Relating the construction of sample spaces and proportional reasoning in Costa Rican students

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Luis A. Hernández-Solís
Universidad Estatal a Distancia, Costa Rica
https://orcid.org/0000-0003-2956-8102
Carmen Batanero
Universidad de Granada, España
https://orcid.org/0000-0002-4189-7139
María M. Gea
Universidad de Granada, España
https://orcid.org/0000-0002-5229-0121

Abstract

The objective of this study was to analyze the relationship between the level of proportional reasoning and the construction of the sample space in Costa Rican students. For this purpose, two questionnaires were used, each with three ratio comparison items, which together consider six different levels of proportional reasoning. In addition, both questionnaires included two identical items on the construction of the sample space compatible with a certain, equiprobable and impossible event. In the responses of 292 students in grades 6-10 (11-15 years old) to these questionnaires we analyze the proportional reasoning shown and the type of event that would occur according to the constructed sample space. The results indicate that few students reach the highest level of proportional reasoning, although the proportion that reaches it increases with the course. The construction of sample spaces associated with the impossible and certain event was difficult and a direct relationship with the level of proportional reasoning is observed, confirmed by the correlation between the level of proportional reasoning and the number of correctly constructed sample spaces.

Item information
Published: Jul 30, 2024
Issue: Vol. 3 (2024): Revista de Educación Estadística
Section: Investigación
DOI: https://doi.org/10.29035/redes.3.1.1
Keywords: Probability, Reasoning, Assessment

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How to Cite
Hernández-Solís, L. A., Batanero, C., & Gea, M. M. (2024). Relating the construction of sample spaces and proportional reasoning in Costa Rican students. Revista De Educação Estatística, 3, 1-28. https://doi.org/10.29035/redes.3.1.1
References

Batanero, C. y Álvarez-Arroyo, R. (2023). Teaching and learning of probability. ZDM Mathematics Education, 56, 5-17. https://doi.org/10.1007/s11858-023-01511-5


Batanero, C., Godino, J. y Navarro-Pelayo, V. (1997). Combinatorial reasoning and its assessment. En I. Gal y J B. Garfield (Eds.), The assessment challenge in statistics education (pp. 239–252). IOS Press.






Batanero, C. y Hernández-Solís, LA. (2023). Razonamiento proporcional en comparación de razones de estudiantes costarricenses y españoles. Uniciencia, 37(1), 1-20. https://doi.org/10.15359/ru.37-1.21


Ben-Chaim, D., Keret, Y. e 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


Boyer, T.W. y 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


Butto, C.M., Fernández, J.D., Araujo, D.C. y Ramírez, A.B. (2019). El razonamiento proporcional en educación básica. Horizontes Pedagógicos, 21(2), 39-52. https://doi.org/10.33881/0123-8264.hop.21204


Burgos, M. y 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


Cañizares, M.J. (1997). Influencia del razonamiento proporcional y combinatorio y de creencias subjetivas en las intuiciones probabilísticas primarias. [Tesis Doctoral, Universidad de Granada].


Carpenter, T.P., Fennema, E. y Romberg, T.A. (Eds.). (2012). Rational numbers: An integration of research. Routledge (2º edición). https://doi.org/10.4324/9780203052624


Chernoff, E. J. (2009). Sample space partitions: An investigative lens. The Journal of Mathematical Behavior, 28(1), 19-29.


Chernoff, E.J. y Zazkis, R. (2011). From personal to conventional probabilities: from sample set to sample space. Educational Studies in Mathematics, 77, 15-33. https://doi.org/10.1007/s10649-0109288-8


English, L.D. (2005). Combinatorics and the development of children's combinatorial reasoning. En G.A. Jones (Ed,), Exploring probability in school: Challenges for teaching and learning (pp. 121-141). Springer.


Etikan, I., Musa, S.A. y Alkassim, R.S. (2016). Comparison of convenience sampling and purposive sampling. American Journal of Theoretical and Applied Statistics, 5(1), 1-4. https://doi.org/10.6224/jn.61.3.10T


González-Forte, J. M., Fernández, C., Van Hoof, J. y 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. (1983). School pupils' probability concepts. Teaching Statistics, 5(2), 34-42.


Groth, R.E., Bergner, J.A. y Austin, J.W. (2020). Dimensions of learning probability vocabulary. Journal for Research in Mathematics Education, 51(1), 75-104. https://doi.org/10.5951/jresematheduc.2019.0008


He, W., Yang, Y. y Gao, D. (2018). Proportional reasoning in 5- to 6-year-olds. Journal of Cognition and Development, 19(4), 389-412. https://doi.org/10.1080/15248372.2018.1495218


Hernández-Salmerón, E., López-Martín, M.M. y Batanero, C. (2017). Estudio exploratorio sobre el lenguaje del azar en educación secundaria obligatoria. En J.M. Muñoz, A. Arnal-Bailera, P. Beltrán-Pellicer, M.L. Callejo y J. Carrillo (Eds.), Investigación en Educación Matemática XXI (pp. 305-314). Universidad de Zaragoza.


Hernández-Solís, L.A., Batanero, C., Gea, M.M. y Álvarez-Arroyo, R. (2021). Construcción de espacios muestrales asociados a distintos tipos de sucesos: un estudio exploratorio con estudiantes de educación primaria. Educación Matemática, 33(1), 181-207. https://doi.org/10.24844/em3301.07


Horvath, J.K. y Lehrer, R. (1998). A model-based perspective on the development of children's understanding of chance and uncertainty. En S. Lajoie (Ed.), Reflections on statistics: learning, teaching, and assessment in grades K-12 (pp. 121-148). Routledge.


Jones, G., Langrall, C., Thornton, C. y Mogill, A. (1997). A framework for assessing and nurturing young children’s thinking in probability. Educational Studies in Mathematics, 32, 101-125. https://doi.org/10.1023/A:1002981520728


Karplus, R., Pulos, S. y 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


Kieren, T.E. (2020). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. En G. Leinhardt, R. Putnam y R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323-371). Routledge. https://doi.org/10.4324/9781315044606


Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59-98. https://doi.org/10.1207/s1532690xci0601_3


Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. En F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (V. 1, pp. 629-667). Information Age.


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


Langrall, C. y Mooney, E. (2005). Characteristics of elementary school students' probabilistic reasoning. En G. Jones (Ed.), Exploring probability in school (pp. 95-119). Springer.


Lima, E. y Borba, R. (2021). Articulando combinatória e probabilidade: jovens e adultos revisitando problemas combinatórios via construção de espaços amostrais. Paradigma, 42(Extra 1), 257-284. https://doi.org/10.37618/PARADIGMA.1011-2251.2021.p257-284.id1025


Lockwood, E., Wasserman, N.H. y Tillema, E.S. (2020). A case for combinatorics: A research commentary. The Journal of Mathematical Behavior, 59, 100783. https://doi.org/10.1016/j.jmathb.2020.100783


Ministerio de Educación Pública (MEP) (2012). Programas de Estudio de Matemáticas. I, II Y III Ciclos de la Educación General Básica y Ciclo Diversificado. MEP.


Noelting, G. (1980a). The development of proportional reasoning and the ratio concept. Part I. Diferentiation 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 succesive stages: Problem solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11(3), 331-363.


Nunes, T., Bryant, P., Evans, D., Gottardis, L. y Terlektsi, M.-E. (2014). The cognitive demands of understanding the sample space. ZDM Mathematics Education, 46(3), 437-448. https://doi.org/10.1007/s11858-014-0581-3


Nilsson, P. (2007). Different ways in which students handle chance encounters in the explorative setting of a dice game. Educational Studies in Mathematics, 66(3), 293-315. https://doi.org/10.1007/s10649-006-9062-0


Piaget, J. e Inhelder, B. (1951). La genèse de l’idée de hasard chez l’enfant. Presses Universitaires de France.


Speiser, R. y Walter, C. (1998). Two dice, two sample spaces. Proceedings of the Fifth International Conference on Teaching Statistics (Vol. 1, pp. 61-66). International Statistical Institute.


Supply, A.S., Vanluydt, E., Van Dooren, W. y Onghena, P. (2023). Out of proportion or out of context? Comparing 8-to 9-year-olds’ proportional reasoning abilities across fair-sharing, mixtures, and probability contexts. Educational Studies in Mathematics, 113, 1-18. https://doi.org/10.1007/s10649-023-10212-5


Van Dooren, W., Vamvakoussi, X. y Verschaffel, L. (2018). Proportional reasoning. International Academy of Education.


Zapata-Cardona, L. (2018). Supporting young children to develop combinatorial reasoning. En A. Leavy, M. Meletiou-Mavrotheris y E. Paparistodemou (Eds.), Statistics in early childhood and primary education (pp. 257-272). Springer: https://doi.org/10.1007/978-981-13-1044-7_15



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