Abstract
Children had to choose one of two urns—each comprising beads of winning and losing colours—from which to draw a winning bead. Three experiments, aimed at diagnosing rules of choice and designed without confounding possible rules with each other, were conducted. The level of arithmetic difficulty of the trials was controlled so as not to distort the effects of the constituent variables of proportion. Children aged 4 to 11 first chose by more winning elements and proceeded with age to choices by greater proportion of winning elements. There were some indications of intermediate choices by fewer losing elements and by greater difference between the two colours. Distinguishing correct choices from favourable draws, namely acknowledging the role of chance in producing the outcome and insisting on the right choice, grew with age. Children switched rather early from considering one dimension to two; they combined the quantities of winning and losing elements either additively by difference or, with age, mostly multiplicatively by proportion. Guided playful activities for young children, based on this research, are suggested for fostering acquisition of the basic constituents of the probability concept: uncertainty of outcome, quantification by proportion, and the reverse relation between the chances of complementary events.
Similar content being viewed by others
Notes
The apparent certainty of success following a correct choice in Noelting’s (1980a, b) task is due to the law of large numbers. The sampled beverage for drinking comprises multitude of molecules. Were the participants to draw a single molecule, the situation would reduce to that of drawing a bead from an urn.
Quoted in Teaching Statistics (1985), 7, 92
The drop in the functions for greater w at age 8 might be accidental or a case of the “U-shaped behavioral growth” in proportional reasoning, as observed by Stavy et al. (1982).
Though Bayesian analysis implies some increase (decrease) of the confidence in the correctness of a choice following even one instance of positive (negative) feedback, this does not entail certainty.
The story goes that Bertrand Russell commented on this apparent oxymoron: “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”
References
Acredolo, C., & O’Connor, J. (1991). On the difficulty of detecting cognitive uncertainty. Human Development, 34(4), 204–223.
Anderson, N. H., & Schlottmann, A. (1991). Developmental study of personal probability. In N. H. Anderson (Ed.), Contributions to information integration theory (Vol. III: Developmental) (pp. 110–134). Hillsdale, NJ: Erlbaum.
Baron, J., & Hershey, J. C. (1988). Outcome bias in decision evaluation. Journal of Personality and Social Psychology, 54(4), 569–579.
Brainerd, C. J. (1981). Working memory and the developmental analysis of probability judgment. Psychological Review, 88(6), 463–502.
Castro, C. S. (1998). Teaching probability for conceptual change. Educational Studies in Mathematics, 35, 233–254.
Chapman, R. H. (1975). The development of children’s understanding of proportions. Child Development, 46, 141–148.
Cosmides, L., & Tooby, J. (1996). Are humans good intuitive statisticians after all? Rethinking some conclusions from the literature on judgment under uncertainty. Cognition, 58(1), 1–73.
Davies, C. M. (1965). Development of the probability concept in children. Child Development, 36(3), 779–788.
Dean, A. L., & Mollaison, M. (1986). Understanding and solving probability problems: A developmental study. Journal of Experimental Child Psychology, 42, 23–48.
Denes-Raj, V., & Epstein, S. (1994). Conflict between intuitive and rational processing: When people behave against their better judgment. Journal of Personality and Social Psychology, 66(5), 819–829.
Even, R., & Kvatinsky, T. (2010). What mathematics do teachers with contrasting teaching approaches address in probability lessons? Educational Studies in Mathematics, 74, 207–222.
Falk, R. (1978). Analysis of the concept of probability in young children. In E. Cohors-Fresenborg & I. Wachsmuth (Eds.), Proceedings of the Second International Conference for the Psychology of Mathematics Education (pp. 144–147). Osnabrück, Germany: Universität Osnabrück.
Falk, R. (1983). Children’s choice behaviour in probabilistic situations. In D. R. Grey, P. Holmes, V. Barnett, & G. M. Constable (Eds.), Proceedings of the First International Conference on Teaching Statistics (Vol. 2, pp. 714–726). Sheffield, England: University of Sheffield, Teaching Statistics Trust.
Falk, R. (2010). The infinite challenge: Levels of conceiving the endlessness of numbers. Cognition and Instruction, 28(1), 1–38.
Falk, R., Falk, R., & Levin, I. (1980). A potential for learning probability in young children. Educational Studies in Mathematics, 11(2), 181–204.
Falk, R., & Konold, C. (1992). The psychology of learning probability. In F. Gordon & S. Gordon (Eds.), Statistics for the twenty-first century (pp. 151–164). Washington, DC: Mathematical Association of America.
Falk, R., & Wilkening, F. (1998). Children’s construction of fair chances: Adjusting probabilities. Developmental Psychology, 34(6), 1340–1357.
Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrecht, Holland: Reidel.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, Holland: Reidel.
Fischbein, E., & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuitions? Educational Studies in Mathematics, 15, 1–24.
Fischbein, E., Nello, M. S., & Marino, M. S. (1991). Factors affecting probabilistic judgements in children and adolescents. Educational Studies in Mathematics, 22, 523–549.
Fischbein, E., Pampu, I., & Minzat, I. (1970). Comparison of ratios and the chance concept in children. Child Development, 41, 377–389.
Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105.
Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2008). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53–96.
Goldberg, S. (1966). Probability judgments by preschool children: Task conditions and performance. Child Development, 37, 157–167.
Hawkins, A. S., & Kapadia, R. (1984). Children’s conceptions of probability—a psychological and pedagogical review. Educational Studies in Mathematics, 15, 349–377.
Hoemann, H. W., & Ross, B. M. (1971). Children’s understanding of probability concepts. Child Development, 42, 221–236.
Hoemann, H. W., & Ross, B. M. (1982). Children’s concepts of chance and probability. In C. J. Brainerd (Ed.), Children’s logical and mathematical cognition: Progress in cognitive development research (pp. 93–121). New York: Springer.
Huber, B. L., & Huber, O. (1987). Development of the concept of comparative subjective probability. Journal of Experimental Child Psychology, 44, 304–316.
Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1997). A framework for assessing and nurturing young children’s thinking in probability. Educational Studies in Mathematics, 32, 101–125.
Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1999). Students’ probabilistic thinking in instruction. Journal for Research in Mathematics Education, 30(5), 487–519.
Kahneman, D., & Frederick, S. (2002). Representativeness revisited: Attribute substitution in intuitive judgment. In T. Gilovich, D. Griffin, & D. Kahneman (Eds.), Heuristics and biases: The psychology of intuitive judgment (pp. 49–81). Cambridge: Cambridge University Press.
Kokis, J. V., Macpherson, R., Toplak, M. E., West, R. F., & Stanovich, K. E. (2002). Heuristic and analytic processing: Age trends and associations with cognitive ability and cognitive styles. Journal of Experimental Child Psychology, 83, 26–52.
Leron, U., & Hazzan, O. (2009). Intuitive vs. analytical thinking: Four perspectives. Educational Studies in Mathematics, 71, 263–278.
Martignon, L., & Krauss, S. (2009). Hands-on activities for fourth graders: A tool box for decision-making and reckoning with risk. International Electronic Journal of Mathematics Education, 4(3), 227–258.
Nickerson, R. S. (2004). Cognition and chance: The psychology of probabilistic reasoning. Mahwah, NJ: Lawrence Erlbaum.
Nilsson, P. (2007). Different ways in which students handle chance encounters in the explorative setting of a dice game. Educational Studies in Mathematics, 66, 293–315.
Noelting, G. (1980a). The development of proportional reasoning and the ratio concept: Part I—differentiation of stages. Educational Studies in Mathematics, 11(2), 217–253.
Noelting, G. (1980b). The development of proportional reasoning and the ratio concept: Part II—problem-structure at successive stages; problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11(3), 331–363.
Offenbach, S. I., Gruen, G. E., & Caskey, B. J. (1984). Development of proportional response strategies. Child Development, 55, 963–972.
Perner, J. (1979). Discrepant results in experimental studies of young children’s understanding of probability. Child Development, 50, 1121–1127.
Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children (L. Leake Jr., P. Burrell & H. D. Fishbein, Trans.). New York: Norton (Original work published 1951).
Reyna, V. F., & Brainerd, C. J. (1994). The origins of probability judgment: A review of data and theories. In G. Wright & P. Ayton (Eds.), Subjective probability (pp. 239–272). Chichester, England: Wiley.
Reyna, V. F., & Brainerd, C. J. (2008). Numeracy, ratio bias, and denominator neglect in judgments of risk and probability. Learning and Individual Differences, 18, 89–107.
Schlottmann, A., & Wilkening, F. (2011). Judgment and decision making in young children. In M. K. Dhami, A. Schlottmann, & M. Waldmann (Eds.), Judgement and decision making as a skill: Learning, development, and evolution (pp. 55–83). Cambridge: Cambridge University Press.
Scholz, R. W., & Waschescio, R. (1986). Children’s cognitive strategies in two-spinner tasks. In L. Burton & C. Hoyles (Eds.), Proceedings of the Tenth International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 463–468). London: University of London.
Shaughnessy, J. M. (1977). Misconceptions of probability: An experiment with a small-group, activity-based, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295–316.
Siegal, M., Waters, L. J., & Dinwiddy, L. S. (1988). Misleading children: Causal attributions for inconsistency under repeated questioning. Journal of Experimental Child Psychology, 45(3), 438–456.
Siegler, R. S. (1981). Developmental sequences within and between concepts. Monographs of the Society for Research in Child Development, 46(2, Serial No. 189).
Siegler, R. S. (1995). Children’s thinking: How does change occur? In F. E. Weinert & W. Schneider (Eds.), Memory performance and competencies: Issues in growth and development (pp. 405–430). Mahwah, NJ: Erlbaum.
Singer, J. A., & Resnick, L. B. (1992). Representations of proportional relationships: Are children part–part or part–whole reasoners? Educational Studies in Mathematics, 23(3), 231–246.
Spinillo, A. G., & Bryant, P. (1991). Children’s proportional judgments: The importance of “half”. Child Development, 62, 427–440.
Spinillo, A. G., & Bryant, P. E. (1999). Proportional reasoning in young children: Part–part comparisons about continuous and discontinuous quantity. Mathematical Cognition, 5(2), 181–197.
Stavy, R., Strauss, S., Orpaz, N., & Carmi, G. (1982). U-shaped behavioral growth in ratio comparisons. In S. Strauss & R. Stavy (Eds.), U-shaped behavioral growth (pp. 11–36). New York: Academic Press.
Stavy, R., & Tirosh, D. (1996). Intuitive rules in science and mathematics: The case of ‘more of A–more of B’. International Journal of Science Education, 18(6), 653–667.
Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181–204.
Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication … and back. The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381.
Yost, P. A., Siegel, A. E., & Andrews, J. M. (1962). Nonverbal probability judgments by young children. Child Development, 33, 769–780.
Acknowledgements
The study was partly supported by the Sturman Center for Human Development, The Hebrew University of Jerusalem. We are grateful to Raphael Falk for his invaluable help in all the stages of the research and to graduate students of the Hebrew University for their assistance in collecting data for Experiment 3. Yael Oren took care of the figures.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(DOC 502 kb)
Rights and permissions
About this article
Cite this article
Falk, R., Yudilevich-Assouline, P. & Elstein, A. Children’s concept of probability as inferred from their binary choices—revisited. Educ Stud Math 81, 207–233 (2012). https://doi.org/10.1007/s10649-012-9402-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-012-9402-1