Problemas de Fermi em testes de avaliação padronizados no 8.º ano
DOI:
https://doi.org/10.48489/quadrante.23587Palavras-chave:
problemas de Fermi, modelação matemática, critérios de tarefas, tarefas de modelação, problemas de testeResumo
Desde 2006, a Alemanha tem vindo a seguir uma estratégia abrangente de monitorização educacional que inclui testes de avaliação comparativos e padronizados (chamados VERA) em matemática. Estes testes são administrados em todo o país e, com algumas exceções, no oitavo ano de todas as escolas de ensino básico. Entre outras competências, estes testes examinam a competência de modelação dos estudantes. Nas tarefas de aplicação e modelação, os vários requisitos associados à construção de tarefas para testes de avaliação criam desafios específicos que muitas vezes levam a que tais tarefas resultem em problemas de palavras em vez de aplicações reais. Uma abordagem possível para criar um problema de modelação adequado para um teste de avaliação é utilizar problemas de Fermi, que se baseiam num contexto real. Tendo por base várias classificações de tarefas matemáticas, este artigo desenvolve uma série de critérios para criar problemas de Fermi para fins de avaliação. Estes critérios são aplicados especificamente aos problemas de Fermi incluídos no referido instrumento de avaliação padronizada, no 8.º ano, na Alemanha. Com base nos resultados, as diferenças e semelhanças entre diversos problemas de Fermi são identificadas e discutidas. Os problemas de Fermi apresentam uma certa homogeneidade como tarefas específicas de modelação, mas estão também associados a um amplo espetro de dificuldades, que parecem estar ligadas ao número de quantidades matemáticas necessárias para obter a solução. Diferentes problemas de Fermi podem abarcar muitos aspetos diferentes de desempenho e são uma boa forma de incorporar situações autênticas em problemas de teste.
Referências
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
Bergman Ärlebäck, J. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Montana Mathematics Enthusiast, 6(3), 331–364.
Bergman Ärlebäck, J., & Albarracín, L. (2019). The use and potential of Fermi problems in the STEM disciplines to support the development of twenty-first century competencies. ZDM Mathematics Education, 51(6), 979–990. https://doi.org/10.1007/s11858-019-01075-3
Bergman Ärlebäck, J., & Bergsten, C. (2010). On the use of realistic Fermi problems in introducing mathematical modelling in upper secondary mathematics. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling mompetencies (pp. 597–609). Boston, MA: Springer US. https://doi.org/10.1007/978-1-4419-0561-1_52
Blum, W., Drüke-Noe, C., Leiß, D., Wiegand, B., & Jordan, A. (2005). Zur rolle von bildungsstandards für die qualitätsentwicklung im mathematikunterricht. Zentralblatt für Didaktik der Mathematik, 37(4), 267–274. https://doi.org/10.1007/BF02655814
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 (ICTMA 12): Education, Engineering and Economics (pp. 222–231). Chichester: Horwood. https://doi.org/10.1533/9780857099419.5.221
Blum, W., vom Hofe, R., Jordan, A., & Kleine, M. (2004). Grundvorstellungen als aufgabenanalytisches und diagnostisches Instrument bei PISA. In M. Neubrand (Ed.), Mathematische kompetenzen von schülerinnen und schülern in Deutschland: Vertiefende analysen im rahmen von PISA 2000 (pp. 145–173). Wiesbaden: VS, Verlag für Sozialwissenschaften.
Büchter, A., & Leuders, T. (2005). Mathematikaufgaben selbst entwickeln: Lernen fördern - Leistung überprüfen. Berlin: Cornelsen.
Drüke-Noe, C. (2014). Aufgabenkultur in klassenarbeiten im fach mathematik. Wiesbaden: Springer Fachmedien. https://doi.org/10.1007/978-3-658-05351-2
Ehmke, T., Köller, O., & Stanat, P. (2017). Äquivalenz der erfassung mathematischer kompetenzen in PISA 2012 und im IQB-Ländervergleich 2012. Zeitschrift für Erziehungswissenschaft, 20(S2), 37–59. https://doi.org/10.1007/s11618-017-0751-5
Ferrando, I., & Albarracín, L. (2021). Students from grade 2 to grade 10 solving a Fermi problem: Analysis of emerging models. Mathematics Education Research Journal, 33, 61–78 https://doi.org/10.1007/s13394-019-00292-z
Greefrath, G. (2010). Analysis of modeling problem solutions with methods of problem solving. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 265–271). Boston, MA: Springer US. https://doi.org/10.1007/978-1-4419-0561-1_23
Greefrath, G. (2019a). Fermi-aufgaben in vergleichsarbeiten in klasse 8 – Kriterien und ergebnisse. In A. Büchter, M. Glade, R. Herold-Blasius, M. Klinger, F. Schacht, & P. Scherer (Eds.), Vielfältige zugänge zum nathematikunterricht (pp. 19–32). Wiesbaden: Springer Fachmedien Wiesbaden. https://doi.org/10.1007/978-3-658-24292-3_2
Greefrath, G. (2019b). Mathematical Modelling—Background and current projects in Germany. In J. M. Marbán, M. Arce, A. Maroto, J. M. Muñoz-Escolano, & Á. Alsina (Eds.), Investigación en Educación Matemática XXIII (pp. 23-41). Valladolid: SEIEM. Retrieved from http://www.seiem.es/docs/actas/23/ActasXXIIISEIEM.pdf
Greefrath, G. (2020). Mathematical modelling competence. Selected current research developments. Avances de Investigación En Educación Matemática, 17, 38–51. https://doi.org/10.35763/aiem.v0i17.303
Greefrath, G., Siller, H.-S., & Ludwig, M. (2017). Modelling problems in German grammar school leaving examinations (Abitur) – Theory and practice. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (pp. 932–939). Dublin: DCU Institute of Education and ERME. Retrieved from https://hal.archives-ouvertes.fr/hal-01933483
Hernandez-Martinez, P., & Vos, P. (2018). “Why do I have to learn this?” A case study on students’ experiences of the relevance of mathematical modelling activities. ZDM Mathematics Education, 50(1–2), 245–257. https://doi.org/10.1007/s11858-017-0904-2
Joint Committee on Standards for Educational Evaluation. (2018). Checklist of the program evaluation standards statements. Retrieved from https://wmich.edu/evaluation/checklists
Kaiser, G. (2007). Modelling and modelling competencies in school. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (ICTMA 12): Education, Engineering and Economics (pp. 110–119). Chichester: Horwood. https://doi.org/10.1533/9780857099419.3.110
KMK (Ed.). (2012). Bildungsstandards im fach mathematik für die allgemeine hochschulreife (Beschluss der Kultusministerkonferenz vom 18.10.2012). Köln: Wolters Kluwer.
Maaß, K. (2006). What are modelling competencies? ZDM Mathematics Education, 38(2), 113–142. https://doi.org/10.1007/BF02655885
Maaß, K. (2010). Classification scheme for modelling tasks. Journal Für Mathematik-Didaktik, 31(2), 285–311. https://doi.org/10.1007/s13138-010-0010-2
Maier, U., Kleinknecht, M., Metz, K., & Bohl, T. (2010). Ein allgemeindidaktisches kategoriensystem zur analyse des kognitiven potenzials von aufgaben. (Fachportal Pädagogik). Retrieved from https://nbn-resolving.org/urn:nbn:de:0111-pedocs-137347
Maier, U., Metz, K., Bohl, T., Kleinknecht, M., & Schymala, M. (2012). Vergleichsarbeiten als instrument der datenbasierten schul- und unterrichtsentwicklung in Gymnasien. In A. Wacker, U. Maier, & J. Wissinger (Eds.), Schul- und Unterrichtsreform durch ergebnisorientierte Steuerung (pp. 197–224). Wiesbaden: VS Verlag für Sozialwissenschaften. https://doi.org/10.1007/978-3-531-94183-7_9
Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Klagenfurt. Retrieved from https://nbn-resolving.org/urn:nbn:de:0168-ssoar-395173
Neubrand, M., Klieme, E., Lüdtke, O., & Neubrand, J. (2002). Kompetenzstufen und schwierigkeitsmodelle für den PISA-Test zur mathematischen grundbildung. Unterrichtswissenschaft, 30(2), 100–119.
Niss, M. (1992). Applications and modelling in school mathematics—Directions for future development. In I. Wirszup & R. Streit (Eds.), Development in school mathematics education around the world (Vol. 3, pp. 346–361). Reston: NCTM.
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Abingdon, Oxon ; New York, NY: Routledge.
Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102(1), 9–28. https://doi.org/10.1007/s10649-019-09903-9
OECD. (2013). PISA 2012 Assessment and analytical framework: Mathematics, reading, science, problem solving and financial literacy. OECD. https://doi.org/10.1787/9789264190511-en
Palm, T. (2007). Features and impact of the authenticity of applied mathematical school tasks. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education. The 14th ICMI Study (pp. 201–208). Boston, MA: Springer US. https://doi.org/10.1007/978-0-387-29822-1_20
Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: Pupils’ interactive modelling processes. Mathematics Education for the Third Millennium: Towards 2010. Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Townsville, 27, 454–461. Sydney: MERGA.
Peter-Koop, A. (2009). Teaching and understanding mathematical modelling through Fermi-problems. In B. Clarke, B. Grevholm, & R. Millman (Eds.), Tasks in primary mathematics teacher education (Vol. 4, pp. 131–146). Boston, MA: Springer US. https://doi.org/10.1007/978-0-387-09669-8_10
Pólya, G. (1981). Mathematical discovery. On understanding, learning, and teaching problem solving. New York: John Wiley & Sons Inc.
Prodromou, L. (1995). The backwash effect: From testing to teaching. ELT Journal, 49(1), 13–25. https://doi.org/10.1093/elt/49.1.13
Reit, X.-R., & Ludwig, M. (2015a). An approach to theory based modelling tasks. In G. A. Stillman, W. Blum, & M. Salett Biembengut (Eds.), Mathematical modelling in education research and practice (pp. 81–91). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-18272-8_6
Reit, X.-R., & Ludwig, M. (2015b). Thought structures as an instrument to determine the degree of difficulty of modelling tasks. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (CERME9, 4-8 February 2015) (pp. 917–922). Prague: Charles University in Prague, Faculty of Education and ERME.
Ross, J., & Ross, M. (1986). Fermi problems or how to make the most of what you already know. In H. L. Schoen & M. J. Zweng (Eds.), Estimation and mental computation (pp. 175–181). Reston, VA: National Council of Teachers of Mathematics.
Schecker, H., & Parchmann, I. (2007). Standards and competence models: The German situation. In D. Waddington, P. Nentwig, & S. Schanze (Eds.), Making it comparable. Standards in science education (pp. 147–164). Münster: Waxmann.
Sriraman, B., & Lesh, R. A. (2006). Modeling conceptions revisited. ZDM Mathematics Education, 38(3), 247–254. https://doi.org/10.1007/BF02652808
van den Heuvel-Panhuizen, M., & Becker, J. (2003). Towards a didactic model for assessment design in mathematics education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 689–716). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-010-0273-8_23
Vos, P. (2018). “How real people really need mathematics in the real world” – Authenticity in mathematics education. Education Sciences, 8(4), 195. https://doi.org/10.3390/educsci8040195
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