한국수학교육학회지시리즈C:초등수학교육 (Education of Primary School Mathematics)

  • 제28권4호
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  • Pages.185-200
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  • 2025
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  • 1226-6914(pISSN)
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  • 2287-9927(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
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수학적 모델링 과제로서 페르미 문제에 대한 초등 저학년의 해결 전략

Younger elementary students' solution strategies for Fermi problems as mathematical modeling tasks

  • 투고 : 2025.09.30
  • 심사 : 2025.10.29
  • 발행 : 2025.10.31
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초록

수학 역량으로서 수학적 모델링의 중요성은 수학교육계의 공감대를 형성하지만 초등 저학년을 대상으로 한 수학적 모델링 연구 및 실제는 상대적으로 미흡한 실정이다. 큰 수의 어림을 겨냥하는 페르미 문제는 모델링 과제의 특성을 띠고 모델링 사이클과 유사한 과정을 거쳐 해결되며 특정 지식을 필요로 하지 않는 등 여러 측면에서 초등 저학년을 위한 수학적 모델링 도입에 적합하다고 평가된다. 이에 본 연구는 초등 3학년을 대상으로 페르미 문제를 제시하여 학생 스스로 생각한 해결 전략을 파악하고, 교사의 지도 개입이나 모둠 활동이 해결 전략에 미치는 영향을 분석하는 것을 목적으로 한다. 연구 결과, 초등학교 3학년 학생들이 사용한 페르미 문제해결 전략은 밀도측정에 치중했으나 지도 후 전략이 다양해져 밀도측정 > 격자분포 > 외부출처 > 모두세기 > 축소·비례의 순으로 나타났고 교사의 개입, 모둠 활동뿐만 아니라 과제 맥락의 영향도 확인되었다. 이로부터 초등 저학년을 대상으로 한 수학적 모델링 지도를 위해 페르미 문제를 활용하기 위한 교수학적 시사점을 탐색하였다.

The importance of mathematical modeling as a mathematical competence is widely recognized in mathematics education. However, research and practice focusing on mathematical modeling for early elementary grades remain relatively insufficient. Fermi problems, which aim at estimating of large quantities, exhibit characteristics of modeling tasks, involve processes similar to the modeling cycle, and do not require specific prior knowledge. For these reasons, they are considered well suited for introducing mathematical modeling to younger elementary students. This study, therefore, aims to identify the problem-solving strategies employed by third-grade elementary students when presented with Fermi problems, and to analyze the effects of teacher intervention and group activities on these strategies. The results revealed that the third graders initially relied mainly on the density measures strategy, but after instruction, their strategies diversified and appeared in the order of density measures > grid distribution > external sources > exhaustive counting > reduction/proportion. The influence of group work was also observed. Based on these findings, this study discusses the potential of utilizing Fermi problems for mathematical modeling instruction in early elementary grades and explores related didactical implications.

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과제정보

이 연구는 2025년도 서울교육대학교 교내연구비를 지원받아 수행되었음

참고문헌

  1. 구나영. (2025). 초등 영재 학생들의 페르미 문제해결 과정 분석. 학교수학, 27(1), 163-183.
  2. 김나희, 장혜원. (2025). 개별 및 모둠별 수학적 모델링에서 초등학생의 과제 맥락 처리 유형 분석. 수학교육, 64(3), 417-436.
  3. 박찬혜. (2022). 초등학교 2학년 수학 교과서 단원별 수학적 모델링 프로그램 개발 및 적용 사례 분석 [석사학위논문, 서울교육대학교 교육전문대학원].
  4. 송설란. (2011). 페르미 문제에 대한 문제해결력과 정당화 과정 [석사학위논문, 한국교원대학교 대학원].
  5. 염재명. (2021). 초등수학에 적용된 수학적 모델링 과제들의 특성 연구 [석사학위논문, 서울교육대학교 교육전문대학원].
  6. 장혜원, 최혜령, 강윤지, 김은혜. (2019). 초등학교 저학년을 위한 수학적 모델링 과제 개발 및 적용 가능성 탐색. 한국초등수학교육학회지, 23(1), 93-117.
  7. 최지선. (2017). 수학적 모델링 수업에 대한 초등 교사의 인식. 수학교육학연구, 27(2), 313-328.
  8. 허정인. (2020). 중등 수학 영재의 창의적 문제해결력에 관한 고찰 [석사학위논문, 부산대학교 대학원].
  9. 허정인, 노지화. (2024). 초등 영재의 페르미 어림을 통한 창의적 문제해결력 분석. East Asian Mathematical Journal, 40(2), 167-181.
  10. Albarracín, L. (2021). Large number estimation as a vehicle to promote mathematical modeling. Early Childhood Education Journal, 49(4), 681-691. https://doi.org/10.1007/s10643-020-01104-x
  11. Albarracín, L., & Ärlebäck, J. B. (2019). Characterising mathematical activities promoted by Fermi problems. For the Learning of Mathematics, 39(3), 10-13.
  12. Albarracín, L., & Ärlebäck, J. B. (2025). Exploring the role of assumptions in mathematical modeling teacher training using Fermi problems. ZDM Mathematics Education 57, 213–228. https://doi.org/10.1007/s11858-025-01677-0
  13. Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi problems involving large numbers. Educational Studies in Mathematics, 86, 79-96. https://doi.org/10.1007/s10649-013-9528-9
  14. 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), 45-57. https://doi.org/10.30722/IJISME.27.02.004
  15. Anhalt, C. O., Staats, S., Cortez, R., & Civil, M. (2018). Mathematical modeling and culturally relevant pedagogy. In Y. J. Dori, Z. R. Mevarech, & D. R. Baker, (Eds.) Cognition, metacognition, and culture in STEM education. (pp. 307-330). Springer, Cham. https://doi.org/10.1007/978-3-319-66659-4_14
  16. 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
  17. Ärlebäck, J. B. (2011). Exploring the solving process of groups solving realistic Fermi problem from the perspective of the anthropological theory of didactics. In Proceedings of the seventh congress of the european society for research in mathematics education (CERME7) (pp. 1010-1019).
  18. Ärlebäck, J. B., & Bergsten, C. (2013). On the use of realistic Fermi problems in introducing mathematical modelling in upper secondary mathematics. Modeling Students' Mathematical Modeling Competencies: ICTMA 13, (pp. 597-609).
  19. Blum, W. (1995). Applications and modelling in mathematics teaching and mathematics education–some important aspects of practice and of research. In C. Sloyer, W. Blum, & I. D. Huntley (Eds.), Advances and perspectives in the teaching of mathematical modeling and applications (pp. 1–20). Water Street Mathematics.
  20. Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling in school and teacher education. Springer Nature.
  21. Brunet-Biarnes, M., & Albarracín, L. (2022). Exploring the negotiation processes when developing a mathematical model to solve a Fermi problem in groups. Mathematics Education Research J ournal, 1-22.
  22. Carlson, M. A., Wickstrom, M. H., Burroughs, E., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary classroom. In C. Hirsch (Ed.), Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics. (pp. 121– 130).
  23. National Council of Teachers of Mathematics. Crites, T. W. (1993). Strategies for estimating discrete quantities. The Arithmetic Teacher, 41(2), 106-108. https://doi.org/10.5951/AT.41.2.0106
  24. English, L. D. (2003). Mathematical modelling with young learners. Mathematical modelling: A way of life : ICTMA, 11. (pp. 3-17).
  25. English, L. D. (2006). Mathematical modeling in the primary school: Children's construction of a consumer guide. Educational Studies in Mathematics, 63, 303-323. https://doi.org/10.1007/s10649-005-9013-1
  26. English, L. D. (2010). Young children's early modelling with data. Mathematics Education Research Journal, 22(2), 24-47. https://doi.org/10.1007/BF03217564
  27. English, L. D. (2012). Data modelling with first-grade students. Educational Studies in Mathematics, 81(1), 15-30. https://doi.org/10.1007/s10649-011-9377-3
  28. English, L. D. (2021). Mathematical and interdisciplinary modeling in optimizing young children's learning. In J. M. Suh, M. H. Wickstrom, & L. D. English (Eds.). Exploring mathematical modeling with young learners. (pp. 3-23). Springer, Cham. https://doi.org/10.1007/978-3-030-63900-6_1
  29. English, L. D. (2025). Spatial modelling in second grade: creating crab colonies. ZDM–Mathematics Education, 57, 259–273. https://doi.org/10.1007/s11858-025-01667-2
  30. English, L. D., & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematics Education Research Journal, 16(3), 58-79. https://doi.org/10.1007/BF03217401
  31. 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 et al. (Eds.), Mathematical modelling and applications, international perspectives on the teaching and learning of mathematical modelling, (pp. 265-275). https://doi.org/10.1007/978-3-319-62968-1_23.
  32. Greer, B., Verschaffel, L., & Mukhopadhyay, S. (2007). Modelling for life: Mathematics and children's experience. In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.) Modelling and applications in mathematics education. New ICMI Study Series, vol 10. (pp. 89–98), Springer. https://doi.org/10.1007/978-0-387-29822-1_7
  33. Kittel, A., & Marxer, M. (2005). Wie viele Menschen passen auf ein Fussballfeld? Mit Fermiaufgaben individuell fördern. Mathematik Lehren, 131, 14-18.
  34. Krawitz, J., Schukajlow, S., Yang, X., & Geiger, V. (2025). A systematic review of international perspectives on mathematical modelling: modelling goals and task characteristics. ZDM Mathematics Education, 57, 193–212. https://doi.org/10.1007/s11858-025-01683-2
  35. Peter-Koop, A. (2005). Fermi problems in primary mathematics classrooms: Fostering children's mathematical modelling processes. Australian Primary Mathematics Classroom, 10(1), 4-8.
  36. 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: Purpose, use and exemplars, (pp. 131-146). Springer. https://doi.org/10.1007/978-0-387-09669-8_10
  37. Schukajlow, S., Krawitz, J., Kanefke, J., Blum, W., & Rakoczy, K. (2023). Open modelling problems: Cognitive barriers and instructional prompts. Educational Studies in Mathematics, 114(3), 417-438. https://doi.org/10.1007/s10649-023-10265-6
  38. Wei, Y., Zhang, Q., & Guo, J. (2022). Can mathematical modelling be taught and learned in primary mathematics classrooms: A systematic review of empirical studies. Education Sciences, 12(12), 923. https://doi.org/10.3390/educsci12120923
  39. Wickstrom, M. H. (2017). Mathematical modeling: challenging the figured worlds of elementary mathematics. In E. Galindo, & J. Newton, (Eds.). Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (pp. 685-692). Hoosier Association of Mathematics Teacher Educators.