한국수학교육학회지시리즈C:초등수학교육 (Education of Primary School Mathematics)

  • 제25권3호
  • /
  • Pages.213-232
  • /
  • 2022
  • /
  • 1226-6914(pISSN)
  • /
  • 2287-9927(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

초등학교 4학년 학생들의 수직선 이해 분석: 분수 개념 및 분수의 덧셈과 뺄셈을 중심으로

An Analysis of Elementary Students' Understanding of Number Line: Focused on Concept of Fractions and Addition and Subtraction of Fractions

  • 투고 : 2022.05.31
  • 심사 : 2022.06.25
  • 발행 : 2022.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

분수 학습에서 수직선 모델의 중요성에 따라, 본 연구에서는 초등학교 4학년 학생들의 수직선에서의 분수 개념 및 분수의 덧셈과 뺄셈에 대한 이해가 어느 정도인지 살펴보았다. 수직선 검사 도구는 분수 나타내기, 분수의 크기 비교, 분수의 덧셈 및 뺄셈과 관련된 문항으로 구성되었으며 구조화된 수직선과 반구조화된 수직선을 제시하였다. 연구 결과, 전반적으로 정답률이 높지 않았으며, 정답보다 오답의 반응이 높게 드러난 문항들도 있었다. 또한 구조화된 수직선에 비하여 반구조화된 수직선이 제시되는 문항에서의 정답률이 낮게 드러났으며, 다양한 오답 반응을 확인할 수 있었다. 본 연구의 결과를 바탕으로 초등학생들의 분수 이해 및 수직선 이해를 위한 교수·학습 방향에 관한 시사점을 논의하였다.

With the importance of number line in learning fractions, this study investigated how 4th grade students understand fractions and its operations in number line. The questionnaire consisted 22 items which were related to representing fractions, comparing the size of fractions, and operating addition and subtraction of fractions. Both structured number line and sub-structured number line were presented in the items. As results of the study, the overall success rates were not high and even some items showed higher incorrect answer rates than the success rates. Also, the students showed a difficulty in solving non-structured number line tasks. It was also noticeable that students showed several types of incorrect answers, which means that students had incomplete understanding of both fractions and number line. This paper is expected to shed light on elementary students' understanding of fractions and number line and to provide implications of how to deal with number line in teaching and learning fractions in the elementary school.

키워드

참고문헌

  1. 교육부(2020a). 수학 2-1. 서울: 비상교육.
  2. 교육부(2020b). 수학 3-1. 서울: 비상교육.
  3. 교육부(2020c). 수학 4-2. 서울: 비상교육.
  4. 교육부(2020d). 수학 3-2. 서울: 비상교육.
  5. 김양권, 홍진곤(2016). 수 개념 학습에서 수직선의 도입과 활용. 한국초등수학교육학회지, 20(3), 431-456.
  6. 김양권, 홍진곤(2017). 초등학생의 수직선 이해와 사용의 어려움. 수학교육논문집, 31(1), 85-101.
  7. 박예은(2017). 역사발생적 원리에 따른 수직선 의미와 지도방안 고찰. 서울대학교 대학원 석사학위논문.
  8. 유진영, 신재홍(2020). 단위 조정에 따른 초등학생의 분수 개념 이해 분석. 초등수학교육, 23(2), 87-116.
  9. 이상미(2010). 초등학교 4, 5, 6학년 학생들의 수직선 이해 실태 조사. 한국교원대학교 대학원 석사학위논문.
  10. Bobis, J., & Bobis, E. (2005). The empty number line: Making children's thinking visible. Making mathematics vital, 66-72.
  11. Bobis, J., & Bobis, E. (2005). The empty number line: Making children's thinking visible. In M. Coupland, J. Anderson, & T. Spencer (Eds.), Making mathematics vital: Proceedings of the 20th biennial conference of the Australian Association of Mathematics Teachers (pp. 66-72). Sydney: AAMT.
  12. Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232. https://doi.org/10.2307/749066
  13. Cramer, K., Ahrendt, S., Monson, D., Wyberg, T., & Miller, C. (2017). Making sense of third-grade students' misunderstandings of the number line. Investigations in Mathematics Learning, 9(1), 19-37. https://doi.org/10.1080/19477503.2016.1245035
  14. Cramer, K., Monson, D., Ahrendt, S., Wyberg, T., Pettis, C., & Fagerlund, C. (2019). Reconstructing the unit on the number line: Tasks to extend fourth graders' fraction understandings. Investigations in Mathematics Learning, 11(3), 180-194. https://doi.org/10.1080/19477503.2018.1434594
  15. Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490-496. https://doi.org/10.5951/mtms.13.8.0490
  16. Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3-26). Columbus, OH: Clearinghouse for Science, Mathematics, and Environmental Education.
  17. Freeman, D. W., & Jorgensen, T. A. (2015). Moving beyond brownies and pizza. Teaching Children Mathematics, 21(7), 412-420. https://doi.org/10.5951/teacchilmath.21.7.0412
  18. Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25(5), 443-471. https://doi.org/10.2307/749485
  19. Gray, E., & Doritou, M. (2008). The number line: Ambiguity and interpretation. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the joint Meeting of PME 32 and PME-NA XXX (Vol. 3, pp. 97-104). Mexico: Cinvestav-UMSNH.
  20. Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587. https://doi.org/10.1037/dev0000252
  21. Hannula, M. S. (2003). Locating fraction on a number line. In N. A. Pateman, B. J. Dougherty & J. Zilliox (Eds.), Proceedings of the 2003 joint meeting of the PME and PMENA (pp. 17-24). Hawaii: University of Hawaii.
  22. Herbst, P. (1997). The number-line metaphor in the discourse of a textbook series. For the Learning of Mathematics, 17(3), 36-45.
  23. Izsak, A., Tillema, E., & Tunc-Pekkan, Z. (2008). Teaching and learning fraction addition on number lines. Journal for Research in Mathematics Education, 39(1), 33-62.
  24. Kara, F., & Incikabi, L. (2018). Sixth grade students' skills of using multiple representations in addition and subtraction operations in fractions. International Electronic Journal of Elementary Education, 10(4), 463-474.
  25. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.
  26. Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates.
  27. Lamon, S. L. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco & F. Curcio (Eds.), The roles of representations in school mathematics (pp. 146-165). Reston, VA: National Council of Teachers of Mathematics.
  28. Lee, M. Y., Choy, B. H., & Mizzi, A. (2021). Exploring the introduction of fractions in Germany, Singapore, and South Korea mathematics textbooks. Korean Society of Mathematical Education, 24(2), 111-130.
  29. Lemmo, A., Branchetti, L., Ferretti, F., Maffia, A., & Martignone, F. (2015). Students' difficulties dealing with number line: A qualitative analysis of a question from national standardized assessment. Quaderni di Ricerca in Didattica (Mathematics), 25(2), 149-156.
  30. Saxe, G. B. (2004). Practices of quantification from a sociocultural perspective. In A. Demetriou & A. Raftopoulos (Eds.), Cognitive developmental change: Theories, models and measurement (pp. 241-263). New York, NY: Cambridge University Press.
  31. Saxe, G. B., Diakow, R., & Gearhart, M. (2013). Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principal representational context. ZDM Mathematics Education, 45, 343-364. https://doi.org/10.1007/s11858-012-0466-2
  32. Saxe, G. B., Shaughnessy, M. M., Shannon, A., Langer-Osuna, J. M., Chinn, R., & Gearhart, M. (2007). Learning about fractions as points on a number line. In W. G. Martin, M. E. Strutchens, & P. C. Elliott (Eds.), The learning of mathematics: 69th yearbook (pp. 221-238). Reston, VA: National Council of Teachers of Mathematics.
  33. Skoumpourdi, C. (2010). The number line: An auxiliary means or an obstacle? International Journal for Mathematics Teaching and Learning.
  34. Teppo, A., & van den Heuvel-Panhuizen, M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM Mathematics Education, 46(1), 45-58. https://doi.org/10.1007/s11858-013-0518-2
  35. Van den Heuvel-Panhuizen, M. (2008). Learning from "Didactikids": An impetus for revisiting the empty number line. Mathematics Education Research Journal, 20(3), 6-31. https://doi.org/10.1007/BF03217528