Fermi Problemlerinde Öğretmen Adayları Tarafından Kullanılan Modelleme Stratejileri: Otomatik Sulama Sistemi Görevi

Yıl 2023, , 980 - 1003, 15.09.2023

Zeynep Çakmak Gürel

https://doi.org/10.17679/inuefd.1235549

Öz

Bu çalışmanın amacı öğretmen adaylarının matematiksel modelleme etkinlikleri sırasında ortaya çıkan model stratejilerinin incelenmesidir. Araştırmada durum çalışması yöntemi kullanılmıştır. Çalışmaya 119 dördüncü sınıf öğretmen adayı katılmıştır. Veriler iki haftada toplanmıştır. Öğretmen adayları kendi isteklerine göre dört ya da beş kişilik olacak şekilde gruplar oluşturmuşlardır. Böylece toplamda 26 farklı grup bulunmaktadır. Modelleme problemi, gerçeklik, özgünlük, yakınlık, açıklık ve modelleme yeterliklerine hitap etme kriterleri dikkate alınarak araştırmacı tarafından tasarlanmıştır. Veri toplama araçlarını öğretmen adaylarının çalışma kağıtları ve sunum yaptıkları kayıtlar oluşturmaktadır. Verilerin analizinde ise tümdengelim ve tümevarım analiz yöntemleri birlikte kullanılmıştır. Elde edilen bulgulara göre, öğretmen adaylarının beş farklı modelleme stratejisi kullandıkları belirlenmiştir. Kapalı olmayan bir yüzeye sığabilecek eleman sayısı bağlamında verilen problemde en fazla “referans noktası” en az “konsantrasyon ölçümü” stratejisi kullanılmıştır. Çözüm uzaylarında fıskiyelerin kapladığı alan yerine bir fıskiye modelinin kapladığı alanın referans alınmasının daha gerçekçi sonuçlar verdiği görülmüştür. Bu stratejiler bağlamında oluşturulan çözüm uzayları modelleme sürecini ayrıntılandırmaktadır. Benzer bir modelleme problemi uygulayacak olan öğretmenlerin oluşan farklı modelleme stratejilerini ve çözüm uzaylarını önceden bilmesi, öğrencilere destek olması açısından önem arz etmektedir. Çözüm uzaylarının öğretmenlere öğrencilerinin bilgilerini takip etmede bir kaynak sağlayacağı düşünülmektedir.

Anahtar Kelimeler

çözüm uzayları, Fermi problemi, matematiksel modelleme, model stratejileri

The Modelling Strategies Used by the Pre-service Teachers in Fermi Problems: The Task of Automatic Irrigation System

Öz

This study aims to explore pre-service teachers’ solution spaces and model strategies in the mathematical modelling tasks involving estimations. A case study method was employed in the research. 119 pre-service teachers participated in the study. The data were collected in two weeks. The pre-service teachers created groups consisting of four or five individuals according to their wishes. Thus, 26 different groups emerged. The modelling problem was designed by the researcher by considering containing the reality, authenticity, closeness, openness, and modelling competencies criteria. The data collection tools consisted of the worksheets and presentation records of the pre-service teachers. The deduction and induction analysis methods were used together in the data analysis. According to the obtained findings, it was determined that the pre-service teachers used five different modelling strategies. In the problem given within the scope of the number of elements that could fit in a surface, the “reference point” was used most; on the other hand, the “concentration measures” were used least. It is found that taking the area covered by a sprinkler model as a reference rather than the area occupied by sprinklers in the solution spaces produces more realistic results. The solutions spaces created within these strategies was used to detail the modelling process. It is significant for teachers who will apply a similar modelling problem to know the created different modelling strategies and solution spaces. It is thought that the solution spaces will provide a source for teachers to keep track of the information related to their students.

Anahtar Kelimeler

Fermi problem, mathematical modelling, model strategies, solution spaces

Kaynakça

• Abay, S. & Gökbulut, Y. (2017). Sınıf öğretmeni adaylarının matematiksel modelleme becerileri: fermi problemleri uygulamaları. Uluslararası Türk Eğitim Bilimleri Dergisi ,9, 65-83. Retrieved from https://dergipark.org.tr/en/pub/goputeb/issue/34356/379918.
• Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi problems involving large numbers. Educ Stud Math, 86, 79–96. https://doi.org/10.1007/s10649-013-9528-9.
• 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.
• Albarracín, L., Ferrando, I. & 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., Segura, C., Ferrando, I., & Gorgorió N. (2022). Supporting mathematical modelling by upscaling real context in a sequence of tasks. Teaching Mathematics and its Applications: An International Journal of the IMA, 41(3), 183–197. https://doi.org/10.1093/teamat/hrab027.
• Anhalt, C. O. Cortez, R. & Bennett, A. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning, 20(3), 202-221.
• Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), 331–364.
• Ärlebäck, J. B., Bergsten, C. (2010). On the Use of Realistic Fermi Problems in Introducing Mathematical Modelling in Upper Secondary Mathematics. In: Lesh, R., Galbraith, P., Haines, C., Hurford, A. (eds) Modeling Students' Mathematical Modeling Competencies. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0561-1_52
• Baran Bulut, D. & Türker, M. (2022). Ortaokul öğrencilerinin üslü ifadeler konusunda modelleme yeterliklerinin incelenmesi: Sarmal kitaplık problemi. Recep Tayyip Erdoğan Üniversitesi Eğitim Fakültesi Dergisi, 2(2), 39-56.
• Blomhøj, M., & Kjeldsen, T. H. (2006). Teaching mathematical modelling through project work. ZDM Mathematics Education, 38(2), 163-177. https://doi.org/10.1007/BF02655887
• Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45-58.
• Blum, W., and Leiss, D. (2007). How do teachers deal with modeling problems? In C. Haines, P. Galbraith, W. Blum and S. Khan (Eds.), Mathematical modeling (ICTMA 12): Education, engineering and economics (pp. 222–231). Chichester: Horwood Publishing.
• Bogdan, R. C. & Biklen, S. K. (2007). Qualitative research for education: an introduction to theory and methods (5. bs.). USA: Pearson Education, Inc
• Borromeo Ferri, R. (2018). Theoretical Competency: For Your Practical Work. In: Learning How to Teach Mathematical Modeling in School and Teacher Education. Springer, Cham. https://doi.org/10.1007/978-3-319-68072-9_2
• Brunet-Biarnes, M., Albarracín, L. (2022). Exploring the negotiation processes when developing a mathematical model to solve a Fermi problem in groups. Math Ed Res J, . https://doi.org/10.1007/s13394-022-00435-9
• Chapman, O. (2015). Mathematics teachers' knowledge for teaching problem solving. LUMAT: International Journal on Math, Science and Technology Education, 3(1), 19-36.
• Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement. 20, 37-46. https://doi.org/10.1177/001316446002000104.
• Common Core State Standards Initiative [CCSI]. (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices and the Council of Chief State School Officers.
• Deniz, D. & Akgün, L. (2018). İlköğretim matematik öğretmeni adaylarının matematiksel modelleme becerilerinin incelenmesi. Akdeniz Eğitim Araştırmaları Dergisi, 12(24), 294-312. doi: 10.29329/mjer.2018.147.16
• Doerr, H. M., & Tripp, J. S. (1999). Understanding how students develop mathematical models. Mathematical Thinking and Learning, 1(3), 231–254.
• Ferrando, I. & Albarracín, L. (2019). 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.
• Ferrando, I., Segura, C., & Pla-Castells, M. (2020). Relations entre contexte, situation et schéma de résolution dans les problèmes d’estimation. Canadian Journal of Science, Mathematics and Technology Education, 20(3), 557-573.
• Ferrando, I., Segura, C., & Pla-Castells, M. (2021). Analysis of the relationship between context and solution plan in modelling tasks involving estimations. In Mathematical Modelling Education in East and West (pp. 119-128). Springer, Cham.
• Gallart, C., Ferrando, I., García-Raffi, L. M., Albarracín, L., & Gorgorió, N. (2017). Design and implementation of a tool for analysing student products when they solve fermi problems. In G. A. Stillman, W. Blum & G. Kaiser (Eds.), Mathematical Modelling and Applications. Crossing and Researching Boundaries in Mathematics Education, (pp. 265–275). Cham, Switzerland: Springer.
• Greefrath, G., and Vorhölter, K. (2016). Teaching and learning mathematical modelling: approaches and developments from German speaking countries. ICME-13 Topical Surveys, 1-42, Switzerland: Springer International Publishing. doi: 10.1007/978-3-319-45004-9_1
• Haberzettl, N., Klett, S., & Schukajlow, S. (2018). Mathematik rund um die Schule—Modellieren mit Fermi-Aufgaben. In K. Eilerts, & K. Skutella (Eds.), Neue Materialien für einen realitätsbezogenen Mathematikunterricht 5. Ein ISTRON-Band für die Grundschule (pp. 31–41). Wiesbaden: Springer Spectrum.https://doi.org/10.1007/978-3-658-21042-7_3
• Kaiser, G., and Stender, P. (2013). Complex modelling problem in cooperative learning environments self-directed. In G. Stillman, G. Kaiser, W. Blum, & J. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 277–294). Dordrecht: Springer.
• Kol, M. (2014). İlköğretim matematik öğretmen adaylarının matematikselleştirme sürecinin bir matematiksel modelleme etkinliği süresince incelenmesi (Yüksek Lisans Tezi). Middle East Technical University, Ankara.
• Krawitz, J., Schukajlow, S., & Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: What are important barriers? Educational Psychology, 38(10), 1221-1238.
• 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.
• 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.
• Mayring, P. (2014). Qualitative content analysis. Theoretical foundation, basic procedures and software solution, http://nbn-resolving.de/urn:nbn:de:0168-ssoar-395173
• Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 365–380). Dordrecht: Springer.
• Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modelling. In R. Lesh, P. L. Galbraith, C. R, Haines, & A. Hurford (Eds.), Modeling students’ mathematical modelling competencies (pp. 43–59). New York: Springer.
• Niss, M. (2015). Prescriptive modelling—Challenges and opportunities. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice: Cultural, social and cognitive influences (pp. 67–79). Cham: Springer.
• Özkan, M. (2021). Ortaokul öğrencilerinin düzgün çokgenler konusunda modelleme yeterlikleri üzerine bir eylem araştırması (Doktora tezi). Marmara Üniversitesi, İstanbul
• Perrenet, J., & Zwaneveld, B. (2012). The many faces of the mathematical modeling cycle. Journal of Mathematical Modelling and Application, 1(6), 3–21.
• Peter-Koop, A. (2004). Fermi problems in primary mathematics classrooms: Pupils’ interactive modelling processes. In I. Putt, R. Faragher, and M. McLean (Eds.), Mathematics Education for the Third Millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville) (pp. 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 (pp. 131–146). Dordrecht: Springer.
• Segura, C., & Ferrando, I. (2021). Classification and analysis of pre-service teachers’ errors in solving Fermi problems. Education Sciences, 11(8), 451.
• Sriraman, B. & Lesh, R. (2006). Beyond traditional conceptions of modelling. Zentralblatt für Didaktik der Mathematik, 38(3), 247-254.
• Stender, S. and Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM Mathematics Education, 47(7). doi:10.1007/s11858-015-0741-0
• Ural, A. (2014). Matematik öğretmen adaylarının matematiksel modelleme becerilerinin incelenmesi, Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 23, 110-141.
• Vorhölter, K., Kaiser, G. & Borromeo Ferri, R. (2014) Modelling in mathematics classroom instruction: an innovative approach for transforming mathematics education. In Y. Li, E. A. Silver & S. Li (Eds). Transforming Mathematics Instruction (pp. 21–36). Cham, Switzerland: Springer.
• Vos, P. & Frejd, P. (2022, February). The modelling cycle as analytic research tool and how it can be enriched beyond the cognitive dimension. Paper accepted for presentation at the 12th Congress of the European Society for Research in Mathematics Education (CERME12), withing the Thematic Working Group 6 Applications and Modelling. Bolzano, Italy
• Wess, R., & Greefrath, G. (2019). Professional competencies for teaching mathematical modelling—Supporting the modelling-specific task competency of prospective teachers in the teaching laboratory. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), European Research in Mathematics: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (pp. 1274–1283). Utrecht, Netherlands.