Students Estimating Large Quantities: From Simple Strategies to the Population Density Model

Lluís Albarracín; Núria Gorgorió

doi:10.29333/ejmste/92285

Full Text (PDF)

Lluís Albarracín ¹ ^* , Núria Gorgorió ²

More Detail

¹ Serra Húnter Fellow, Universitat Autònoma de Barcelona, SPAIN² Universitat Autònoma de Barcelona, SPAIN^* Corresponding Author

Abstract

We study how a group of 16-year-old students solve a sequence of problems that require estimating large quantities. The problem sequence was designed to foster students creating mathematical models. We characterise the models constructed by the students while working both individually and in small teams and analyse how they evolve throughout the sequence because of having to deal with the restrictions imposed by the real-life context of the different problems. We observe that: a) the students progressively abandon the poorest strategies of the initial individual schemes to embrace strategies adaptable to a greater number of scenarios; b) students change the conceptual strategies that support their models, not as a result of a conceptual discussion, but as a consequence of a procedural obstacle or limitation; c) they adjust their models by introducing nuances, because of a better mathematization of the given situation. As a group, they finally reach a model – a generalization of the idea of population density – that is not only useful for solving a particular problem, but can apply to other situations.

Keywords

models
modelling-eliciting activities
estimation
population density
compulsory secondary education

License

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 14, Issue 10, October 2018, Article No: em1579

https://doi.org/10.29333/ejmste/92285

Publication date: 14 Jun 2018

Article Views: 2554

Article Downloads: 1331

Open Access References How to cite this article

References

Albarracín, L., & Gorgorió, N. (2013). Problemas de estimación de grandes cantidades: modelización e influencia del contexto. Revista Latinoamericana de Investigación en Matemática Educativa, 16(3), 289-315. https://doi.org/10.12802/relime.13.1631.
Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi problems involving large numbers. Educational Studies in Mathematics, 86(1), 79-96. https://doi.org/10.1007/s10649-013-9528-9.
Albarracín, L., & Gorgorió, N. (2015). On the Role of Inconceivable Magnitude Estimation Problems to Improve Critical Thinking. In U. Gellert, J. Giménez, C. Hahn and S. Kafoussi (eds.), Educational Paths to Mathematics (pp. 263-277). Dordrecht: Springer International Publishing.
Anderson, P. M., & Sherman, C. A. (2010). Applying the Fermi estimation technique to business problems. The Journal of Applied Business and Economics, 10(5), 33.
Ärlebäck, J. B., & Doerr, H. M. (2015a). 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). Dordrecht: Springer International Publishing.
Ärlebäck, J. B., & Doerr, H. M. (2015b). At the core of modelling: Connecting, coordinating and integrating models. In K. Krainer & N. Vondrová (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 802-808). Prague, Czech Republic: Charles University in Prague, Faculty of Education and ERME.
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. https://doi.org/10.1007/978-1-4419-0561-1_52.
Ärlebäck, J. B. (2011). Exploring the solving process of groups solving realistic Fermi problem from the perspective of the anthropological theory of didactics. In M. Pytlak, E. Swoboda & T. Rowland (Eds.) Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (CERME7) (pp. 1010–1019). Rzeszów: University of Rzeszów, Poland.
Aymerich, À., & Albarracín, L. (2016). Complejidad en el proceso de modelización de una tarea estadística. Modelling in Science Education and Learning, 9(1), 5-24. https://doi.org/10.4995/msel.2016.4121.
Blum, W. & Leiss, D. (2007). How do students’ and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum & S. Khan (Eds.), Mathematical Modelling: Education, Engineering and Economics. (pp. 222-231). Chichester, UK: Horwood Publishing.
Blum, W., & Borromeo Ferri, R. B. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45-58. https://doi.org/10.1007/978-94-007-0910-2_3.
Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM, 38(2), 86–95. https://doi.org/10.1007/BF02655883.
Clayton, J. (1996). A criterion for estimation tasks. International Journal of Mathematical Education in Science and Technology, 27(1), 87-102.
Czocher, J. A. (2016). Introducing Modeling Transition Diagrams as a Tool to Connect Mathematical Modeling to Mathematical Thinking. Mathematical Thinking and Learning, 18(2), 77-106. https://doi.org/10.1080/10986065.2016.1148530.
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. https://doi.org/10.2307/30034902.
Efthimiou, C. J., & Llewellyn, R. A. (2007). Cinema, Fermi problems and general education. Physics education, 42(3), 253. https://doi.org/10.1088/0031-9120/42/3/003.
Joram, E., Gabriele, A. J., Bertheau, M., Gelman, R., & Subrahmanyam, K. (2005). Children’s use of the reference point strategy for measurement estimation. Journal for Research in Mathematics Education, 36(1), 4-23.
Lesh, R. (2010). Tools, researchable issues and conjectures for investigating what it means to understand statistics (or other topics) meaningfully. Journal of Mathematical Modelling and Application, 1(2), 16-48. https://doi.org/10.2307/30034918.
Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking and Learning, 5(2), 157-189. https://doi.org/10.1080/10986065.2003.9679998.
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, UK: Routledge.
Matsuzaki, A., & Saeki, A. (2013). Evidence of a dual modelling cycle: Through a teaching practice example for pre-service teachers. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 195–205). Dordrecht: Springer International Publishing.
Perrenet, J., & Zwaneveld, B. (2012). The many faces of the mathematical modeling cycle. Journal of Mathematical Modelling and Application, 1(6), 3-21.
Pollak, H. O. (1979). The interaction between mathematics and other school subjects. In UNESCO (eds.) New Trends in Mathematics Teaching IV, (pp. 232–248). Paris: UNESCO.
Reif, F., & St. John, M. (1979). Teaching physicists’ thinking skills in the laboratory. American Journal of Physics, 47(11), 950-957. https://doi.org/10.1119/1.11618.
Robinson, A. W. (2008). Don’t just stand there—teach Fermi problems! Physics Education, 43(1), 83.
Sriraman, B., & Knott, L. (2009). The mathematics of estimation: Possibilities for interdisciplinary pedagogy and social consciousness. Interchange, 40(2), 205-223. https://doi.org/10.1007/s10780-009-9090-7.
Sriraman, B., & Lesh, R. (2006). Modeling conceptions revisited. ZDM The International Journal on Mathematics Education, 38(3), 247-254. https://doi.org/10.1007/BF02652808.
Wessels, H. M. (2014). Levels of mathematical creativity in model-eliciting activities. Journal of Mathematical Modelling and Application, 1(9), 22-40.
White, H. B. (2004). Math literacy. Biochemistry and Molecular Biology Education, 32(6), 410-411. https://doi.org/10.1002/bmb.2004.494032060415.
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. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

How to cite this article

APA

Albarracín, L., & Gorgorió, N. (2018). Students Estimating Large Quantities: From Simple Strategies to the Population Density Model. Eurasia Journal of Mathematics, Science and Technology Education, 14(10), em1579. https://doi.org/10.29333/ejmste/92285

Vancouver

Albarracín L, Gorgorió N. Students Estimating Large Quantities: From Simple Strategies to the Population Density Model. EURASIA J Math Sci Tech Ed. 2018;14(10):em1579. https://doi.org/10.29333/ejmste/92285

AMA

Albarracín L, Gorgorió N. Students Estimating Large Quantities: From Simple Strategies to the Population Density Model. EURASIA J Math Sci Tech Ed. 2018;14(10), em1579. https://doi.org/10.29333/ejmste/92285

Chicago

Albarracín, Lluís, and Núria Gorgorió. "Students Estimating Large Quantities: From Simple Strategies to the Population Density Model". Eurasia Journal of Mathematics, Science and Technology Education 2018 14 no. 10 (2018): em1579. https://doi.org/10.29333/ejmste/92285

Harvard

Albarracín, L., and Gorgorió, N. (2018). Students Estimating Large Quantities: From Simple Strategies to the Population Density Model. Eurasia Journal of Mathematics, Science and Technology Education, 14(10), em1579. https://doi.org/10.29333/ejmste/92285

MLA

Albarracín, Lluís et al. "Students Estimating Large Quantities: From Simple Strategies to the Population Density Model". Eurasia Journal of Mathematics, Science and Technology Education, vol. 14, no. 10, 2018, em1579. https://doi.org/10.29333/ejmste/92285