, 167-189.Google Scholar
  5. Borovcnik, M., and Peard, R. (1996). Probability. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, and C. Laborde (Eds.) International Handbook in Mathematics Education (Part 1, 239-288). Dordrecht: Kluwer.Google Scholar
  6. Cabriá, S. (1992). Filosofía de la probabilidad (Philosophy of probability). Valencia: Tirant le Blanc.Google Scholar
  7. Callaert, H. (2003). In search of the specificity and the identifiability of stochastic thinking and reasoning. Proceedings of CERME III,Bellaria, Italy, 2003. On line: http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG5/TG5_list3.html.Google Scholar
  8. Cardano, G. (1961). The book on games of chances. New York: Holt, Rinehart and Winston. (Original work Liber de Ludo Aleae; included in Opera Omnia, vol.1, published in 1663).Google Scholar
  9. Carnap, R. (1950). Logical foundations of probability, Chicago: University of Chicago Press.Google Scholar
  10. Díaz, C., and de la Fuente, I. (2006). Assessing psychology students’ difficulties with conditional probability and bayesian reasoning. In A. Rossman y B. Chance (Eds.) Proceedings of the Seventh International Conference on Teaching Statistics. Salvador (Bahia), Brazil: International Association for Statistical Education. CD ROM.Google Scholar
  11. Eco, U. (1979). Tratado de semiótica general (General semiotics). Barcelona: Lumen.Google Scholar
  12. Falk, R. (1979). Revision of probability and the time axis. In Proceedings of the third International Conference for the Psychology of Mathematics Education (64-66). Warwick, U. K: Organising Committee.Google Scholar
  13. Falk, R. (1989). Inference under uncertainty via conditional probabilities. In R. Morris (Ed.) Studies in mathematics education: The teaching of statistics, 7, 175-184, Paris: UNESCOGoogle Scholar
  14. Finetti, de B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institute Henri Poincaré, 7, 1-68.Google Scholar
  15. Gal, I. (2005). Democratic access to probability: Issues of probability literacy. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning. (39-63). New York: Springer.Google Scholar
  16. Gillies, D. (2000). Varieties of propensities. British Journal of Philosophy of Science, 51, 807-835.CrossRefGoogle Scholar
  17. Godino, J. D. (2002). Un enfoque ontológico y semiótico de la cognición matemática. (A semiotic and ontological approach to mathematical cognition). Recherches en Didactiques des Mathematiques, 22, 2-3.Google Scholar
  18. Godino, J. D., and Batanero, C. (1994). Significado personal e institucional de los objetos matemáticos (Institutional and personal meaning of mathematical objects). Recherches en Didactiques des Mathématiques, 14 (3), 325-355.Google Scholar
  19. Godino, J. D., and Batanero, C. (1998) Clarifying the meaning of mathematical objects as a priority area of research in Mathematics Education. In A. Sierpinska, and J. Kilpatrick (Eds.) Mathematics Education as a Research Domain: A Search for Identity (177-195). Dordrecht: Kluwer.Google Scholar
  20. Godino, J. D., and Batanero, C. (1999). The meanings of mathematical objects as análisis units for didactics of mathematics. In I. Schwank (Ed.) Proceedings of the First Conference of the European Society for Research in Mathematics Education (Vol. 2, 232-244). Osnabrück: Forschungsinstitut für Mathematikdidaktik. Hacking, I. (1965). The logic of statistical inference. Cambridge: Cambridge University Press.Google Scholar
  21. Godino, J. D., Batanero, C., and Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students. Educational Studies in Mathematics, 60 (1), 3-36.CrossRefGoogle Scholar
  22. Godino, J. D., Contreras, A., and Font, V. (In press). Análisis de procesos de instrucción basado en el enfoque ontológico-semiótico de la cognición matemática (Analysing instruction processes with base on the onto-semiotic approach to mathematical cognition). Recherches en Didactiques des Mathematiques.Google Scholar
  23. Hacking, I. (1975). The emergence of probability. Cambridge, MA: Cambridge University Press.Google Scholar
  24. Henry, M. (1997). L’enseignement des statistiques et des probabilities [Teaching of statistics and probability]. In P. Legrand (Coord.), Profession enseignant: Les maths en collège et en lycée (254-273). Paris: Hachette-Éducation,Google Scholar
  25. Huygens, C. (1998). L’art de conjecturer [The art of conjecturing]. (J. Peyroux, Trans.). Paris: A. Blanchard, (Original work Ratiociniis in aleae ludo, published 1657).Google Scholar
  26. Kahneman, D., and Tversky, A. (1982). Judgments of and by representativeness. In D. Kahhenam, P. Slovic and A. Tversky (Eds.) Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.Google Scholar
  27. Keynes, J. M. (1921). A treatise on probability. New York: Macmillan.Google Scholar
  28. Laplace P. S. (1985). Ensayo filosöfico sobre las probabilidades (Philosophical essay on probability). Madrid: Alianza Editorial (Original work published in 1814).Google Scholar
  29. M.E.C. (1992). Decretos de Ense≁anza Secundaria Obligatoria (Secondary school guidelines). Madrid: Ministerio de Educación y Ciencia.Google Scholar
  30. Mellor, D. H. (1971). The matter of chance. Cambridge. Cambridge University Press.Google Scholar
  31. Mises, R. von (1952). Probabilidad, estadística y verdad [Probability, statistics and truth]. Madrid: Espasa Calpe (Original work published 1928).Google Scholar
  32. Moivre, A. de (1967). The doctrine of chances (3rd ed.). New York: Chelsea Publishing (Original work published in 1718).Google Scholar
  33. N.C.T.M. (2000). Principles and standards for school mathematics. Reston, VA; N.C.T.M. http://standards.nctm.org/Google Scholar
  34. Parzysz, B. (2003). L’Enseignement de la statistique et des probabilités en France: Évolution au cours d’une carriére d’enseignant (période 1965-2002) (Teaching of statistics and probability in France. Evolution along a teacher’s professional work (period 1965-2002). In B. Chaput (Coord), Probabilités au lycée. Paris: Commission Inter-Irem Statistique et Probabilités.Google Scholar
  35. Pascal, B. (1963a). Correspondance avec Fermat (Correspondence with Fermat). In Oeuvres Complètes (43-49). París: ed. Seuil. (Original letter in 1654).Google Scholar
  36. Pascal, B. (1963b). Traité du triangle arithmétique et traités connexes (Traitise of arithmetical triangle and related treatises). In Oeuvres Complètes (50-100). París: ed. Seuil. (Original letter in 1654).Google Scholar
  37. Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31, 602-625.CrossRefGoogle Scholar
  38. Peirce, C. S. (1932). Notes on the doctrine of chances. In Collected papers of C. S. Peirce (Vol 2. 404-414). Havard University Press. (Original work published in 1910).Google Scholar
  39. Popper, K. R. (1957). The propensity interpretation of the calculus of probability and the quantum theory. In S. Körner (Ed.) Observation and interpretation The Colston Papers (Vol. 9, 65-70). University of Bristol.Google Scholar
  40. Popper, K. R. (1959). The propensity Interpretation of probability. British Journal of the Philosophy of Science, 10, 25-42.CrossRefGoogle Scholar
  41. Ramsey, F. P. (1926). Truth and probability. In F. P. Ramsey (1931), The Foundations of Mathematics and other Logical Essays (Ch. VII, 156-198), edited by R.B. Braithwaite, London: Kegan, Paul, Trench, Trubner and Co.Google Scholar
  42. Sides, A., Osherson, D., Bonini, N., and Viale, R. (2002). On the reality of the conjunction fallacy. Memory and Cognition, 30(2), 191-198.Google Scholar
  43. Székely, G. J. (1986). Paradoxes in probability theory and mathematical statistics. Dordrecht, the Netherlands: Reidel.Google Scholar
  44. Tentory, K., Bonini, N., and Osherson, D. (2004). The conjunction fallacy: a misunderstanding about conjunction? Cognitive Science, 28, 467-477.CrossRefGoogle Scholar
  45. Truran, K. M. (1995). Animism: A view of probability behaviour. In B. Atweh and S. Flavel (Eds.) Proceedings of the Eighteenth Annual Conference of the Mathematics Education Group of Australasia (MERGA) (537-541). Darwin, N. T.: MERGA.Google Scholar
  46. Vergnaud, G. (1982). Cognitive and developmental psychology and research in mathematics education: some theoretical and methodological issues. For the Learning of Mathematics 3 (2), 31-41.Google Scholar
  47. Vergnaud, G. (1990). La théorie des champs conceptuels (Theory of conceptual fields). Recherches en Didactiques des Mathématiques, 10 (2-3), 133-170.Google Scholar
  48. Vygotski, L. S. (1934). Pensamiento y lenguaje. (Thought and language). Madrid: Visor, 1993.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Carmen Batanero
    • 1
  • Carmen Díaz
    • 1
  1. 1.University of GranadaSpain

Personalised recommendations