한국수학교육학회지시리즈A:수학교육 (The Mathematical Education)

  • 제63권3호
  • /
  • Pages.393-409
  • /
  • 2024
  • /
  • 1225-1380(pISSN)
  • /
  • 2287-9633(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

초등학교 6학년의 분수와 소수의 크기에 대한 수직선 표상의 정확성 및 사용 전략 분석

Analyses of the precision and strategies for representing the magnitude of fractions and decimals on the number line among 6th graders

  • 허진영 (한양대학교) ;
  • 임수현 (한양대학교)
  • Jinyoung Heo (Hanyang University) ;
  • Soo-hyun Im (Hanyang University)
  • 투고 : 2024.06.26
  • 심사 : 2024.08.05
  • 발행 : 2024.08.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

분수와 소수가 나타내는 크기에 대한 이해를 돕기 위해, 수의 크기를 공간상에 직관적으로 표시하는 수직선 모델이 널리 활용된다. 본 연구에서는 수직선 추정 과제를 활용하여 초등학교 6학년 학생들의 분수와 소수에 대한 이해도 및 문제 해결 전략을 살펴보고, 다양한 전략 사용의 유연성이 분수와 소수의 표상의 정확성, 연산 능력, 수학 학업성취도의 개인차와 연관성이 있는지를 분석하였다. 분석 결과, 학생들은 자연수에 비해 분수와 소수의 수직선 표상 정확성이 상대적으로 낮았으며, 특별히 분수의 경우 분모가 짝수인 분수보다 홀수인 분수에서, 소수의 경우 소수 세 자리 수 보다 소수 두 자리 수에서 수직선 표상의 정확성이 더 낮게 나타났다. 전략 사용의 측면에서 학생들은 분수를 수직선에 표상할 때 기준점 참고 전략, 분할 전략, 어림 전략 순으로 많이 사용하였으며, 소수를 표상할 때에는 기준점 참고 전략, 반올림 전략, 자연수 변환 전략 순으로 많이 사용하였다. 마지막으로 분수를 표상할 때 사용하는 전략이 다양할수록 분수의 표상 정확성과 수학 학업성취 점수가 높게 나타났다. 이러한 결과를 토대로 분수의 다양한 개념과 자연수 대비 소수의 자릿값 및 0의 개념에 대한 세심한 지도의 필요성을 제언하였다. 또한 분수와 소수에 대한 이해를 돕기 위한 방안으로, 수직선 모델 활용과 함께 표상 전략을 병행해서 지도하는 방법에 대해 논의하였다.

The number line model, which intuitively marks numerical magnitudes in space, is widely utilized to help in understanding the magnitudes that fractions and decimals represent. The study analyzed 6th graders' understanding of fractions and decimals, their problem solving strategies, and whether individual differences in the flexibility of various strategy uses are associated with the accuracy of numerical representation, calculation fluency, and overall mathematical achievement. As a result of the study, students showed relatively lower accuracy in representing fractions and decimals on a number line compared to natural numbers, especially for fractions with odd denominators compared to even denominators, and for two-digit decimals compared to three-digit decimals. Regarding strategy use, students primarily used benchmark, segmentation, and approximation strategies for fractions, and benchmark, rounding, and transformation strategies for decimals sequentially. Lastly, as students used various representation strategies for fractions, their accuracy in representing fractions and their overall mathematical achievement scores showed significantly better outcomes. Taken together, we suggest the need for careful instruction on different interpretations of fractions, the place value of decimals, and the meaning of zero in decimal places. Moreover, we discuss instructional methods that integrate the number line model and its diverse representation strategies to enhance students' understanding of fractions and decimals.

키워드

과제정보

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF. 2021S1A5A8061509).

참고문헌

  1. Barbieri, C. A., Rodrigues, J., Dyson, N., & Jordan, N. C. (2020). Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies. Journal of Educational Psychology, 112(3), 628-648. https://doi.org/10.1037/edu0000384
  2. Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91-125). Academic Press.
  3. Braithwaite, D. W., & Siegler, R. S. (2018). Developmental changes in the whole number bias. Developmental Science, 21(2), e12541. https://doi.org/10.1111/desc.12541
  4. Chang, H. (2017). Research on teaching method for the properties of arithmetic based on analysis of elementary school mathematics textbooks. Journal of Elementary Mathematics Education in Korea, 21(1), 1-22.
  5. Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21-29. https://doi.org/10.1016/j.learninstruc.2014.08.003
  6. Fitzsimmons, C. J., Thompson, C. A., & Sidney, P. G. (2020). Do adults treat equivalent fractions equally? Adults' strategies and errors during fraction reasoning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 46(11), 2049-2074. https://doi.org/10.1037/xlm0000839
  7. Foley, T. E., & Cawley, J. F. (2014). About the mathematics of division: Implications for students with disabilities. In J. F. Cawley, & L. J. Cawley. (Eds.), Mathematics instruction for students with disabilities (pp. 131-149). Routledge.
  8. Girit, D., & Akyuz, D. (2016). Pre-service middle school mathematics teachers' understanding of students' knowledge: Location of decimal numbers on a number line. International Journal of Education in Mathematics, Science and Technology, 4(2), 84-100. https://doi.org/10.18404/ijemst.74290
  9. Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In H. Stevenson, H. Azuma, & K. Hakuta (Eds.), Child development and education in Japan (pp. 262-272). Freeman.
  10. Heo, J., & Im, S.-h. (2024). An analysis of learning experience opportunities presented in the 2015 revised elementary mathematics textbooks: Focused on visual models of fractions and decimals and their connection. Journal of Elementary Mathematics Education in Korea, 28(2), 225-246. https://doi.org/10.54340/kseme.2024.28.2.3
  11. Im, S.-h., & Varma, S. (2024). The application of arithmetic principles predicts mathematical achievement in college students. Learning and Instruction, 91, 101889. https://doi.org/10.1016/j.learninstruc.2024.101889
  12. Kang, H. K. (2011). A Study on the learning-teaching plan about a essential concept of decimal fraction based on decimal positional notation. Journal of Elementary Mathematics Education in Korea, 15(1), 199-219.
  13. Kim, J. W. (2022a). An analysis of elementary students' understanding of number line: Focused on concept of fractions and addition and subtraction of fractions. Education of Primary School Mathematics, 25(3), 213-232. https://doi.org/10.7468/jksmec.2022.25.3.213
  14. Kim, J. W. (2022b). An analysis of understanding of prospective elementary teachers on students' strategies for fraction tasks with number lines. The Mathematical Education, 61(3), 375-396, https://doi.org/10.7468/mathedu.2022.61.3.375
  15. Kim, Y. S., & Nam, J. (2023). The understanding of decimals by 5th-grade elementary school students. The Journal of Education, 43(2), 73-88. http://doi.org/10.25020/je.2023.43.2.73
  16. Kwon, S. Y. (2003). A study on elementary school students' understanding of fractions. School Mathematics, 5(2), 259-273.
  17. Ministry of Education. (2015). Mathematics curriculum (Ministry of Education Notice No. 2015-74 [Supplement 8]). https://www.moe.go.kr/boardCnts/viewRenew.do?boardID=141&lev=0&statusYN=C&s=moe&m=0404&opType=N&boardSeq=60747
  18. Newman, R. S., & Berger, C. F. (1984). Children's numerical estimation: Flexibility in the use of counting. Journal of Educational Psychology, 76(1), 55-64. https://doi.org/10.1037/0022-0663.76.1.55
  19. Ni, Y., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52. https://doi.org/10.1207/s15326985ep4001_3
  20. Park, H., Li, H. C., & Kim, S. W. (2016). Verifying use of reference points in the number line estimation task. Korean Journal of Cognitive and Biological Psychology, 28(4), 653-673. http://doi.org/10.22172/cogbio.2016.28.4.003
  21. Peeters, D., Degrande, T., Ebersbach, M., Verschaffel, L., & Luwel, K. (2016). Children's use of number line estimation strategies. European Journal of Psychology of Education, 31, 117-134. https://doi.org/10.1007/s10212-015-0251-z
  22. Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14(5). 307-317. https://doi.org/10.2307/748675
  23. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. https://doi.org/10.1037/0022-0663.93.2.346
  24. Rittle-Johnson, B., Star, J. R., Durkin, K., & Loehr, A. (2019). Compare and discuss to promote deeper learning. In E. Malano (Ed.), Deeper learning, dialogic learning, and critical thinking (pp. 48-64). Routledge.
  25. Schiller, L. K., Abreu-Mendoza, R. A., & Rosenberg-Lee, M. (2024). Adults systematically underestimate decimals and whole number exposure induces further magnitude-based underestimation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 50(3), 484-499. https://doi.org/10.1037/xlm0001235
  26. Schneider, M., Heine, A., Thaler, V., Torbeyns, J., De Smedt, B., Verschaffel, L., Jacobs, A. M., & Stern, E. (2008). A validation of eye movements as a measure of elementary school children's developing number sense. Cognitive Development, 23(3), 409-422. https://doi.org/10.1016/j.cogdev.2008.07.002
  27. Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., & Luwel, K. (2018). Associations of number line estimation with mathematical competence: A meta-analysis. Child Development, 89(5), 1467-1484. https://doi.org/10.1111/cdev.13068
  28. Schneider, M., & Siegler, R. S. (2010). Representations of the magnitudes of fractions. Journal of Experimental Psychology: Human Perception and Performance, 36(5), 1227-1238. https://doi.org/10.1037/a0018170
  29. Sidney, P. G., Thalluri, R., Buerke, M. L., & Thompson, C. A. (2019). Who uses more strategies? Linking mathematics anxiety to adults' strategy variability and performance on fraction magnitude tasks. Thinking & Reasoning, 25(1), 94-131. https://doi.org/10.1080/13546783.2018.1475303
  30. Siegler, R. S., & Braithwaite, D. W. (2017). Numerical development. Annual Review of Psychology, 68, 187-213. https://doi.org/10.1146/annurev-psych-010416-044101
  31. Siegler, R. S., & Lemaire, P. (1997). Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126(1), 71-92. https://doi.org/10.1037/0096-3445.126.1.71
  32. Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26(4), 346-351. https://doi.org/10.1177/ 0963721417700129
  33. Siegler, R. S., & Thompson, C. A. (2014). Numerical landmarks are useful-except when they're not. Journal of Experimental Child Psychology, 120(1), 39-58. https://doi.org/10.1016/j.jecp.2013.11.014
  34. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296. https://doi.org/10.1016/j.cogpsych.2011.03.001
  35. Sowinski C., Dunbar K., & LeFevre J. (2014). Calculation fluency test [Unpublished technical report]. Math Lab, Carleton University. https://carleton.ca/cacr/?p=316
  36. Stafylidou, S., & Vosniadou, S. (2004). The development of students' understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503-518. https://doi.org/10.1016/j.learninstruc.2004.06.015
  37. Varma, S., & Karl, S. R. (2013). Understanding decimal proportions: Discrete representations, parallel access, and privileged processing of zero. Cognitive Psychology, 66(3), 283-301. https://doi.org/10.1016/j.cogpsych.2013.01.002
  38. Verschaffel, L. (2024). Strategy flexibility in mathematics. ZDM Mathematics Education, 56, 115-126. https://doi.org/10.1007/s11858-023-01491-6
  39. Yoon, C., & Chang, H. (2023). An analysis on reasoning of 4th-grade elementary school students in comparing unlike fraction magnitudes. Journal of the Korean Society of Mathematical Education Series C: Education of Primary School Mathematics, 26(3), 181-197. https://doi.org/10.7468/jksmec.2023.26.3.181
  40. Yuan, Y., & Chen, K. (2023). Whole number bias of students in fraction number line tasks. International Journal of Science and Mathematics Education, 21(5), 1433-1449. https://doi.org/10.1007/s10763-022-10315-0