Costa Rican students’ proportional reasoning and comparing probabilities in spinners

Luis A. Hernández-Solís; Carmen Batanero; María M. Gea

doi:10.29333/ejmste/13869

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Luis A. Hernández-Solís ¹ , Carmen Batanero ² ^* , María M. Gea ²

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¹ Universidad Estatal a Distancia, Mercedes, San Jose, COSTA RICA² University of Granada, Granada, SPAIN^* Corresponding Author

Abstract

This research aimed to relate Costa Rican students (11-16-year-olds) competence to compare probabilities in spinners and proportional reasoning in the comparison of ratios. We gave one of two questionnaires to a sample of 292 students (grade 6 to grade 10) with three probability comparison and three ratio comparison problems each. Globally both questionnaires cover six different proportional reasoning levels for each type of problem. Additionally, each questionnaire contains two comparison probabilities items intended to discover a specific bias. We analyze the percentages of correct responses to the items, strategies used to compare probabilities per school grade, and students’ probabilistic reasoning level. The results confirm more difficulty in comparing ratio than in comparing probability and suggest that the reasoning level achieved is lower than established in previous research. The main bias in the students’ responses was to consider the physical distribution of colored sectors in the spinners. Equiprobability and outcome approach were very scarce.

Keywords

comparing probabilities in spinners
proportional reasoning
reasoning levels
11-16-year-olds students

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 19, Issue 12, December 2023, Article No: em2373

https://doi.org/10.29333/ejmste/13869

Publication date: 01 Dec 2023

Online publication date: 07 Nov 2023

Article Views: 862

Article Downloads: 543

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References

ACARA. (2020). The Australian curriculum: Mathematics. Australian Curriculum, Assessment and Reporting Authority.
American Educational Research Association, American Psychological Association, National Council on Measurement in Education (2014). Standards for educational and psychological testing. AERA, APA & NCME.
Batanero, C., & Álvarez-Arroyo, R. (2023). Teaching and learning of probability. ZDM-Mathematics Education. https://doi.org/10.1007/s11858-023-01511-5
Batanero, C., & Hernández-Solís (2023). Razonamiento proporcional en comparación de razones de estudiantes costarricenses y españoles [Proportional reasoning in comparison of ratios of Costa Rican and Spanish students]. Uniciencia [Unscience], 37(1), 1-20. https://doi.org/10.15359/ru.37-1.21
Batanero, C., Hernández-Solís, L., & Gea, M. M. (In press). Analyzing Costa Rican and Spanish students´ comparison of probabilities and ratios. Statistics Education Research Journal.
Begolli, K. N., Dai, T., McGinn, K. M., & Booth, J. L. (2021). Could probability be out of proportion? Self-explanation and example-based practice help students with lower proportional reasoning skills learn probability. Instructional Science, 49, 441-473. https://doi.org/10.1007/s11251-021-09550-9
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
Berrocal, P. F. (1990). Relaciones teórico-empíricas entre los esquemas de proporción: Probabilidad y covariación [Theoretical-empirical relationships between proportion schemes: probability and covariation]. Revista de Psicología General y Aplicada [Journal of General and Applied Psychology], 43(3), 331-337.
Borovcnik, M. (2016). Probabilistic thinking and probability literacy in the context of risk. Educação Matemática Pesquisa [Mathematics Education Research], 18(3), 1491-1516.
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
Burgos, M., & Godino, J. D. (2020). Modelo ontosemiótico de referencia de la proporcionalidad. Implicaciones para la planificación curricular en primaria y secundaria [Onto-semiotic reference model of proportionality. Implications for curriculum planning in primary and secondary schools]. Avances de Investigación en Educación Matemática [Research Advances in Mathematics Education], 18, 1-20. https://doi.org/10.35763/aiem.v0i18.255
Burgos, M., & Godino, J. D. (2021). Conocimiento didáctico-matemático de la proporcionalidad en futuros maestros de educación primaria [Didactic-mathematical knowledge of proportionality in future primary education teachers]. Profesorado [Teaching Staff], 25(2), 281-306. https://doi.org/10.30827/profesorado.v25i2.8725
Burgos, M., & Godino, J. D. (2022). Assessing the epistemic analyses competence of prospective primary school teachers on proportionality tasks. International Journal of Science and Mathematics Education, 20, 367-389. https://doi.org/10.1007/s10763-020-10143-0
Butto, C. M., Fernández, J. D., Araujo, D. C., & Ramírez, A. B. (2019). El razonamiento proporcional en educación básica [Proportional reasoning in basic education]. Horizontes Pedagógicos [Pedagogical Horizons], 21(2), 39-52. https://doi.org/10.33881/0123-8264.hop.21204
Byant, P., & Nunes, T. (2012). Children’s understanding of probability: A literature review. Nuffield Foundation.
Cañizares, M. J. (1997). Influencia del razonamiento proporcional y combinatorio y de creencias subjetivas en las intuiciones probabilísticas primarias [Influence of proportional and combinatorial reasoning and subjective beliefs on primary probabilistic intuitions] [PhD thesis, University of Granada].
Cañizares, M. J., Batanero, C., Serrano, L., & Ortiz, J. J. (1997). Subjective elements in children’s comparison of probabilities. In E. Pehkonen (Ed.), Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (pp. 49-56). Eric Clearinghouse for Science, Mathematics, and Environmental Education.
Carpenter, T. P., Fennema, E., & Romberg, T. A. (Eds.). (2012). Rational numbers: An integration of research. Routledge. https://doi.org/10.4324/9780203052624
Davies, H. (1965). Development of the probability concept in children. Child Development, 99, 29-39. https://doi.org/10.2307/1126923
Falk, R., Falk, R., & Levin, I. (1980). A potential for learning probability in young children. Educational Studies in Mathematics, 11, 181-204. https://doi.org/10.1007/BF00304355
Gal, I. (2005). Towards ‘probability literacy’ for all citizens. In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 43-71). Springer.
Goldberg, E. (1966). Probability judgment by preschool children. Child Development, 37, 157-167. https://doi.org/10.1111/j.1467-8624.1966.tb05376.x
González-Forte, J. M., Fernández, C., Van Hoof, J., & Van Dooren, W. (2022). Profiles in understanding operations with rational numbers. Mathematical Thinking and Learning, 24(3), 230-247. https://doi.org/10.1080/10986065.2021.1882287
Green, D. R. (1982). Probability concepts in school pupils aged 11-16 years [PhD thesis, University of Loughborough].
Gurbuz, R., Erdem, E., & Firat, S. (2014). The effect of activity-based teaching on remedying the probability-related misconceptions: A cross-age comparison. Creative Education, 5(01), 18-30. https://doi.org/10.4236/ce.2014.51006
Hernández-Solís, L., Gea, M. M., Batanero, C., & Álvarez-Arroyo, R. (2023). Investigación sobre el razonamiento de los niños en la comparación de probabilidades [Research on children’s reasoning in comparing probabilities]. Boletín de Estadística e Investigación Operativa [Bulletin of Statistics and Operational Research], 39(1).
Hernández-Solís, L., Gea, M.M., Batanero, C., & Álvarez-Arroyo, R. (2021). Resolución de tareas probabilísticas en contexto geométrico por estudiantes de educación primaria [Resolution of probabilistic tasks in geometric context by primary education students]. Educação e Realidade [Education and Reality], 46(3), e105401. https://doi.org/10.1590/2175-6236105401
Hoemann, H. W., & Ross, B. M. (1971). Children’s understanding of probability concepts. Child Development, 42(1), 221-236. https://doi.org/10.2307/1127077
Jeong, Y., Levine, S. C., & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities. Journal of Cognition and Development, 8(2), 237-256. https://doi.org/10.1080/15248370701202471
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 (pp. 909-955). Information Age Publishing & NCTM.
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
Kazak, S., & Leavy, A. (2022). The emerging interplay between subjective and objective notions of probability in young children. Canadian Journal of Science, Mathematics and Technology Education, 22, 538-557. https://doi.org/10.1007/s42330-022-00227-0
Kieren, T. E. (2020). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.), Analyses of arithmetic for mathematics teaching (pp. 323-371). Routledge. https://doi.org/10.4324/9781315044606
Krippendorff, K. (2018). Content analyses: An introduction to its methodology. SAGE. https://doi.org/10.4135/9781071878781
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 Publishing & NCTM.
Lecoutre, M. P. & Durand, J. L. (1988). Judgements probabilistes et modèles cognitifs: Étude d’une situation aleatoire [Probabilistic judgments and cognitive models: Study of a random situation]. Educational Studies in Mathematics, 19, 357-368. https://doi.org/10.1007/BF00312452
Maury, S. (1984). La quantification des probabilités: Analyze des arguments utilisés par les élèves de classe de seconde [The quantification of probabilities: Analysis of the arguments used by second grade students]. Recherches en Didactique des Mathématiques [Research in Mathematics Didactics], 5(2), 187-214.
MEFP. (2022). Real Decreto 157/2022, de 1 de marzo, por el que se establecen la ordenación y las enseñanzas mínimas de la educación primaria [Royal Decree 157/2022, of March 1, which establishes the organization and minimum teachings of primary education]. Ministerio de Educación y Formación Profesional [Ministry of Education and Vocational Training].
MEP. (2012). Programas de estudio de matemáticas. I, II, y III ciclos de la educación general básica y ciclo diversificado [Mathematics study programs. I, II, and III cycles of basic general education and diversified cycle]. Ministerio de Educación Pública [Ministry of Public Education].
Muñiz-Rodríguez, L., Rodríguez-Muñiz, L. J., & Alsina, Á. (2020). Deficits in the statistical and probabilistic literacy of citizens: Effects in a world in crisis. Mathematics, 8(11), 1872. https://doi.org/10.3390/math8111872
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
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://doi.org/10.1007/BF00697744
Obando, G., Vasco, C. E., & Arboleda, L. C. (2014). Enseñanza y aprendizaje de la razón, la proporción y la proporcionalidad: Un estado del arte [Teaching and learning reason, proportion and proportionality: A state of the art]. Revista Latinoamericana de Matemática Educativa [Latin American Journal of Educational Mathematics], 17(1), 59-81. https://doi.org/10.12802/relime.13.1713
Pérez-Echeverría, M. P., Carretero, M., & Pozo, J. I. (1986). Los adolescentes ante las matemáticas: Proporción y probabilidad [Adolescents facing mathematics: Proportion and probability]. Cuadernos de Pedagogía [Pedagogy Notebooks], 133, 9-13.
Piaget, J., & Inhelder, B. (1951). La genése de l’idée de hasard chez l’enfant [The genesis of the idea of chance in children]. Presses Universitaires de France.
Pratt, D., & Kazak, S. (2018), Research on uncertainty. In D. Ben-Zvi, K. Makar, & J. Garfield (Eds.), International handbook of research in statistics education (pp. 193-227). Springer. https://doi.org/10.1007/978-3-319-66195-7_6
Supply, A. S., Van Dooren, W., Lem, S., & Onghena, P. (2020). Assessing young children’s ability to compare probabilities. Educational Studies in Mathematics, 103(1), 27-42. https://doi.org/10.1007/s10649-019-09917-3
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
Van Dooren, W., Vamvakoussi, X., Verschaffel, L., Marope, M., Vosniadou, S., Anderson, L., de Ibarrola, M., & Popa, S. (2018). Proportional reasoning. International Academy of Education.
Vásquez, C., García-Alonso, I., Seckel, M. J., & Alsina, Á. (2021). Education for sustainable development in primary education textbooks–An educational approach from statistical and probabilistic literacy. Sustainability, 13(6), 3115. https://doi.org/10.3390/su13063115
Watson, J., & Shaughnessy, J. (2004). Proportional reasoning: Lessons from research in data and chance. Mathematics Teaching in the Middle School, 10(2), 104-109. https://doi.org/10.5951/MTMS.10.2.0104
Yost, P., Siegel, A. & Andrews, J. (1962). Non-verbal probability judgement by young children. Child Development, 33, 769-780. https://doi.org/10.1111/j.1467-8624.1962.tb05113.x

How to cite this article

APA

Hernández-Solís, L. A., Batanero, C., & Gea, M. M. (2023). Costa Rican students’ proportional reasoning and comparing probabilities in spinners. Eurasia Journal of Mathematics, Science and Technology Education, 19(12), em2373. https://doi.org/10.29333/ejmste/13869

Vancouver

Hernández-Solís LA, Batanero C, Gea MM. Costa Rican students’ proportional reasoning and comparing probabilities in spinners. EURASIA J Math Sci Tech Ed. 2023;19(12):em2373. https://doi.org/10.29333/ejmste/13869

AMA

Hernández-Solís LA, Batanero C, Gea MM. Costa Rican students’ proportional reasoning and comparing probabilities in spinners. EURASIA J Math Sci Tech Ed. 2023;19(12), em2373. https://doi.org/10.29333/ejmste/13869

Chicago

Hernández-Solís, Luis A., Carmen Batanero, and María M. Gea. "Costa Rican students’ proportional reasoning and comparing probabilities in spinners". Eurasia Journal of Mathematics, Science and Technology Education 2023 19 no. 12 (2023): em2373. https://doi.org/10.29333/ejmste/13869

Harvard

Hernández-Solís, L. A., Batanero, C., and Gea, M. M. (2023). Costa Rican students’ proportional reasoning and comparing probabilities in spinners. Eurasia Journal of Mathematics, Science and Technology Education, 19(12), em2373. https://doi.org/10.29333/ejmste/13869

MLA

Hernández-Solís, Luis A. et al. "Costa Rican students’ proportional reasoning and comparing probabilities in spinners". Eurasia Journal of Mathematics, Science and Technology Education, vol. 19, no. 12, 2023, em2373. https://doi.org/10.29333/ejmste/13869