Fundamental concepts and their key properties in probability: how to identify them and provide intuitions that support them
DOI:
https://doi.org/10.5007/1981-1322.2019.e67500Abstract
Probability is highlighted by theoretical concepts, which are far from being intuitive. The first step is to identify key concepts; the second is to clarify these concepts not only using mathematical tools by illustrating either their value in context or their specific properties. We focus on facets of probability that allow for linking its various interpretations, or that link probability to statistical inference. We re-state fundamental problems with probability that arise from the specificity of the concept and the purpose for which it has been “designed”. This specificity requires meta strategies that go far beyond the instruction of the mathematical details. In this paper, we focus on theoretical aspects of key ideas of probability and how to provide intuitions and images that are sustainable.
References
Batanero, C. (2016). Understanding randomness: Challenges for research and teaching. In K. Krainer and N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 34-49). Prague: ERME.
Batanero, C. & Borovcnik, M. (2016). Statistics and probability in high school. Rotterdam: Sense.
Batanero, C., Chernoff, E., Engel, J. Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. ICME-13 Topical Surveys. Cham, Switzerland: Springer.
Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In: A. G. Jones (Ed.), Exploring probability in school (pp. 15-37). New York: Springer.
Bennett, D. J. (1999). Randomness. Cambridge, MA: Harvard University Press.
Borovcnik, M. (1992). Stochastik im Wechselspiel von Intuitionen und Mathematik (Stochastics in the interplay between intuitions and mathematics). Mannheim: Bibliographisches Institut.
Borovcnik, M. (2006). Probabilistic and statistical thinking. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 484-506). Barcelona: European Society for Research in Mathematics Education.
Borovcnik, M. (2011). Strengthening the role of probability within statistics curricula. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics–challenges for teaching and teacher education. (pp. 71-83). New York: Springer.
Borovcnik, M. (2012). Multiple perspectives on the concept of conditional probability. Avances de Investigación en Didactica de la Matemática, 1(2), 5-27.
Borovcnik, M. (2015a). Central theorems of probability theory and their impact on probabilistic intuitions. In J. M. Contreras, et al. (Eds.), Didáctica de la Estadística, Probabilidad y Combinatoria, 2 (pp. 15-35). Granada, Universidad de Granada.
Borovcnik, M. (2015b). Risk and decision making: The “logic” of probability. The Mathematics Enthusiast, 12(1, 2 & 3), 113-139.
Borovcnik, M., & Bentz, H. J. (1990, 2003). Intuitive Vorstellungen von Wahrscheinlichkeits-konzepten: Fragebögen und Tiefeninterviews (Intuitive conceptions of probability: Questionnaire and in-depth interviews). Technical Reports. Klagenfurt University.
Borovcnik, M., Kapadia, R. (2011). Modelling in probability and statistics – Key ideas and innovative examples. In J. Maaß, & J. O‘Donoghue (Eds.), Real-World Problems for Secondary School Students–Case Studies (pp. 1-44). Rotterdam: Sense Publishers.
Borovcnik, M., & Kapadia, R. (2014). A historical and philosophical perspective on probability. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (pp. 7-34). New York: Springer.
Borovcnik, M. & Kapadia, R. (2018). Reasoning with risk: Teaching probability and risk as twin concepts. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics – Advances in Probability Education Research (pp. 3-22). ICME 13 Topic Series. Cham, Switzerland: Springer International Publishing.
Carranza, P. & Kuzniak, A. (2008). Duality of probability and statistics teaching in French education. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Joint ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges for Teaching and Teacher Education. Monterrey: ICMI and IASE.
Chernoff, E. J. (2013). Probabilistic relativism: A multivalentological investigation of normatively incorrect relative likelihood comparisons. Philosophy of Mathematics Education Journal, 27, 1-30.
Chernoff, E. & Sriraman, B. (Eds.) (2014). Probabilistic thinking: presenting plural perspectives. Advances in Mathematics Education (Vol. 7). New York: Springer.
de Finetti, B. (1937/1992). La prévision: ses lois logiques, ses sources subjectives. Annales Institut Henri Poincaré, 7, 1-68 (1937); English translation: Foresight: Its logical laws, its subjective sources. In S. Kotz, & N. L. Johnson (Eds.), Breakthroughs in statistics. (pp. 134-174). New York, Berlin: Springer.
de Finetti, B. (1974). Theory of probability. New York: Wiley (Translated by A. Machi & A. Smith).
David, F. N. (1962). Games, gods and gambling. London: Griffin.
Dubben, H.-H., & Beck-Bornholdt, H.-P. (2010). Mit an Sicherheit grenzender Wahrscheinlichkeit. Logisches Denken und Zufall. Reinbek: Rowohlt.
Dürr, D., Goldstein, S., Tumulka, R., & Zanghi, N. (2004). Bohmian mechanics and quantum field theory. In: Physical Review Letters, 93(9).
Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrecht: Reidel.
Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Reidel,
Gal, I. (2002). Adults’ statistical literacy: Meanings, components, responsibilities (with discussion). International Statistical Review, 70(1), 1-51.
Gal, I. (2005). Towards “probability literacy” for all citizens: Building blocks and instructional dilemmas. In G.A. Jones (Ed.), Exploring probability in school (pp. 39-63). Dordrecht: Kluwer.
Gigerenzer, G. (2002). Calculated risks: How to know when numbers deceive you. New York: Simon & Schuster.
Hacking, I. (1975). The emergence of probability. Cambridge: Cambridge University Press.
Heitele, D. (1975). An epistemological view on fundamental stochastic ideas. Educational Studies in Mathematics, 6, 187-205.
Jones, G. A., Langrall, C.W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning, Vol. 2 (pp. 909-956). Greenwich, CT: Information Age Publishing.
Kahneman, D. & Tversky, A. (1972). Subjective probability. A judgment of representativeness. Cognitive Psychology 3(3), 430-454.
Kahneman, D. & Tversky, A. (1973). Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 5(2), 207-232
Kahneman, D. & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-292.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.
Kapadia, R., Borovcnik, M. (Eds.) (1991). Chance encounters. Probability in education. Dordrecht: Kluwer.
Kolmogorov, A. N. (1933/1956). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin; English translation: Foundations of the theory of probability. New York: Chelsea.
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59-98.
Laplace, P. S. de (1814/1995). Théorie analytique des probabilités, 2nd ed. Paris: Courcier; 1st ed. 1812; reprinted 1995. Paris: Jacques Gabay.
Lecoutre, M. P. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23(6), 557-568.
Mises, R. v. (1919). Grundlagen der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 5, 52–99.
Piaget, J. & Inhelder, B. (1951). La genèse de l’idée de hasard chez l’enfant (The origin of the idea of chance in children). Paris: Presses Universitaires de France.
Simons, P. R. (1984). Instructing with analogies. Journal of Educational Psychology, 76(3), 513–527.
Spiegelhalter, D. (2014a). Probabilistic thinking. In: E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (Back cover). New York: Springer.
Spiegelhalter, D. (2014b, April). What can education learn from real-world communication of risk and uncertainty? Eight British Congress on Mathematical Education, Nottingham.
Steinbring, H. (1991). The theoretical nature of probability in the classroom. In: R. Kapadia & M. Borovcnik (Eds.). Chance encounters (pp. 135-167). Dordrecht: Kluwer.
Styer, D. F. (2000). The strange world of quantum mechanics. Cambridge: Cambridge University Press.
Székely, G.J. (1986). Paradoxes in probability and mathematical statistics. Dordrecht: Reidel.
Tversky, A. & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124- 1130.
von Plato, J. (1994). Creating modern probability: its mathematics, physics and philosophy in historical perspective. Cambridge: Cambridge University Press.
Downloads
Published
Issue
Section
License
Authors hold the copyright and grant the journal the right for their articles' first publication, being their works simultaneously licensed under the Creative Commons Attribution License (CC BY), which allows the sharing of such works with its authorship acknowledged and its initial publication in this journal.
Authors are allowed to enter into separate additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or as a book chapter, with an acknowledgment of its initial publication in this journal).
