Which is the Best Strategy for a Fermi Problem? Adaptability of Pre-Service Teachers
Abstract
Fermi problems, suitable for primary school, pose a real and open situation that allows the development and comparison of multiple strategies. This requires teachers to be adaptive (able to choose the most appropriate strategy). The aim of this paper is to characterise and analyse the adaptability of prospective teachers when solving these problems. For this purpose, the research is divided into two studies. Study 1 presents a survey addressed to experts in mathematics education; the analysis of their answers makes it possible to link the contextual characteristics of the problems with strategies, and these with appropriateness criteria (accuracy, speed and rigour). These results lead to a characterisation of adaptability that allows us to approach Study 2 with pre-service teachers, finding that most adaptive solvers use strategies non-systematically.
Keywords
Adaptability, Flexibility, Estimation, Modelling, Fermi problemsReferences
Albarracín, L., Ferrando, I. y Gorgorió, N. (2021). The Role of Context for Characterising Students’ Strategies when Estimating Large Numbers of Elements on a Surface. International Journal of Science and Mathematics Education, 19, 1209-1227. https://doi.org/10.1007/s10763-020-10107-4
Albarracín, L. y 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), 45-57. https://doi.org/10.30722/IJISME.27.02.004
Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), 331-364. https://doi.org/10.54870/1551-3440.1157
Blöte, A. W., Van der Burg, E. y Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: instruction effects. Journal of Educational Psychology, 93, 627-638. https://doi.org/10.1037/0022-0663.93.3.627
Carlson, J. E. (1997). Fermi problems on gasoline consumption. The Physics Teacher, 35(5), 308-309. https://doi.org/10.1119/1.2344696
Chapman, O. (2015). Mathematics teachers’ knowledge for teaching problem solving. LUMAT: International Journal on Math, Science and Technology Education, 3(1), 19-36. https://doi.org/10.31129/lumat.v3i1.1049
Durkin, K., Star, J. R. y Rittle-Johnson, B. (2017). Using comparison of multiple strategies in the mathematics classroom: lessons learned and next steps. ZDM Mathematics Education, 49, 585-597. https://doi.org/10.1007/s11858-017-0853-9
Efthimiou, C. J. y 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
English, L. D. (2011). Data modeling in the beginning school years. En P. Sullivan y M. Goos (Eds.), Proceedings of the 34th Annual Conference of the Mathematics Education Research Group of Australia (pp. 226-234). MERGA.
Ferrando, I., Segura, C. y Pla-Castells, M. (2021). Analysis of the relationship between context and solution plan in modelling tasks involving estimations. En F. K. S. Leung, G. A. Stillman, G. Kaiser y K. L. Wong (Eds.), Mathematical Modelling Education in East and West (pp. 119-128). Cham: Springer. https://doi.org/10.1007/978-3-030-66996-6_10
Garcia Coppersmith, J. y Star, J. R. (2022). A Complicated Relationship: Examining the Relationship Between Flexible Strategy Use and Accuracy. Journal of Numerical Cognition, 8(3), 382-397. https://doi.org/10.5964/jnc.7601
Haberzettl, N., Klett, S. y Schukajlow, S. (2018). Mathematik rund um die Schule—Modellieren mit Fermi-Aufgaben. En K. Eilerts y K. Skutella (Eds.), Neue Materialien für einen realitätsbezogenen Mathematikunterricht 5. Ein ISTRON-Band für die Grundschule (pp. 31-41). Springer Spectrum. https://doi.org/10.1007/978-3-658-21042-7_3
Henze, J. y Fritzlar, T. (2010). Primary school children’s model building processes by the example of Fermi questions. En A. Ambrus y E. Vásárhelyi (Eds.), Problem Solving in Mathematics Education. Proceedings of the 11th ProMath conference (pp. 60-75). Eötvös Loránd University.
Heinze, A., Star, J. R. y Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535-540. https://doi.org/10.1007/s11858-009-0214-4
Hickendorff, M. (2022). Flexibility and Adaptivity in Arithmetic Strategy Use: What Children Know and What They Show. Journal of Numerical Cognition, 8(3), 367-381. https://doi.org/10.5964/jnc.7277
Hickendorff, M., McMullen, J. y Verschaffel, L. (2022). Mathematical Flexibility: Theoretical, Methodological, and Educational Considerations. Journal of Numerical Cognition, 8(3), 326-334. https://doi.org/10.5964/jnc.10085
Huincahue-Arcos, J., Borromeo-Ferri, R. y Mena-Lorca, J. J. F. (2018). El conocimiento de la modelación matemática desde la reflexión en la formación inicial de profesores de matemática. Enseñanza de las Ciencias. Revista de investigación y experiencias didácticas, 36(1), 99-115. https://doi.org/10.5565/rev/ensciencias.2277
Klock, H. y Siller, H.-S. (2020). A Time-Based Measurement of the Intensity of Difficulties in the Modelling Process. En H. Wessels, G. A. Stillman, G. Kaiser, y E. Lampen (Eds.), International perspectives on the teaching and learning of mathematical modelling (pp. 163-173). Springer. https://doi.org/10.1007/978-3-030-37673-4_15
Ko, P. Y. y Marton, F. (2004). Variation and the secret of the virtuoso. En F. Marton y A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 43-62). Erlbaum. https://doi.org/10.4324/9781410609762
Krawitz, J., Schukajlow, S. y Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: what are important barriers? Educational Psychology, 38(10), 1221-1238. https://doi.org/10.1080/01443410.2018.1502413
Lemaire, P. y Siegler, R. S. (1995). Four aspects of strategic change: contributions to children’s learning of multiplication. Journal of experimental psychology: General, 124(1), 83. https://doi.org/10.1037/0096-3445.124.1.83
Levav-Waynberg, A. y Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. The Journal of Mathematical Behavior, 31(1), 73-90. https://doi.org/10.1016/j.jmathb.2011.11.001
Lu, X. y Kaiser, G. (2022). Creativity in students’ modelling competencies: conceptualisation and measurement. Educational Studies in Mathematics, 109(2), 287-311. https://doi.org/10.1007/s10649-021-10055-y
Newton, K. J., Lange, K. y Booth, J. L. (2020). Mathematical flexibility: Aspects of a continuum and the role of prior knowledge. The Journal of Experimental Education, 88(4), 503-515. https://doi.org/10.1080/00220973.2019.1586629
Nistal A. A., Van Dooren W. y Verschaffel L. (2012). Flexibility in Problem Solving: Analysis and Improvement. En N. M. Seel (Eds.), Encyclopedia of the Sciences of Learning. Springer. https://doi.org/10.1007/978-1-4419-1428-6_540
Obersteiner, A., Alibali, M. W. y Marupudi, V. (2022). Comparing fraction magnitudes: Adults’ verbal reports reveal strategy flexibility and adaptivity, but also bias. Journal of Numerical Cognition, 8(3), 398-413. https://doi.org/10.5964/jnc.7577
Okamoto, H., Hartmann, M. y Kawasaki, T. (2023). Analysis of the Relationship between Creativity in Fermi Problems Measured by Applying Information Theory, Creativity in Psychology, and Mathematical Creativity. Education Sciences, 13(3), 315. https://doi.org/10.3390/educsci13030315
Peter-Koop, A. (2009). Teaching and Understanding Mathematical Modelling through Fermi-Problems. En B. Clarke, B. Grevholm y R. Millman (Eds.), Tasks in primary mathematics teacher education (pp. 131-146). Springer. https://doi.org/10.1007/978-0-387-09669-8_10
Pla-Castells, M., Melchor, C. y Chaparro, G. (2021). MAD+. Introducing misconceptions in the temporal analysis of the mathematical modelling process of a Fermi problem. Education Sciences, 11(11), 747. https://doi.org/10.3390/educsci11110747
Segura, C., Ferrando, I. y Albarracín, L. (2021). Análisis de los factores de complejidad en planes de resolución individuales y resoluciones grupales de problemas de estimación de contexto real. Quadrante, 30(1), 31-51. https://doi.org/10.48489/quadrante.23592
Segura, C., Ferrando, I. y Albarracín, L. (2023). Does collaborative and experiential work influence the solution of real-context estimation problems? A study with prospective teachers. The Journal of Mathematical Behavior, 70, 101040. https://doi.org/10.1016/j.jmathb.2023.101040
Segura, C. y Ferrando, I. (2023). Pre-service teachers’ flexibility and performance in solving Fermi problems. Educational Studies in Mathematics, 113(2), 207-227. https://doi.org/10.1007/s10649-023-10220-5
Robinson, A. W. (2008). Don’t just stand there—teach Fermi problems! Physics Education, 43(1), 83-87. https://doi.org/10.1088/0031-9120/43/01/009
Sáenz, C. (2007). La competencia matemática (en el sentido de PISA) de los futuros maestros. Enseñanza de las Ciencias, 25(3), 355-366. https://doi.org/10.5565/rev/ensciencias.3701
Schoenfeld, A. H. (1982). Measures of problem-solving performance and of problem-solving instruction. Journal for Research in Mathematics Education, 13(1), 31-49. https://doi.org/10.2307/748435
Schukajlow, S., Krug, A. y Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393-417. https://doi.org/10.1007/s10649-015-9608-0
Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM Mathematics Education, 41, 619-625. https://doi.org/10.1007/s11858-009-0203-7
Siegler, R. S. (1996). Emerging minds: The process of change in children ‘s thinking. Oxford: Oxford University Press. https://doi.org/10.1093/oso/9780195077872.001.0001
Sriraman, B. y Lesh, R. (2006). Modeling conceptions revisited. Zentralblatt für Didaktik derMathematik, 38, 247-254. https://doi.org/10.1007/BF02652808
Sriraman, B. y 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
Star, J. R. y Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and instruction, 18(6), 565-579. https://doi.org/10.1016/j.learninstruc.2007.09.018
Taggart, G. L., Adams, P. E., Eltze, E., Heinrichs, J., Hohman, J. y Hickman, K. (2007). Fermi Questions. Mathematics Teaching in the Middle School, 13(3), 164-167. https://doi.org/10.5951/MTMS.13.3.0164
Threlfall, J. (2002). Flexible Mental Calculation. Educational Studies in Mathematics, 50, 29-47. https://doi.org/10.1023/A:1020572803437
Van Dooren, W., Verschaffel, L. y Onghena, P. (2003). Preservice teachers’ preferred strategies for solving Arithmetic and Algebra word problems. Journal of Mathematics Teacher Education, 6(1), 27-52. https://doi.org/10.1023/A:1022109006658
Verschaffel, L., Luwel, K., Torbeyns, J. y Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335-359. https://doi.org/10.1007/BF03174765
Published
Downloads
Copyright (c) 2023 Carlos Segura Cordero, Irene Ferrando
This work is licensed under a Creative Commons Attribution 4.0 International License.