How Do Elementary School Students Understand '='? - Performance on Various Item Types -

초등학생들은 '='를 어떻게 이해하는가? - 문항유형별 실태조사 -

  • Received : 2016.01.10
  • Accepted : 2016.02.04
  • Published : 2016.02.28

Abstract

Understanding the equal sign is of great significance to the development of algebraic thinking. Given this importance, this study investigated in what ways a total of 695 students from second to sixth graders understand the equal sign. The results showed that students were successful in solving standard problems whereas they were poor at items demanding high relational thinking. It was noticeable that some of the students were based on computational thinking rather than relational understanding of the equal sign. The students had a difficulty both in understanding the structure of equations and in solving equations in non-standard problem contexts. They also had incomplete understanding of the equal sign. This paper is expected to explore the understanding of the equal sign by elementary school students in multiple problem contexts and to provide implications of how to promote students' understanding of the equal sign.

등호에 대한 이해는 대수적 사고 발달에 핵심이 되는 바, 본 연구에서는 우리나라 초등학교 2~6학년 학생 695명의 등호 이해가 어느 정도인지 살펴보았다. 연구 결과 전반적으로 정답 반응이 오답 반응에 비하여 높게 드러났으나, 정답 반응 가운데 등호의 관계적 관점이 아닌 계산에 치중하는 등호의 연산적 관점 또한 적지 않게 발견할 수 있었다. 또한 표준 문맥 이외의 등식 문맥에서 등식 구조를 판단하거나 등식을 해결하는데 어려움을 겪고 있으며, 등호 개념에 관한 불안전한 이해를 가지고 있다는 것도 확인할 수 있었다. 본 연구를 통하여 우리나라 초등학교 학생들의 등호 이해의 실태를 파악하고 앞으로의 지도 방향에 대한 시사점을 모색할 수 있을 것이라 기대한다.

Keywords

References

  1. 강명희(2010). 양변 연산식에서 문제풀이전략 유형과 학생들의 등호개념 발달 연구: 정답반응은 등호의 관계적 개념을 뜻하는가?. 학습자중심교과교육연구, 10(2), 15-33.
  2. 강명희(2011). 비표준 구조 연산식 정답반응 분석을 통한 초등학생들의 등호개념 이해 연구. 열린교육실행연구, 14, 17-30.
  3. 교육부(2015a). 수학 1-1. 서울: (주)천재교육.
  4. 교육부(2015b). 수학 6-2. 서울: (주)천재교육.
  5. 교육부(2015c). 수학 1-1 교사용 지도서. 서울: (주)천재교육.
  6. 기정순.정영옥(2008). 등호 문맥에 따른 초등학생의 등호 개념 이해와 지도 방법 연구. 학교수학, 10(4), 537-555.
  7. 이종희.김선희(2003). 등호 개념의 분석 및 학생들의 등호 이해 조사. 수학교육학연구, 13(3), 287-307.
  8. Bell, M., et al. (2012a). Everyday mathematics 1st grade student's math journal volume 1(CCSS edition). Chicago, IL: McGraw-Hill.
  9. Bell, M., et al. (2012b). Everyday mathematics 1st grade teacher's lesson guide volume 1 (CCSS edition). Chicago, IL: McGraw-Hill.
  10. Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511-558. https://doi.org/10.5951/jresematheduc.46.5.0511
  11. Blanton, M., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
  12. Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G. (2015). A specific misconception of the equal sign acts as a barrier to children's learning of early algebra. Learning and Individual Differences, 38, 61-67. https://doi.org/10.1016/j.lindif.2015.01.001
  13. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Port smouth, NH: Heinemann.
  14. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
  15. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232.
  16. Freiman, V., & Lee, L. (2004). Tracking primary students' understanding of the equality sign. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (pp. 415-422). Bergen, Norway: Bergen University College.
  17. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326. https://doi.org/10.1007/BF00311062
  18. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester(Ed.), Second handbook of research on mathematics teaching and learning (pp. 707-762). Charlotte, NC: Information Age.
  19. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for research in Mathematics Education, 37(4), 297-312.
  20. Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children's understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316-350. https://doi.org/10.5951/jresematheduc.43.3.0316
  21. McNeil, N. M., & Alibali, M. W. (2005). Why don't you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883-899. https://doi.org/10.1111/j.1467-8624.2005.00884.x
  22. McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E. (2015). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology, 107(2), 423. https://doi.org/10.1037/a0037687
  23. Molina, M. & Ambrose, R. (2008). From an operational to a relational conception of the equal sign: Third graders developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61-80.
  24. Stephens, A. C., Knuth, E. J., Blanton, M. L., Isler, I., Gardiner, A. M., & Marum, T. (2013). Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students' understandings. The Journal of Mathematical Behavior, 32(2), 173-182. https://doi.org/10.1016/j.jmathb.2013.02.001
  25. Schliemann, A., Carraher, D., Brizuela, B., & Jones, W. (1998). Solving algebra problems before algebra instruction. Arlington, VA: National Science Foundation.