Article Sidebar

Download
PDF
Statistic
Read Counter : 265 Download : 120

Downloads

Download data is not yet available.

Main Article Content

Abstract

This paper analyzes the relationship between proportional reasoning and understanding fair games in Costa Rican students. We conducted a quantitative and qualitative analysis of the answers to six items on comparing ratios of increasing difficulty level and another item on prize estimation in a fair game. We describe the strategies employed and the semiotic conflicts detected in 292 Costa Rican students from Grades 6 to 10 (11-16-year-olds), comparing the findings with those established in previous research. The results show an increase in the level of proportional reasoning with the grade, although the age at which the higher levels are reached is lower than that assumed by Noelting. The percentage of students applying correct strategies in the fair game problem also increases with grade, and a relationship between the understanding of fair game and the level of proportional reasoning is observed.

Keywords

Evaluation Fair Game Understanding Proportional Reasoning Level

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite
Gea, M. M., Hernández-Solís, L. A., Batanero, C., & Álvarez-Arroyo, R. (2023). Relating students’ proportional reasoning level and their understanding of fair games. Journal on Mathematics Education, 14(4), 663–682. https://doi.org/10.22342/jme.v14i4.pp663-682
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Azcárate, P. (1995). El conocimiento profesional de los profesores sobre las nociones de aleatoriedad y probabilidad. Su estudio en el caso de la educación primaria. [Unpublished Ph.D.]. University of Cádiz, Spain.
  2. Batanero, C., & Álvarez-Arroyo, R. (2023). Teaching and learning of probability. ZDM Mathematics Education. https://doi.org/10.1007/s11858-023-01511-5
  3. Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 15–37). Springer.
  4. Batanero, C., Ortiz, J., Gómez, E., & Gea, M. M. (2019). Les jeux équitables comme contexte pour l’enseignement des probabilités et la formation des enseignants. In V. Martin, M. Thibault, & L. Theis (Eds.), Enseigner les premiers concepts de probabilités (pp. 219-244). Presses de L’Université de Québec.
  5. Batista, R., Borba, R., & Henriques, A. (2022). Fairness in games: a study on children’s and adults’ understanding of probability. Statistics Education Research Journal, 21(1), 13-13. https://doi.org/10.52041/serj.v21i1.79
  6. Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91-125). Academic Press.
  7. Ben-Chaim, D., Keret, Y., & Ilany, B. S. (2012). Ratio and proportion: Research and teaching in mathematics teachers’ education. Sense Publisher. https://doi.org/10.1007/978-94-6091-784-4_2
  8. Boyer, T. W., & Levine, S. C. (2015). Prompting children to reason proportionally: Processing discrete units as continuous amounts. Developmental Psychology, 51(5), 615–620. https://doi.org/10.1037/a0039010.
  9. Burgos, M., & Godino, J. D. (2020). Modelo ontosemiótico de referencia de la proporcionalidad. Implicaciones para la planificación curricular en primaria y secundaria. Avances de Investigación en Educación Matemática, 18, 1-20. https://doi.org/10.35763/aiem.v0i18.255
  10. Cañizares, M. J. (1997). Influencia del razonamiento proporcional y combinatorio y de creencias subjetivas en las intuiciones probabilísticas primarias. [Unpublished Ph.D.]. University of Granada, Spain.
  11. Cañizares, M. J., Batanero, C., Serrano, L., & Ortiz, J. J. (2004). Children’s understanding of fair games. In M. A. Mariotti (Ed.), European research in mathematics education III: Proceedings of 3rd Conference of the European Society for Research in Mathematics Education. CERME. http://erme.site/cerme-conferences/cerme3/cerme-3-proceedings/
  12. Gal, I. (2005). Towards "probability literacy" for all citizens: Building blocks and instructional dilemmas. In G. Jones (Ed.), Exploring probability in school (pp. 39-63). Springer.
  13. Godino, J., & Batanero, C. (1994). Significado institucional y personal de los objetos matemáticos. Recherches en Didactiques des Mathématiques, 14(3), 325-355.
  14. Godino, J., Batanero, C., & Font, V. (2007). The ontosemiotic approach to research in mathematics education. ZDM. The International Journal on Mathematics Education, 39(1-2), 127-135. https://doi.org/10.1007/s11858-006-0004-1
  15. Godino, J. D., Batanero, C., & Font, V. (2019). The onto-semiotic approach: Implications for the prescriptive character of didactics. For the Learning of Mathematics, 39(1), 38-43.
  16. Green, D. R. (1982). Probability concepts in school pupils aged 11-16 years. [Unpublished Ph.D.] University of Loughborough. UK.
  17. Hernández-Solís, L. A., Batanero, C., Álvarez-Arroyo, R., & Gea, M. (2021). Significados personales del concepto de juego equitativo en niños y niñas costarricenses. Innovaciones Educativas, 23(34), 228-243. http://dx.doi.org/10.22458/ie.v23i34.3429
  18. Inhelder, B. & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence, Basic Books.
  19. Jones, G., Langrall, C., & Mooney, E. (2007). Research in probability: responding to classroom realities. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (vol. 2, pp. 909-956). Information Age & NCTM.
  20. Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents' proportional reasoning on ‘rate’ problems. Educational Studies in Mathematics, 14(3), 219-233. https://doi.org/10.1007/BF00410539
  21. Krippendorff, K. (2013). Content analysis: An introduction to its methodology. SAGE.
  22. Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6, 59- 98. https://doi.org/10.1207/s1532690xci0601_3
  23. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (vol. 1, pp. 629-667). Information Age.
  24. Lecoutre, M. P. (1992). Cognitive models and problem spaces in "purely random" situations. Educational Studies in Mathematics, 23, 557-568. https://doi.org/10.1007/BF00540060
  25. Lidster, S. T., Watson, J. M., Collis, K. F., & Pereira-Mendoza, L. (1996). The relationship of the concept of fair to the construction of probabilistic understanding. In P. C. Clarkson (Ed.), Technology in Mathematics Education, Proceedings of the Nineteenth Annual Conference of the Mathematics Education Research Group of Australasia, Melbourne. MERGA.
  26. Ministerio de Educación Pública (MEP) (2012). Programas de Estudio de Matemáticas. I, II Y III Ciclos de la Educación General Básica y Ciclo Diversificado. Author.
  27. Ministerio de Educación y Formación Profesional (2022). Real Decreto 157/2022, de 1 de marzo, por el que se establecen la ordenación y las enseñanzas mínimas de la Educación Primaria. Madrid: Author.
  28. Mohamed, N. & Ortiz, J. (2012). Evaluación de conocimientos de profesores en formación sobre el juego equitativo. Números, 80, 103-117.
  29. 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. https://doi.org/10.1007/BF00304357
  30. 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. https://www.jstor.org/stable/3481806
  31. Ortiz, J., Batanero, C., & Contreras, J. (2012). Conocimiento de futuros profesores sobre la idea de juego equitativo. Revista Latinoamericana de Investigación en Matemática Educativa, 15(1), 63-69.
  32. Paparistodemou, E., Noss, R., & Pratt, D. (2002). Exploring sample space: Developing young children’s knowledge of randomness. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching of Statistics. International Statistical Institute. https://iase-web.org/documents/papers/icots6/2a3_papa.pdf?1402524960
  33. Pérez-Echeverría, M. P., Carretero, M., & Pozo, J. I. (1986). Los adolescentes ante las matemáticas: Proporción y probabilidad. Cuadernos de Pedagogía, 133, 9-13.
  34. Piaget, J., & Inhelder, B. (1951). La genése de l'idée de hasard chez l'enfant. Paris: Presses Universitaires de France.
  35. Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31(5), 602–625. https://doi.org/10.2307/749889
  36. Schlottmann, A., & Anderson, N. H. (1994). Children's judgements of expected value. Developmental Psychology, 30(1), 56-66. https://doi.org/10.1037/0012-1649.30.1.56
  37. Sharma, S. (2016). Probability from a socio-cultural perspective. Statistics Education Research Journal, 15(2), 126-144. https://doi.org/10.52041/serj.v15i2.244
  38. Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204. https://doi.org/10.1007/BF02400937
  39. Van Dooren, W. (2014). Probabilistic thinking: Analyses from a psychological perspective. In E. J. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking: Presenting plural perspectives. (pp. 123–126). Springer.
  40. Vidakovic, D., Berenson, S., & Brandsma, J. (1998). Children’s intuitions of probabilistic concepts emerging from fair play. In L. Pereira-Mendoza, L. S. Kea, T. W. Kee, & W. Wong (Eds.), Proceedings of the 5th International Conference on Teaching Statistics (Vol. 1, pp. 67–73). International Statistical Institute.
  41. Watson, J. & ColIis, K. F. (1994). Multimodal functioning in understanding chance and data concepts. In J.P. Ponte, & J. P. Matos (Eds), Proceedings of the XVIII International Conference for the Psychology of Mathematics Education. Universidad de Lisboa.
  42. Watson, J. M., & Moritz, J. B. (2003). Fairness of dice: A longitudinal study of students’ beliefs and strategies for making judgments. Journal for Research in Mathematics Education, 34(4), 270–303. https://doi.org/10.2307/30034785