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The Role of Context for Characterising Students’ Strategies when Estimating Large Numbers of Elements on a Surface

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Abstract

This paper presents a qualitative study developed with a group of 16-year-old students who were asked to estimate large numbers of elements on a bounded surface. Taking the realistic mathematics education framework as a reference, we presented the students with an activity sequence comprised of four different tasks—each one with a different realistic context, which were variations of the same estimation problem. Our data comprises the written reports that the students submitted on completing the tasks. The strategies deployed by the students to solve the tasks consisted in the incardination of procedures related to the concepts of measurement and proportionality. For each task, we present the characterization of the strategies through a visual resource in the form of a tree-like diagram. We integrated the treelike diagrams of the four tasks to generate a tree-like diagram for the general problem. This allowed us to discuss the role of context in the strategies the students reported having used.

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Notes

  1. In both schools, the playground is rectangular.

References

  • Albarracín, L., & Gorgorió, N. (2013). Problemas de estimación de grandes cantidades: Modelización e influencia del contexto [Problematizing the estimation of large numbers: Modeling and the influence of context]. Revista Latinoamericana de Investigación en Matemática Educativa, 16(3), 289–315.

  • 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), 45–47.

  • 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.

    Google Scholar 

  • Ärlebäck, J. B., & Albarracín, L. (2019). The use and potential of Fermi problems in the STEM disciplines to support the development of twenty-first century competencies. ZDM, 51(6), 979–990.

  • Carlson, J. E. (1997). Fermi problems on gasoline consumption. The Physics Teacher, 35, 308–309. https://doi.org/10.1119/1.2344696.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Czocher, J. A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99(2), 137–159.

    Article  Google Scholar 

  • De Lange, J. (1995). Assessment: No change without problems. In T. A. Romberg (Ed.), Reform in school mathematics (pp. 87–172). Albany, NY: SUNY Press.

  • De Lange, J. (1996). Using and applying mathematics in education. In A. J. Bishop (Ed.), International handbook of mathematics education, Part I (pp. 49–97). Utrecht, The Netherlands: Kluwer Academia Press.

  • Doerr, H. M. (2006). Teachers’ ways of listening and responding to students’ emerging mathematical models. ZDM The International Journal on Mathematics Education, 38(3), 255–268.

    Article  Google Scholar 

  • Efthimiou, C. J., & Llewellyn, R. A. (2007). Cinema, Fermi problems and general education. Physics Education, 42(3), 253–261.

    Article  Google Scholar 

  • Ferrando, I. & Albarracín, L. (2019). Students from grade 2 to grade 10 solving a Fermi problem: Analysis of emerging models. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-019-00292-z

  • Ferrer, M., Fortuny, J. M., & Morera, L. (2014). Efectos de la actuación docente en la generación de oportunidades de aprendizaje matemático [Effects of teaching activity on the generation of mathematical learning opportunities]. Enseñanza de las Ciencias: Revista de Investigación y Experiencias Didácticas, 32(3), 385–405.

    Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, The Netherlands: Kluwer Academic Publishers.

  • Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25, 443–471.

    Article  Google Scholar 

  • Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1–3), 111–129.

    Article  Google Scholar 

  • Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293–307.

    Article  Google Scholar 

  • Haberzettl, N., Klett, S., & Schukajlow, S. (2018). Mathematik rund um die Schule – Modellieren mit Fermi-Aufgaben [Mathematics around school - Modelling with Fermi tasks]. In K. Eilerts & K. Skutella (Eds.), Neue Materialien für einen realitätsbezogenen Mathematikunterricht 5. Ein ISTRON-Band für die Grundschule (pp. 31–41). Wiesbaden, Germany: Springer Spectrum.

  • Keune, M., & Henning, H. (2003). Modelling and spreadsheet calculation. In Q. X. Ye, W. Blum, S.-K. Houston, & Q. Y. Yian (Eds.), Mathematical modelling in education and culture (pp. 101–110). Chichester, UK: Horwood Publishing Limited.

    Chapter  Google Scholar 

  • Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the fifth congress of the European Society for Research in mathematics education (pp. 2330–2339). Nicosia, Cyprus: Department of Education, University of Cyprus.

    Google Scholar 

  • Leikin, R., & Levav-Waynberg, A. (2008). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge. Canadian Journal of Science, Mathematics, and Technology Education, 8(3), 233–251.

    Article  Google Scholar 

  • Palm, T. (2006). Word problems as simulations of real-world situations: A proposed framework. For the Learning of Mathematics, 26(1), 42–47.

    Google Scholar 

  • Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58.

    Article  Google Scholar 

  • 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, NY: Springer.

  • Pollak, H. (1969). How can we teach application of mathematics? Educational Studies in Mathematics, 2, 393–404.

    Article  Google Scholar 

  • Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

  • Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.

  • Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311–343). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334–370). New York, NY: MacMillan.

  • Sriraman, B., & Knott, L. (2009). The mathematics of estimation: Possibilities for interdisciplinary pedagogy and social consciousness. Interchange, 40(2), 205–223.

    Article  Google Scholar 

  • Tangney, B., & Bray, A. (2013). Mobile technology, maths education & 21C learning. In Proceedings of the 12th world conference on mobile and contextual learning (pp. 20–27). Doha, Qatar: College of the North Atlantic – Qatar.

    Google Scholar 

  • Van Den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.

    Article  Google Scholar 

  • Van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics, 25(2), 2–9.

    Google Scholar 

  • Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Dordrecht, The Netherlands: Springer.

  • Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577–601.

    Article  Google Scholar 

  • Wijaya, A., van den Heuvel-Panhuizen, M., & Doorman, M. (2015). Opportunity-to-learn context-based tasks provided by mathematics textbooks. Educational Studies in Mathematics, 89(1), 41–65.

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Authors and Affiliations

  1. Universitat Autònoma de Barcelona, Barcelona, Spain

    Lluís Albarracín & Núria Gorgorió

  2. Universitat de València, Valencia, Spain

    Irene Ferrando

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  1. Lluís Albarracín
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  2. Irene Ferrando
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  3. Núria Gorgorió
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Correspondence to Lluís Albarracín.

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Albarracín, L., Ferrando, I. & Gorgorió, N. The Role of Context for Characterising Students’ Strategies when Estimating Large Numbers of Elements on a Surface. Int J of Sci and Math Educ 19, 1209–1227 (2021). https://doi.org/10.1007/s10763-020-10107-4

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