Analysis of primary school students’ answers to fair game tasksAn experience with preservice teachers

  1. María Burgos
  2. María del Mar López-Martín
  3. Verónica Albanese
  4. Carmen Gloria Aguayo-Arriagada
Revista:
BEIO, Boletín de Estadística e Investigación Operativa

ISSN: 1889-3805

Año de publicación: 2023

Volumen: 39

Número: 3

Tipo: Artículo

Otras publicaciones en: BEIO, Boletín de Estadística e Investigación Operativa

Resumen

In order for teachers to understand how mathematical learning develops in their students, they must be able to interpret and judge the mathematical activity put at stake. This will allow teachers to make action decisions enabling them to overcome the difficulties that students have in solving mathematical tasks. This paper describes the results of the evaluation of the knowledge and competences of pre-service teachers to interpret students’ responses to probability tasks in the context of fair game and to propose strategies to surpass the limitations in those students who gave an inadequate solution. Among the results obtained, we highlight that pre-service teachers show a biased knowledge of the notion of fair game and encounter difficulties in interpreting and justifying the correctness of students’ answers, as well as in identifying proportional reasoning in their responses. Intervention proposals to help students overcome the errors made were vague and not very explicit in relation to the mathematical content involved.

Referencias bibliográficas

  • Alonso-Castaño, M., P. Alonso, M. Mellone, and L. J. Rodríguez-Muñiz. 2019. “Conocimiento Matemático de Maestros En Formación Cuando Crean y Resuelven Una Tarea de Probabilidad.” In Investigación En Educación Matemática XXIII, edited by J. M. Marbán, M. Arce, A. Maroto, J. M. MuñozEscolano, and A. Alsina, 606. Valladolid: SEIEM.
  • Alsina, A., and C. A. Vásquez. 2016. “De La Competencia Matemática a La Alfabetización Probabilística En El Aula: Elementos Para Su Caracterización y Desarrollo.” UNIÓN. Revista Iberoamericana de Educación Matemática 48: 41–58.
  • Batanero, C., R. Álvarez-Arroyo, L. A. Hernández-Solís, and M. M. Gea. 2021. “El Inicio Del Razonamiento Probabilístico En Educación Infantil.” PNA 15 (4): 267–88. https://doi.org/10.30827/pna.v15i4.22349.
  • Batanero, C., P. Arteaga, L. Serrano, and B. Ruiz. 2014. “Prospective Primary School Teachers’ Perception of Randomness.” In Probabilistic Thinking: Presenting Plural Perspectives, edited by E. J. Chernoff and B. Sriraman, 345–66. Springer.
  • Batanero, C., E. J. Chernoff, H. S. Engel J. Lee, and E. Sánchez. 2016. Res. Teach. Learning Probability. ICME-13. Topical Survey Series. Springer.
  • Batanero, C., E. Gómez, J. M. Contreras, and C. Díaz. 2015. “Conocimiento Matemático de Profesores de Primaria En Formación Para La Enseñanza de La Probabilidad: Un Estudio Exploratorio.” Práxis Educativa 10 (1): 11–34. https://doi.org/10.5212/PraxEduc.v.10i1.0001.
  • Batanero, C., E. Gómez, L. Serrano, and J. M. Contreras. 2012. “Comprensión de La Aleatoriedad Por Futuros Profesores de Educación Primaria.” Redimat 1 (3). https://doi.org/10.4471/redimat.2012.13.
  • Begolli, K. N., T. Dai, K. M. McGinn, and J. L. Booth. 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: 441–73. https://doi.org/10.1007/s11251-021-09550-9.
  • Boyer, T. W., and S. C. Levine. 2015. “Prompting Children to Reason Proportionally: Processing Discrete Units as Continuous Amounts.” Developmental Psychology 51 (5): 615–20.
  • Bryant, P., and T. Nunes. 2012. “Children’s Understanding of Probability: A Literature Review (Full Report).” The Nuffield Foundation.
  • Cañizares, M., C. Batanero, L. Serrano, and J. J. Ortiz. 1999. “Comprensión de La Idea de Juego Equitativo En Los Niños.” Números 37: 37–55.
  • Chernoff, E. J., and G. L. Russell. 2012. “The Fallacy of Composition: Prospective Mathematics Teachers’ Use of Logical Fallacies.” Canadian J. Science, Mathematics and Technology Education 12 (3): 259–71. https://doi.org/10.1080/14926156.2012.704128.
  • Choy, B. H. 2016. “Snapshots of Mathematics Teacher Noticing During Task Design.” Mathematics Education Research Journal 28 (3): 421–40. https://doi.org/10.1007/s13394-016-0173-3.
  • Common Core State Standards Initiative (CCSSI). 2010. “Common Core State Standards for Mathematics.” 2010. http://www.corestandards.org/Math/.
  • Fernández, C., G. Sánchez-Matamoros, J. Valls, and M. L. Callejo. 2018. “Noticing Students’ Mathematical Thinking: Characterization, Development and Contexts.” Avances de Investigación En Educación Matemática 13: 39–61.
  • Gal, I. 2005. “Towards ‘Probability Literacy’ for All Citizens: Building Blocks and Instructional Dilemmas.” In Exploring Probability in School: Challenges for Teaching and Learning, edited by G. A. Jones, 39–63. Springer.
  • Gea, M. M., R. Parraguez, and C. Batanero. 2017. “Comprensión de La Probabilidad Clásica y Frecuencial Por Futuros Profesores.” In Investigación En Educación Matemática XXI, edited by J. M. Muñoz-Escolano et al., 267–76. SEIEM.
  • Godino, J. D., C. Batanero, and V. Font. 2007. “The onto-Semiotic Approach to Research in Mathematics Education.” ZDM. The International Journal on Mathematics Education 39 (1-2): 127–35. https://doi.org/10.1007/s11858-006-0004-1.
  • Godino, J. D., B. Giacomone, C. Batanero, and V. Font. 2017. “Enfoque Ontosemiótico de Los Conocimientos y Competencias Del Profesor de Matemáticas.” Bolema 31 (57): 90–113. https://doi.org/10.1590/1980-4415v31n57a05.
  • Gómez, E., C. Batanero, and J. M. Contreras. 2013. “Conocimiento Matemático de Futuros Profesores Para La Enseñanza de La Probabilidad Desde El Enfoque Frecuencial.” Bolema 28 (48): 209–29. https://doi.org/10.1590/1980-4415v28n48a11.
  • Hernández, L. A., C. Batanero, R. Álvarez, and M. M. Gea. 2021. “Significados Personales Del Concepto de Juego Equitativo En Niños y Niñas Costarricenses.” Innovaciones Educativas 23 (34): 228–43. https://doi.org/10.22458/ie.v23i34.3429.
  • Jacobs, V. R., L. C. Lamb, and R. Philipp. 2010. “Professional Noticing of Children’s Mathematical Thinking.” Journal for Research in Mathematics Education 41 (2): 169–202. https://doi.org/10.5951/jresematheduc.41.2.0169.
  • Jones, G. A., C. W. Langrall, and E. S. Mooney. 2007. “Research in Probability (Responding to Classroom Realities).” In Second Handbook of Research on Mathematic Teaching and Learning, edited by F. Lester, 909–56. National Council of Teachers of Mathematics (NCTM).
  • Kazak, S., R. Wegerif, and T. Fujita. 2015. “Combining Scaffolding for Content and Scaffolding for Dialogue to Support Conceptual Breakthroughs in Understanding Probability.” ZDM, Mathematics Education 47: 1269–83. https://doi.org/10.1007/s11858-015-0720-5.
  • Lamon, S. 2007. “Rational Number and Proportional Reasoning: Toward a Theoretical Framework for Research.” In Second Handbook of Research on Mathematic Teaching and Learning, edited by F. K. Lester, 629–67. NCTM.
  • Langrall, C. W., and E. S. Mooney. 2005. “Characteristics of Elementary School Students’ Probabilistic Reasoning.” In Exploring Probability in School, edited by G. Jones, 95–119. Springer. https://doi.org/10.1007/0-387-24530-8\_5.
  • Lee, H. S., R. L. Angotti, and J. Tarr. 2010. “Making Comparisons Between Observed Data and Expected Outcomes: Students’ Informal Hypothesis Testing with Probability Simulation Tools.” Statistics Education Research Journal 9 (1): 68–96. https://doi.org/10.52041/serj.v9i1.388.
  • Lidster, S. T., J. M. Watson, K. F. Collis, and L. Pereira-Mendoza. 1996. “The Relationship of the Concept of Fair to the Construction of Probabilistic Understanding.” In Technology in Mathematics Education, Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia, 352–59. MERGA.
  • Llinares, S., and K. Krainer. 2006. “Mathematics (Student) Teachers and Teacher Educators as Learners.” In Handbook of Research on the Psychology of Mathematics Education. Past, Present and Future, edited by A. Gutierrez and P. Boero, 429–60. Sense Publishers.
  • Mason, J. 2016. “Perception, Interpretation and Decision Making: Understanding Gaps Between Competence and Performance-a Commentary.” ZDM, Mathematics Education 48 (1-2): 219–26. https://doi.org/10.1007/s11858-016-0764-1.
  • Mohamed, N. 2012. “Evaluación Del Conocimiento de Los Futuros Profesores de Educación Primaria Sobre Probabilidad.” PhD thesis, Universidad de Granada, Melilla. https://www.ugr.es/~batanero/pages/ARTICULOS/TESISMOHAMED.pdf.
  • Mohamed, N., and J. J. Ortiz. 2012. “Evaluación de Conocimientos de Profesores En Formación Sobre El Juego Equitativo.” Números 80: 103–17.
  • Mohamed, N., J. J. Ortiz, and L. Serrano. 2013. “Evaluación Del Conocimiento Sobre Juego Equitativo En Futuros Profesores.” In Acta Latinoamericana de Matemática Educativa, 1359–67. Comité Latinoamericano de Matemática Educativa.
  • Ortiz, J. J., C. Batanero, and J. M. Contreras. 2012. “Conocimiento de Futuros Profesores Sobre La Idea de Juego Equitativo.” Revista Latinoamericana de Investigación En Matemática Educativa 15 (1): 63–91.
  • Parraguez, R., M. M. Gea, D. Díaz-Levicoy, and C. Batanero. 2017. “¿Conectan Los Futuros Profesores Las Aproximaciones Frecuencial y Clásica de La Probabilidad?” Revista Digital: Matemática, Educación e Internet 17 (2): 1–15. https://doi.org/10.18845/rdmei.v17i2.3077.
  • Pratt, D. 2000. “Making Sense of the Total of Two Dice.” Journal of Research in Mathematics Education 31: 602–25. https://doi.org/10.2307/749889. “Principles and Standards for School Mathematics.” 2000. https://www.nctm.org/standards/standards-for-school-mathematics/; National Council of Teachers of Mathematics (NCTM). 2000.
  • Prodromou, T. 2012. “Connecting Experimental Probability and Theoretical Probability.” ZDM, Mathematics Education 44 (7): 855–68. https://doi.org/10.1007/s11858-012-0469-z. “Real Decreto 126/2014, de 28 de Febrero, Por El Que Se Establece El Currículo Básico de La Educación Primaria.” 2014. https://www.boe.es/eli/es/rd/2014/02/28/126. 2014.
  • Schlottmann, A., and N. H. Anderson. 1994. “Children’s Judgements of Expected Value.” Developmental Psychology 30 (1): 56–66. https://doi.org/10.1037/0012-1649.30.1.56.
  • Van Dooren, W. 2014. “Probabilistic Thinking: Analyses from a Psychological Perspective.” In Probabilistic Thinking, edited by Chernoff E. J. and Sriraman B., 123–26. Springer. https://doi.org/10.1007/978-94-007-7155-0\_7.
  • Vásquez, C. A., and A. Alsina. 2015. “El Conocimiento Del Profesorado Para Enseñar Probabilidad: Un Análisis Global Desde El Modelo Del Conocimiento Didáctico-Matemático.” Avances de Investigación En Educación Matemática 7: 27–48. https://doi.org/10.35763/aiem.v1i7.104.
  • Vásquez, C. A., and A. Alsina. 2017. “Aproximación Al Conocimiento Común Del Contenido Para Enseñar Probabilidad Desde El Modelo Del Conocimiento Didáctico-Matemático.” Revista Educación Matemática 29: 79–108. https://doi.org/10.24844/em2903.03.
  • Vásquez, C. A., and A. Alsina. 2019. “Diseño, Construcción y Validación de Una Pauta de Observación de Los Significados de La Probabilidad En El Aula de Educación Primaria.” REVEMAT: Revista Eletrônica de Educação Matemática 14: 1–20. https://doi.org/10.5007/1981-1322.2019.e62434.
  • Watson, J. M., and K. F. ColIis. 1994. “Multimodal Functioning in Understanding Chance and Data Concepts.”
  • Watson, J. M., K. F. Collis, and J. B. Moritz. 2007. The Development of Chance Measurement. Brill Sense.