Abstract
One hundred four students aged 8 to 16 worked on one Fermi problem involving estimating the number of people that can fit in their school playground. We present a qualitative analysis of the different mathematical models developed by the students. The analysis of the students’ written productions is based on the identification of the model of elements distribution and the strategy used. The results show how the students adapt their solutions in order to tackle the problem from their available knowledge. Indeed, younger students have important difficulties to deal with two-dimensional mathematical contents, but they overcome them by simplifying the problem. Finally, we also discuss the possibilities of using the proposed problem as part of a sequence to promote mathematical modelling in each educational stage, in basis of the potentialities identified in our analysis.
Similar content being viewed by others
References
ACARA. (2016). Australian curriculum: mathematics aims. Retrieved July 2018. https://australiancurriculum.edu.au.
Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi problems involving large numbers. Educational Studies in Mathematics, 86(1), 79–96.
Albarracín, L., & Gorgorió, N. (2019). Using large number estimation problems in primary education classrooms to introduce mathematical modelling. International Journal of Innovation in Science and Mathematics Education, 27(2), 33–45.
Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Montana Mathematics Enthusiast, 6(3), 331–364.
Ärlebäck, J. B., & Doerr, H. M. (2015). Moving beyond a single modelling activity. In G. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical Modelling in Education Research and Practice (pp. 293–303). Cham: Springer International Publishing.
Blomhøj, M. (2009). Different perspectives in research on the teaching and learning mathematical modelling. In M. Blomhøj & S. Carreira (Eds.), Mathematical applications and modelling in the teaching and learning of mathematics. Proceedings from Topic Study Group 21 at the 11th International Congress on Mathematical Education in Monterrey, México (pp. 1–17). Roskilde: Roskilde Universitet.
Blum, W. (2002). ICMI Study 14: applications and modelling in mathematics education–discussion document. Educational Studies in Mathematics, 51(1), 149–171.
Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.
Borromeo Ferri, R. (2018). Leaning how to teach mathematical modeling in school and teacher education. Cham: Springer.
Borromeo Ferri, R., & Blum, W. (2013). Barriers and motivations of primary teachers implementing modelling in mathematical lessons. In B. Ubuz, Ç. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 1000–1010). Ankara: Middle East Technical University.
CCSI. (2010). Common core state standards for mathematics. Retrieved July 2018. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf.
Efthimiou, C. J., & Llewellyn, R. A. (2007). Cinema, Fermi problems and general education. Physics Education, 42(3), 253–261.
English, L. D. (2013). Reconceptualizing statistical learning in the early years. In L. English & J. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 67–82). Dordrecht: Springer.
English, L. D. (2015). Learning through modelling in the primary years. In N. H. Lee & D. K. E. Ng (Eds.), Mathematical modelling: from theory to practice (pp. 99–124). Singapore: National Institute of Education.
English, L. D., & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematics Education Research Journal, 16(3), 58–79.
Fernández, C., De Bock, D., Verschaffel, L., & Van Dooren, W. (2014). Do students confuse dimensionality and “directionality”? The Journal of Mathematical Behavior, 36, 166–176.
Ferrando, I., Albarracín, L., Gallart, C., García-Raffi, L. M., & Gorgorió, N. (2017). Análisis de los modelos matemáticos producidos durante la resolución de problemas de Fermi. BOLEMA, 31(57), 220–242.
Geiger, V., Stillman, G., Brown, J., Galbraith, P., & Niss, M. (2018). Using mathematics to solve real world problems: the role of enablers. Mathematics Education Research Journal, 30(1), 7–19.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 1–28). Erlbaum: Hillsdale.
Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302–310.
Kazak, S., Pratt, D., & Gökce, R. (2018). Sixth grade students’ emerging practices of data modelling. ZDM Mathematics Education, 50(7), 1151–1163.
Lesh, R., & Doerr, H. M. (2003). A modeling perspective on teacher development. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 125-140). New Jersey: Lawrence Erlbaum Associates.
Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2-3), 157–189.
Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). London: Routledge.
Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequence. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35-58). New Jersey: Lawrence Erlbaum Associates.
Marsh, H. W., Abduljabbar, A. S., Parker, P. D., Morin, A. J., Abdelfattah, F., & Nagengast, B. (2014). The big-fish-little-pond effect in mathematics: a cross-cultural comparison of US and Saudi Arabian TIMSS Responses. Journal of Cross-Cultural Psychology, 45(5), 777–804.
McLean, J. A., & Doerr, H. M. (2016). A bootstrapping approach to elicit students’ informal inferential reasoning through model development sequences. In C. R. Hirsch & A. R. McDuffie (Eds.), Annual Perspectives in Mathematics Education 2016: Mathematical Modeling and Modeling Mathematics (pp. 163–173). Reston: National Council of Teachers of Mathematics.
Ng, K. E. D. (2013). Teacher readiness in mathematical modelling: are there differences between pre-service and experienced teachers? In G. Stillman, G. Kaiser, W. Blum, & J. Brown (Eds.), Connecting to practice: teaching practice and the practice of applied mathematicians (pp. 339–348). Dordrecht: Springer.
Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: pupils’ interactive modelling processes. In S. Ruwisch & A. Peter-Koop (Eds.), Mathematics education for the third millennium: Towards 2010 (pp. 454–461). Sydney: MERGA.
Peter-Koop, A. (2009). Teaching and understanding mathematical modelling through Fermi-problem. In B. Clarke, B. Grevholm, & R. Millman (Eds.), Tasks in Primary Mathematics Teacher Education (pp. 131–146). New York: Springer.
Real Decreto 1105/2014, de 26 de diciembre, por el que se establece el currículo básico de la Educación Secundaria Obligatoria y del Bachillerato. Boletín Oficial del Estado, Madrid, Spain, December 26th, 2014. Retrieved from https://www.boe.es/boe/dias/2015/01/03/pdfs/BOE-A-2015-37.pdf.
Sriraman, B., & Knott, L. (2009). The mathematics of estimation: possibilities for interdisciplinary pedagogy and social consciousness. Interchange, 40(2), 205–223.
Stender, P., & Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM Mathematics Education, 47(7), 1255–1267.
Stohlmann, M., & Albarracín, L. (2016). What is known about elementary grades mathematical modelling. Education Research International, 2016, Article ID 5240683, 1–9.
Zawojewski, J. S., Lesh, R. A., & 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 problem solving, learning, and teaching (pp. 337–358). London: Lawrence Erlbaum Associates.
Funding
This research is supported by the projects EDU2017-84377-R and EDU2017-82427-R (Ministerio de Economía, Industria y Competitividad, Spain) and also 2017 SGR 497 (AGAUR, Generalitat de Catalunya).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ferrando, I., Albarracín, L. Students from grade 2 to grade 10 solving a Fermi problem: analysis of emerging models. Math Ed Res J 33, 61–78 (2021). https://doi.org/10.1007/s13394-019-00292-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13394-019-00292-z