Primary-school children’s knowledge of transitivity of probability and of bounds of probability

References

• Acredolo, C., O’Connor, J., Banks, L., & Horobin, K. (1989). Children’s ability to make probability estimates: Skills revealed through application of Anderson’s functional measurement methodology. Child Development, 933–945.
   PubMed Web of Science ®Google Scholar
• Alxatib, S., & Pelletier, F. J. (2011). The psychology of vagueness: Borderline cases and contradictions. Mind & Language, 26(3), 287–326. https://doi.org/10.1111/j.1468-0017.2011.01419.x
   Web of Science ®Google Scholar
• Ameel, E., Verschueren, N., & Schaeken, W. (2007). The relevance of selecting what’s relevant: A dual process approach to transitive reasoning with spatial relations. Thinking & Reasoning, 13(2), 164–187. https://doi.org/10.1080/13546780600780671
   Web of Science ®Google Scholar
• Andrews, G., & Halford, G. S. (1998). Children’s ability to make transitive inferences: The importance of premise integration and structural complexity. Cognitive Development, 13(4), 479–513. https://doi.org/10.1016/S0885-2014(98)90004-1
   Web of Science ®Google Scholar
• Baratgin, J., Coletti, G., Jamet, F., & Petturiti, D. (2017). Experimental evaluation of the understanding of qualitative probability and probabilistic reasoning in young children. International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, 27-30 June 2017, 107–112. Springer.
   Google Scholar
• Biehl, M., & Halpern-Felsher, B. L. (2001). Adolescents’ and adults’ understanding of probability expressions. Journal of Adolescent Health, 28(1), 30–35. https://doi.org/10.1016/S1054-139X(00)00176-2
   PubMed Web of Science ®Google Scholar
• Blutner, R., Pothos, E. M., & Bruza, P. (2013). A quantum probability perspective on borderline vagueness. Topics in Cognitive Science, 5(4), 711–736. https://doi.org/10.1111/tops.12041
   PubMed Web of Science ®Google Scholar
• Bonini, N., Osherson, D., Viale, R., & Williamson, T. (1999). On the psychology of vague predicates. Mind & Language, 14(4), 377–393. https://doi.org/10.1111/1468-0017.00117
   Web of Science ®Google Scholar
• Brooks, P. J., & Braine, M. D. (1996). What do children know about the universal quantifiers all and each? Cognition, 60(3), 235–268. https://doi.org/10.1016/0010-0277(96)00712-3
   PubMed Web of Science ®Google Scholar
• Chu, J., Cheung, P., Schneider, R. M., Sullivan, J., & Barner, D. (2020). Counting to infinity: Does learning the syntax of the count list predict knowledge that numbers are infinite? Cognitive Science, 44(8), e12875. https://doi.org/10.1111/cogs.12875
   PubMed Web of Science ®Google Scholar
• Cliff, N. (1992). Article commentary: Abstract measurement theory and the revolution that never happened. Psychological Science, 3(3), 186–190. https://doi.org/10.1111/j.1467-9280.1992.tb00024.x
   Web of Science ®Google Scholar
• Davidson, D. (1995). The representativeness heuristic and the conjunction fallacy effect in children’s decision making. Merrill-Palmer, 328–346.
   Web of Science ®Google Scholar
• De Finetti, B. (1931). Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17, 298–329. https://doi.org/10.4064/fm-17-1-298-329
   Google Scholar
• Delgrande, J. P., Renne, B., & Sack, J. (2019). The logic of qualitative probability. Artificial Intelligence, 275, 457–486. https://doi.org/10.1016/j.artint.2019.07.002
   Web of Science ®Google Scholar
• Falk, R. (2010). The infinite challenge: Levels of conceiving the endlessness of numbers. Cognition and Instruction, 28(1), 1–38. https://doi.org/10.1080/07370000903430541
   Web of Science ®Google Scholar
• Falk, R., Yudilevich-Assouline, P., & Elstein, A. (2012). Children’s concept of probability as inferred from their binary choices—revisited. Educational Studies in Mathematics, 81(2), 207–233. https://doi.org/10.1007/s10649-012-9402-1
   Web of Science ®Google Scholar
• Fischbein, E., Nello, M. S., & Marino, M. S. (1991). Factors affecting probabilistic judgements in children and adolescents. Educational Studies in Mathematics, 22(6), 523–549. https://doi.org/10.1007/BF00312714
   Google Scholar
• Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105.
   Web of Science ®Google Scholar
• Greenacre, M. (2017). Correspondence analysis in practice. Chapman and Hall/CRC.
   Google Scholar
• Groth, R. E. (2018). Unpacking implicit disagreements among early childhood standards for statistics and probability. In Aisling Leavy, Maria Meletiou-Mavrotheris, & Efi Paparistodemou (Eds.), Statistics in early childhood and primary education (pp. 149–162). Springer.
   Google Scholar
• Hoffner, C., Cantor, J., & Badzinski, D. M. (1990). Children’s understanding of adverbs denoting degree of likelihood. Journal of Child Language, 17(1), 217–231. https://doi.org/10.1017/S0305000900013192
   PubMed Web of Science ®Google Scholar
• Howie, P., & Roebers, C. M. (2007). Developmental progression in the confidence-accuracy relationship in event recall: Insights provided by a calibration perspective. Applied Cognitive Psychology: The Official Journal of the Society for Applied Research in Memory and Cognition, 21(7), 871–893.
   Web of Science ®Google Scholar
• Huber, B. L., & Huber, O. (1987). Development of the concept of comparative subjective probability. Journal of Experimental Child Psychology, 44(3), 304–316. https://doi.org/10.1016/0022-0965(87)90036-1
   Web of Science ®Google Scholar
• Ibrahim, Z. M., Ngom, A., & Tawfik, A. Y. (2010). A dynamic qualitative probabilistic network approach for extracting gene regulatory network motifs. 2010 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), 18-21 December 2010, 380–385. IEEE.
   Google Scholar
• Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge University Press.
   Google Scholar
• Jones, G. A. (2005). Exploring probability in school: Challenges for teaching and learning, Volume 40. Springer Science & Business Media.
   Google Scholar
• Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. Second Handbook of Research on Mathematics Teaching and Learning, 2, 909–955.
   Google Scholar
• 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(2), 101–125. https://doi.org/10.1023/A:1002981520728
   Google Scholar
• Kipman, U., Kühberger, A., & Pletzer, B. (2019). Activity-oriented teaching of stochastics in elementary school. British Journal of Educational Psychology, 89(1), 131–145. https://doi.org/10.1111/bjep.12226
   PubMed Web of Science ®Google Scholar
• Kolmogorov, A. (1950). Foundations of the theory of probability. Chelsea Publishing Company.
   Google Scholar
• Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement: Additive and polynomial representations, volume 1. Academic Press.
   Google Scholar
• Kultusministerkonferenz. (2022). Bildungsstandards für das fach mathematik primarbereich. Retrieved November 4, 2022, from https://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/2022/2022_06_23-Bista-Primarbereich-Mathe.pdf.
   Google Scholar
• Leavy, A., Meletiou-Mavrotheris, M., & Paparistodemou, E. (2018). Statistics in early childhood and primary education. Springer.
   Google Scholar
• Luce, R. D. (1967). Sufficient conditions for the existence of a finitely additive probability measure. The Annals of Mathematical Statistics, 38(3), 780–786. https://doi.org/10.1214/aoms/1177698871
   Google Scholar
• Markovits, H., Dumas, C., & Malfait, N. (1995). Understanding transitivity of a spatial relationship: A developmental analysis. Journal of Experimental Child Psychology, 59(1), 124–141. https://doi.org/10.1006/jecp.1995.1005
   Web of Science ®Google Scholar
• Meder, B., Mayrhofer, R., & Ruggeri, A. (2022). Developmental trajectories in the understanding of everyday uncertainty terms. Topics in Cognitive Science, 14(2), 258–281. https://doi.org/10.1111/tops.12564
   PubMed Web of Science ®Google Scholar
• Mullet, E., & Rivet, I. (1991). Comprehension of verbal probability expressions in children and adolescents. Language & Communication, 11(3), 217–225. https://doi.org/10.1016/0271-5309(91)90007-I
   Web of Science ®Google Scholar
• Odrowaz-Sypniewska, J. (2010). Vagueness and contextualism. 169–183. Ontos Verlag.
   Google Scholar
• Savage, L. J. (1954). The foundations of statistics. Wiley.
   Google Scholar
• Skoumpourdi, C. (2004). The teaching of probability theory as a new trend in Greek primary education. Proceedings of 10th International Congress on Mathematical Education, Copenhagen, Denmark, 4-11 July 2004.
   Google Scholar
• Smith, C. L. (1980). Quantifiers and question answering in young children. Journal of Experimental Child Psychology, 30(2), 191–205. https://doi.org/10.1016/0022-0965(80)90057-0
   Web of Science ®Google Scholar
• Smith, C. L. (2007). Bootstrapping processes in the development of students’ commonsense matter theories: Using analogical mappings, thought experiments, and learning to measure to promote conceptual restructuring. Cognition and Instruction, 25(4), 337–398. https://doi.org/10.1080/07370000701632363
   Web of Science ®Google Scholar
• Smith, C. L., Solomon, G. E., & Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51(2), 101–140. https://doi.org/10.1016/j.cogpsych.2005.03.001
   PubMed Web of Science ®Google Scholar
• Stigler, S. M. (1986). John Craig and the probability of history: From the death of Christ to the birth of Laplace. Journal of the American Statistical Association, 81(396), 879–887.
   Web of Science ®Google Scholar
• Tirosh, D., & Stavy, R. (1996). Intuitive rules in science and mathematics: The case of ‘everything can be divided by two’. International Journal of Science Education, 18(6), 669–683. https://doi.org/10.1080/0950069960180603
   Web of Science ®Google Scholar
• Vamvakoussi, X., & Vosniadou, S. (2012). Bridging the gap between the dense and the discrete: The number line and the “rubber line” bridging analogy. Mathematical Thinking and Learning, 14(4), 265–284. https://doi.org/10.1080/10986065.2012.717378
   Web of Science ®Google Scholar
• Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1–4.
   Web of Science ®Google Scholar
• Wasserman, L. (2004). All of statistics. Springer Science.
   Google Scholar
• Wellman, M. P. (1990). Fundamental concepts of qualitative probabilistic networks. Artificial Intelligence, 44(3), 257–303. https://doi.org/10.1016/0004-3702(90)90026-V
   Web of Science ®Google Scholar
• Wright, B. C. (2021). Towards a resolution of some outstanding issues in transitive research: An empirical test on middle childhood. Learning & Behavior, 49(2), 204–221. https://doi.org/10.3758/s13420-020-00440-7
   PubMed Web of Science ®Google Scholar
• Wright, B. C., & Dowker, A. D. (2002). The role of cues to differential absolute size in children’s transitive inferences. Journal of Experimental Child Psychology, 81(3), 249–275. https://doi.org/10.1006/jecp.2001.2653
   PubMed Web of Science ®Google Scholar
• Wright, B. C., Robertson, S., & Hadfield, L. (2011). Transitivity for height versus speed: To what extent do the under-7s really have a transitive capacity? Thinking & Reasoning, 17(1), 57–81. https://doi.org/10.1080/13546783.2010.544548
   Web of Science ®Google Scholar
• Wright, B. C., & Smailes, J. (2015). Factors and processes in children’s transitive deductions. Journal of Cognitive Psychology, 27(8), 967–978. doi:10.1080/20445911.2015.1063641
   PubMed Web of Science ®Google Scholar