Education of Primary School Mathematics (한국수학교육학회지시리즈C:초등수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

Aspects of Understandings on Statistical Variability across Varying Degrees of Task Structuring

과제의 구조화 정도에 따른 초등학생들의 통계적 변이성 이해 양상에 대한 사례 연구

  • Han, Chaereen (Seoul Singok Elementary School) ;
  • Lee, Kyungwon (Graduate School of Department of Mathematics Education, Seoul National University) ;
  • Kim, Doyen (Graduate School of Department of Mathematics Education, Seoul National University) ;
  • Bae, Mi Seon (Seoul Global Highschool) ;
  • Kwon, Oh Nam (Department of Mathematics Education, Seoul National University)
  • Received : 2018.03.30
  • Accepted : 2018.04.23
  • Published : 2018.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

The structure of a mathematics task shapes the aspects of learning of those who solve the task. This study explores the process of understandings on the statistical variability of primary school students. Students were given two problems with different degrees of structuring - a well-structured problem (WSP) and an ill-structured problem (ISP) - and discussed in a group to solve each task. The highest level of development achieved in both cases appeared to be similar. However, when given the ISP, students dynamically proposed ideas and justified the conclusion based on their hypothesis. Furthermore, all students actively participated in solving the ISP until the end whereas some students were marginalized while solving the WSP. This discrepancy results from the difference in the degrees of task structuring.

수학 과제의 구조는 이를 해결하는 학생들의 배움의 양상에 영향을 미친다. 이 연구에서는 구조화된 정도가 다른 두 가지 문제를 소집단 토론 활동으로 해결하는 초등학생들의 통계적 변이성 이해 양상을 탐색하였다. 비구조화된 문제와 구조화된 문제에서 학생들의 통계적 변이 추론 발달 정도는 비슷하였지만 비구조화된 문제에서 학생들은 보다 다양한 아이디어를 전 과정에 걸쳐 역동적으로 제시하였으며, 구조화된 문제에서는 나타나지 않았던 가설에 기반한 추론의 양상을 보였다. 또한 비구조화된 문제에서 모든 학생이 끝까지 활발하게 참여하는 모습을 보였으며, 구조화된 문제에서는 일부 학생이 소외되는 현상이 나타났다. 이러한 차이는 과제의 구조화된 정도에서 비롯되었음을 확인하였다.

Keywords

References

  1. 고은성. 이경화 (2011). 일반학급 학생들과의 비교를 통한 수학영재학급 학생들의 표본 개념 이해 수준 연구. 영재교육연구, 21(2), 287-307. (Ko, E-S & Lee, K. H. (2011). Study on levels of mathematically gifted students' understanding of statistical samples through comparison with non-gifted students. Journal of Gifted/Talented Education, 21(2), 287-307.) https://doi.org/10.9722/JGTE.2011.21.2.287
  2. 고은성 (2012). 통계적 변이성 사고 요소 간의 관계 연구. 학교수학, 14(4), 495-516. (Ko, E-S. (2012). The relationship among components of thinking related to statistical variability. School Mathematics, 14(4), 495-516.)
  3. 고은성 (2013). 수학영재학급 학생들과 일반학급 학생들의 통계적 변이성 인식 수준 비교 연구. 영재교육연구, 23(3), 287-406. (Ko, E-S. (2013). A comparison of mathematically talented students and non-talented students' level of statistical thinking: The noticing of statistical variability. Journal of Gifted/Talented Education, 23(3), 287-406.)
  4. 김동희. 김민경 (2016). 초등학생의 창의.융합적 사고 및 문제해결력에 관한 연구. 학교수학, 18(3), 541-569. (Kim, D. & Kim, M. K. (2016). A study on creativity integrated thinking and problem solving of elementary school students in ill-structured mathematics problems. School Mathematics, 18(3), 541-569.)
  5. 김민경. 이지영. 홍지연. 김은경 (2011). 초등학교 수학 교과서에서 나타난 '문제'의 비구조성 (ill-structured)에 관한 연구. 학습자중심교과교육연구, 11(2), 1-21. (Kim, M. K., Lee, J-Y, Hong, J. Y., & Kim, E. K. (2011). A study of 'ill-structured' status from mathematics problems in elementary school textbooks. Journal of Learner-Centered Curriculum and Instruction, 11(2), 1-21.)
  6. 김민경. 조미경. 박윤미. 허지연 (2012). 초등학교 4학년 학생들의 비구조화된 문제에서 나타난 해결 과정 및 추론 분석. 한국수학교육학회지 시리즈 A <수학교육>, 51(2), 95-114. (Kim, M. K., Heo, J. Y., Cho, M. K., & Park, Y. M. (2012). An analysis on the 4th graders' ill-structured problem solving and reasoning. J. Korea. Soc. Math. Ed. Ser. A: The Mathematical Education, 51(2), 95-114.)
  7. 김민경. 이지영. 홍지연. 주현정 (2012). 자료분석에 관한 비구조화된 문제해결모형 적용에서 나타난 초등학교 5학년 학생들의 의사결정에 관한 연구. 한국수학교육학회지 시리즈 E <수학교육논문집>, 26(2), 221-247. (Kim, M. K., Lee, J. Y., Hong, J. Y., & Joo, H. J. (2012). Decision making from the 5th graders' ill-structured problem of data analysis. J. Korea. Soc. Math. Ed. Ser. E: Communications of Mathematical Education, 26(2), 221-247.)
  8. 김민경. 박은정 (2013). 비구조화된 정도에 따른 비례 문제 유형에서 나타난 초등학생의 비례추론에 관한 연구. 한국학교수학회논문집, 16(4), 719-743. (Kim, M. K. & Park, E. J. (2013). Children's proportional reasoning on problem type of proportion according to ill-structured degree. Journal of the Korean School Mathematics Society, 16(4), 719-743.)
  9. 김민경. 허지연. 박은정 (2014). 초등수학에서의 비구조화된 문제해결 모형 설계, 적용 및 그 교육적 의미. 한국초등수학교육학회지, 18(2), 189-209. (Kim, M. K., Heo, J. Y., & Park E, J. (2014). Design, application and its educational implication of ill-structured problem solving in elementary mathematics education. Journal of Elementary Mathematics Education in Korea, 18(2), 189-209.)
  10. 김영미. 박영희 (2006). 초등학교 5학년 학생의 통계적 변이성 개념의 이해와 그 지도에 관한 연구. 수학교육학연구, 16(3), 221-249. (Kim, Y. M. & Park, Y. H. (2006). Understanding of statistical variation concept of elementary school 5th graders and study on its lesson plans. Journal of Educational Research in Mathematics, 16(3), 221-249.)
  11. 나미영. 조형미. 권오남 (2017). 미래학교 수학교실 의 교육방법론에 대한 탐색: 비구조화된 문제에서 학생들의 질문 만들기를 중심으로. 한국수학교육학회 시리즈 A <수학교육>, 56(3), 301-318. (Na, M., Cho, H., & Kwon, O. N. (2017). Teaching methodology for future mathematics classroom: Focusing on students' generative question in ill-structured problem. J. Korean Soc. Math. Ed. Ser. A: The Mathematical Education, 56(3), 301-318.)
  12. 박유나. 박만구 (2015). 문제해결에서 생산적 실패의 경험이 초등학생의 수학적 문제해결력 및 수학적 성향에 미치는 영향. 한국수학교육학회 시리즈 C <초등수학교육>, 18(2), 123-139.(Park, Y., & Park, M. (2015). The influences of experiences of productive failures on mathematical problem solving abilities and mathematical dispositions. J. Korea Soc. Mathe Ed. Ser. C: Education of Primary School Mathematics, 18(2), 123-139.)
  13. 박태학 (2003). 학교통계교육의 문제점과 개선방향. 교육학연구, 41(2), 401-430. (Park, T. H. (2003). The problems and reform directions of statistics education at school levels. Korean Journal of Educational Research, 41(2), 401-430.)
  14. 송선아. 이경화 (2007). 중학교 3학년 학생들의 변이성 이해에 대한 사례 연구. 학교수학, 9(1), 29-44. (Song, S. A., & Lee, K. H. (2007). A case study aimed at junior high school 3rd grade students understanding of variability. School Mathematics, 9(1), 29-44.)
  15. Ben-Zvi, D. (2004). Reasoning about variability in comparing distributions. Statistics Education Research Journal, 3(2), 42-63.
  16. Brousseau, G. (1997). Theory of didactical Situations in Mathematics. Dordrecht: Kluwer Academic Publishers.
  17. Cobb, G. W. & Moore, D. S. (1997). Mathematics, statistics, and teaching. The American Mathematical Monthly, 104(9), 801-823. https://doi.org/10.1080/00029890.1997.11990723
  18. Chi, M. T. H. & Glaser, R. (1985). Problem solving ability. In R. J. Sternberg (Ed.), Human abilities: An information processing approach (pp. 227-250). New York: W. H. Freeman and Company.
  19. Ge, X. & Land, S. M. (2003). Scaffolding students’ problem-solving processes in an ill-structured task using question prompts and peer interactions. Educational Technology Research and Development, 51(1), 21-38. https://doi.org/10.1007/BF02504515
  20. Hiebert, J. (1992). Reflection and communication: Cognitive considerations in school mathematics reform. International Journal of Educational Research, 17(5), 439-456. https://doi.org/10.1016/S0883-0355(05)80004-7
  21. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth: Heinemann.
  22. Hogan, K., Nastasi, B. K., & Pressley, M. (1999). Discourse patterns and collaborative scientific reasoning in peer and teacher-guided discussions. Cognition and Instruction, 17(4), 379-432. https://doi.org/10.1207/S1532690XCI1704_2
  23. Jonassen, D. H. (1997). Instructional design models for well-structured and ill-structured problem-solving learning outcomes. Educational Technology Research and Development, 45(1), 1042-1629.
  24. Kapur, M. (2006). Productive failure: A hidden efficacy of seemingly unproductive production. In R. Sun (Ed.), Proceedings of the Annual Meeting of the Cognitive Science Society, 28(28), (pp. 1587-1592). Mahwah: Erlbaum.
  25. Kapur, M. (2008). Productive failure. Cognition and Instruction, 26(3), 379-425. https://doi.org/10.1080/07370000802212669
  26. Kapur, M. & Kinzer, C. K. (2009). Productive failure in CSCL groups. International Journal of Computer-Supported Collaborative Learning, 4(1), 21-46. https://doi.org/10.1007/s11412-008-9059-z
  27. Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523-550. https://doi.org/10.1007/s11251-009-9093-x
  28. Kapur, M. (2012). Productive failure in learning the concept of variance. Instructional Science, 40(4), 651-672. https://doi.org/10.1007/s11251-012-9209-6
  29. Kapur, M. & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45-83. https://doi.org/10.1080/10508406.2011.591717
  30. Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38, 1008-1022. https://doi.org/10.1111/cogs.12107
  31. Ko, E-S. & Lee, K. H. (2010). Are mathematically talented elementary students also talented in statistics?. In B. Sriraman & K. H. Lee (Eds.), The Elements of Creativity and Giftedness in Mathematics. Rotterdam: Sense Publishers.
  32. NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics.
  33. Mason, J., & Johnston-Wilder, S (2006). Designing and Using Mathematical Tasks. St. Albans: Tarquin.
  34. Moore, D. S. (1990). Uncertainty. In Steen, L. A (Ed.) On the shoulders of giants: A new approaches to numeracy (pp. 95-137). Washington, D. C.: National Academy Press.
  35. Moore, D. S. (1992). Teaching statistics as a respectable subject. In F. S. Gordon & S. P. Gordon (Eds.) Statistics For the Twenty-First Century (pp. 14-25). Washington, D. C.: Mathematical Association of America.
  36. Moore, D. S. (1997). New pedagogy and new content: The case of statistics. International Statistical Review 65(2), 123-137. https://doi.org/10.1111/j.1751-5823.1997.tb00390.x
  37. Pfannkuch, M. & Wild, C. (2004). Towards an understanding of statistical thinking. In D. Ben-Zvi & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning and Thinking (pp. 17-46). New York: Kluwer Academic Publishers.
  38. Shaughnessy, J. M. (1997). Missed opportunities on the teaching and learning of data and chance. In J. Garfield & J. Truran (Eds.), Research Papers on Stochastics Education (pp. 129-145). Minneapolis: The University of Minnesota.
  39. Simon, H. A. (1973). The structure of ill-structured problems. Artificial Intelligence, 4, 181-201. https://doi.org/10.1016/0004-3702(73)90011-8
  40. Snee, R. D. (1990). Statistical thinking and its contribution to total quality. The American Statistician, 44(2), 116-121. https://doi.org/10.2307/2684144
  41. Stake, R. E. (1995). The Art of Case Study Research. Thousand Oaks: Sage Publications.
  42. Truxaw, M. P. & DeFranco, T. (2008). Mapping mathematics classroom discourse and its implications for models of teaching. Journal for Research in Mathematics Education 39(5), 489-525.
  43. Reading, C. & Shaughnessy, J. M. (2004). Reading About Variation. In: Ben-Zvi, D & Garfield, J (Eds) The Challenge of Developing Statistical Literacy, Reasoning and Thinking (pp. 201-226). Dordrecht: Springer.
  44. Rice, J. A. (2007). Mathematical Statistics and Data Analysis (3rd ed.). Belmont: Thompson Brooks/Cole.
  45. Watson, J. M., Kelly, B. A., Callingham, R. A., & Shaughnessy, J. M. (2003). The measurement of school students’ understanding of statistical variation. International Journal of Mathematical Education in Science and Technology, 34(1), 1-29. https://doi.org/10.1080/0020739021000018791
  46. Watson, J. M. (2006). Issues for statistical literacy in the middle school. In A. Rossman, & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. Salvador: International Association for Statistical Education and International Statistical Institute.
  47. Wild, C. J. & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-248. https://doi.org/10.1111/j.1751-5823.1999.tb00442.x