Mathematical Modeling in Statistical Activities: Key Episodes for Model Generation
DOI:
https://doi.org/10.15359/ru.36-1.16Keywords:
mathematical modeling, model-eliciting activities, Modeling Activity Diagrams, secondary education, statistics, variabilityAbstract
This work is intended to assist 15-year-old secondary school students to learn about statistics using mathematical modeling. Specifically, open statistical problems are presented in which real social phenomena are studied by students using large data sets that comply with the design principles of model-eliciting activities. The tasks assigned are aimed at presenting the concept of variability and its application to understanding the situations studied using a mathematical model. The study focuses on identifying key episodes in the activities in which progress is made in the construction of mathematical models, and the elements that promote them. To do so, a qualitative analysis is carried out based on records of the students’ group work in the classroom using Modeling Activity Diagrams. The results obtained show that decisions about the design of a problem, such as using large amounts of data, or the ambiguity of social concepts such as “fair taxation,” are essential for promoting the development of mathematical models. The conclusions of this investigation have implications for the design of statistical tasks, and also for identifying the role of mathematical modeling in the learning of statistical concepts.
References
Albarracín, L., Arleback, J., Civil, E. & Gorgorió, N. (2019). Extending Modelling Activity Diagrams as a tool to characterise mathematical modelling processes. The Mathematics Enthusiast, 16(1), 211-230.
Albarracín, L., Aymerich, À. & Gorgorió, N. (2017). An open task to promote students to create statistical concepts through modelling. Teaching Statistics, 39(3), 100-105.
Ärlebäck, J. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), 331-364.
Aymerich, À., Gorgorió, N. & Albarracín, L. (2017). Modelling with statistical data: Characterisation of student models. In Mathematical Modelling and Applications (pp. 37-47). Springer.
Batanero, C., Díaz, C., Contreras, J.M. & Roa, R. (2013). El sentido estadístico y su desarrollo. Números, 83, 7-18.
Batanero, C., Estepa, A., Godino, J.D. & Green, D.R. (1996). Intuitive strategies and preconceptions about association in contingency tables. Journal for Research in Mathematics Education, 27, 151-169.
Ben-Zvi, D. & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representation. In J. Garfield, D. Ben-Zvi, & C. Reading (Eds.), Background Readings of the Second International Research Forum on Statistical Reasoning, Thinking, and Literacy (pp. 73-110). Centre for Cognition Research in Learning and Teaching, University of New England.
Blum, W., & Leiß, D. (2006). How do students and teachers deal with modeling problems? In C. Haines, P. Galbraith, W. Blum & S. Khan (Eds.), Mathematical Modeling (ICTMA12): Education, Engineering and Economics (pp. 222-231). Horwood Publishing.
Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86-95.
Burrill, G., & Biehler R. (2011). Fundamental statistical ideas in the school curriculum and in training teachers. In C. Batanero, G. Burrill & C. Reading (Eds.), Teaching statistics in school mathematics, Challenges for teaching and teacher education (pp. 57-69). Springer.
Carreira, S., Amado, N., & Lecoq, F. (2011). Mathematical Modeling of Daily Life in Adult Education: Focusing on the Notion of knowledge. In G. Kaiser, W. Blum, R. Borromeo Ferri & G. Stillman (Eds.), Trends in teaching and Learning of Mathematical Modeling (pp. 199-210). Springer.
Crites, T., & Laurent, R. T. (2015). Putting essential understanding of statistics into practice, Grades 9-12. National Council of Teachers of Mathematics.
delMas, R. C. (2002). Statistical literacy, reasoning, and thinking: A commentary. Journal of Statistics Education, 10(3). doi.org/10.1080/10691898.2002.11910674
delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 79-95). Kluwer.
Doerr, H. M. & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110-136.
Doerr, H. M. & Tripp, J. S. (1999). Understanding how students develop mathematical models. Mathematical Thinking and Learning, 1, 231-254.
Hahn, C. (2015). La recherche internationale en éducation statistique: état des lieux et questions vives. Statistique et Enseignement, 6(2), 25-39.
Gal, I. (2002). Adults’ Statistical Literacy: Meanings, Components, Responsibilities. International Statistical Review, 70(1), 1-25.
Gal, I. (2004). Statistical Literacy: Meanings, Components, Responsabilities. En D. Ben-Zvi, & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning and Thinking (pp. 47-78). Kluwer Academic Publishers.
Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modeling process. ZDM, 38(2), 143-162.
Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review, 75(3), 372-396.
Greefrath, G. (2011). Using Technologies: New Possibilities of Teaching and Learning Modeling – Overview. In G. Kaiser, W. Blum, R. Borromeo Ferri, R. & G. Stillman (Eds.), Trends in teaching and Learning of Mathematical Modeling (pp. 301-304). Springer.
Hernández-Solís, L., Batanero, C., Gea, M., & Álvarez-Arroyo, R. (2021). Comparación de probabilidades en urnas: Un estudio con estudiantes de educación primaria. Uniciencia, 35(2), 1-18. https://doi.org/10.15359/ru.35-2.9
Lehrer, R. & Romberg, T. (1996). Exploring children’s data modeling. Cognition and Instruction, 14, 69-108.
Lehrer, R. & Schauble, L. (2000). Inventing data structures for representational purposes: Elementary grade children's classification models. Mathematical Thinking and Learning, 2, 51-74.
Lesh, R. (1997). Matematización: La necesidad "real" de la fluidez en las representaciones. Enseñanza de las Ciencias, 15(3), 377-391.
Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2-3), 157-189.
Lesh, R. E., & Doerr, H. M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Lawrence Erlbaum Associates Publishers.
Lesh, R., Amit, M., & Schorr, R. Y. (1997). Using “real-life” problems to prompt students to construct statistical models for statistical reasoning. In I. Gal & J. Garfield (Eds.), The assessment challenge in statistics education (pp. 65-84). IOS Press.
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.
Kaiser, G. & Stender, P. (2013). Complex modeling problem in cooperative learning environments self-directed. In G. A. Stillman, G. Kaiser, W. Blum, J. & Brown (Eds.), Teaching Mathematical Modeling: Connecting to Research and Practice. International Perspectives on the Teaching and Learning of Mathematical Modeling (pp. 277-294). Springer.
Pla-Castells, M. & García-Fernández, I. (2020). TaskTimeTracker: A tool for temporal analysis of the problem solving process. Investigación en Entornos Tecnológicos en Educación Matemática, 1, 9-14.
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Ubilla, F. (2019). Componentes del sentido estadístico identificados en un ciclo de investigación estadística desarrollado por futuras maestras de primaria. In J. M. Marbán, M. Arce, A. Maroto, J. M. Muñoz-Escolano & Á. Alsina (Eds.), Investigación en Educación Matemática XXIII (pp. 583-592). SEIEM.
Ubilla, F. (2020). ¿Qué rol juegan los datos en el ciclo de investigación estadística? UNO, 91, 63-68.
Watson, J. M. (2011). Foundations for improving statistical literacy. Statistical Journal of the IAOS, 27(3-4), 197-204. https://doi.org/10.3233/SJI-2011-0728
Wild, C. J., & Pfannkuchen, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-248.
Zawojewski, J., Lesh, R., & English, L. D. (2003). A models and modeling perspective on the role of small group learning activities. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics teaching, learning, and problem solving. Lawrence Erlbaum Associates, Inc.
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