Journal of Gifted/Talented Education (영재교육연구)

The Korean Society for the Gifted (한국영재학회)
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Study on Levels of Mathematically Gifted Students' Understanding of Statistical Samples through Comparison with Non-Gifted Students

일반학급 학생들과의 비교를 통한 수학영재학급 학생들의 표본 개념 이해 수준 연구

  • Received : 2011.05.06
  • Accepted : 2011.06.07
  • Published : 2011.06.30
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Abstract

The purpose of this study is to investigate levels of mathematically gifted students' understanding of statistical samples through comparison with non-gifted students. For this purpose, rubric for understanding of samples was developed based on the students' responses to tasks: no recognition of a part of population (level 0), consideration of samples as subsets of population (level 1), consideration of samples as a quasi-proportional, small-scale version of population (level 2), recognition of the importance of unbiased samples (level 3), and recognition of the effect of random sampling (level 4). Based on the rubric, levels of each student's understanding of samples were identified. t tests were conducted to test for statistically significant differences between mathematically gifted students and non-gifted students. For both of elementary and middle school graders, the t tests show that there is a statistically significant difference between mathematically gifted students and non-gifted students. Table of frequencies of each level, however, shows that levels of mathematically gifted students' understanding of samples were not distributed at the high levels but were overlapped with levels of non-gifted students' understanding of samples.

본 연구에서는 일반학급 학생들과의 비교를 통해 수학영재학급 학생들의 표본 개념 이해 수준을 살펴본다. 먼저 조사 과제에 대한 학생들의 반응을 토대로 표본 개념 이해 수준을 평가하기 위한 기준을 개발하였다. 학생들의 반응을 분석한 결과 표본이 모집단의 일부분이라는 것에 대한 인식이 부족한 0수준, 표본을 모집단의 부분집합으로 인식하는 1수준, 표본을 모집단의 준비례적 축소버전으로 인식하는 2수준, 편의없는 표본의 중요성을 인식하는 3수준, 무작위 추출이 표본에 미치는 영향을 이해하는 4수준으로 구분할 수 있었다. 개발된 평가 기준을 근거로 각 학생의 이해 수준을 조사한 후, 수학영재학급 학생들과 일반학급 학생들의 표본에 대한 이해 수준의 차이를 알아보기 위해 두 독립표본 t 검정을 실시하였다. 검정결과 초등학교와 중학교 모두에서 수학영재학급 학생들과 일반학급 학생들 두 그룹 간에 통계적으로 유의한 차이가 있는 것으로 나타났다. 그러나 수준별 빈도를 조사한 결과 수학영재학급 학생들의 이해 수준이 상위 수준에 분포되기보다는 일반학급 학생들의 이해 수준과 상당부분 중첩됨을 확인할 수 있었다.

Keywords

References

  1. 고은성, 이경화 (2007). GSP 환경에서의 중학교 수학영재 학생들의 문제 해결 과정 분석 시각적 추론과 논리적 추론을 중심으로. 영재교육연구, 17(3), 521-539.
  2. 고은성, 이경화 (2011). 예비교사들의 통계적 표집에 대한 이해. 수학교육학연구, 21(1), 17-32.
  3. 고은성, 이경화, 송상헌 (2008a). 시각적 사고와 분석적 사고 사이에서 이미지의 역할. 학교수학, 18(1), 63-78.
  4. 고은성, 이경화, 송상헌 (2008b). 수학영재 학생들의 정다면체 정의 구성 활동 분석. 영재교육연구, 18(1), 53-77.
  5. 김민정, 이경화, 송상헌 (2008). 초등 수학영재의 대수적 사고 특성에 관한 분석. 학교수학, 10(1), 23-42.
  6. 김지원, 송상헌 (2004). 한 수학영재아의 수학적 사고 특성에 관한 사례 연구. 수학교육학연구, 14(1), 89-110.
  7. 나귀수, 이경화, 한대희, 송상헌 (2007). 수학 영재 학생들의 조건부 확률 문제해결 방법. 학교수학, 9(3), 397-408.
  8. 류현아, 정영옥, 송상헌 (2007). 입체도형에 대한 6-7학년 수학영재들의 공간시각화 능력 분석. 학교수학, 9(2), 277-289.
  9. 성태제 (2002). 타당도와 신뢰도. 학지사: 서울
  10. 송상헌, 신은주 (2007). 수학 영재의 추상화 학습에서 기호의 의미 작용 과정 사례 분석. 학교수학, 9(1), 161-180.
  11. 송상헌, 임재훈, 정영옥, 권석일, 김지원 (2007). 초등수학영재들이 페그퍼즐 과제에서 보여주는 대수적 일반화 과정 분석. 수학교육학연구, 17(2), 163-178.
  12. 송상헌, 장혜원, 정영옥 (2006a). 초등학교 6학년 수학영재들의 기하 과제 증명 능력에 관한 사례 분석. 수학교육학연구, 16(4), 327-344.
  13. 송상헌, 허지연, 임재훈 (2006b). 도형의 최대 분할 과제에서 초등학교 수학 영재들이 보여주는 정당화의 유형 분석. 수학교육학연구, 16(1), 79-94.
  14. 이경화, 최남광, 송상헌 (2007). 수학영재들의 아르키메데스 다면체 탐구 과정 - 정당화 과정과 표현 과정을 중심으로. 학교수학, 9(4), 487-505.
  15. 이경화, 유연주, 홍진곤, 박민선, 박미미 (2010). 수학 우수아의 통계적 개념 이해도 조사. 학교수학, 12(4), 547-561.
  16. 이외숙, 임용빈, 성내경, 소병수 (2000). 통계학 입문(제2판). 서울: 경문사.
  17. Australian Education Council. (1991). A national statement on mathematics for Australian schools. Australia, Carlton, Vic.: Author. [http://www.eric.ed.gov/PDFS/ED428947.pdf]
  18. Batanero, C., Henry, M., & Parzysz, B. (2005). The nature of chance and probability. In G. A. Jones(Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 15-37). New York, NY: Springer.
  19. Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The Solo Taxonomy. New York: Academic Press.
  20. Carmona, M. J. (2004). Mathematical background and attitudes toward statistics in a sample of undergraduate students. Paper presented at the 10th International Congress on Mathematics Education, Copenhagen, Denmark. [http://www.stat.auckland.ac.nz/-iase/publications/ 11/Carmona.doc]
  21. Chance, B., delMas, R., & Garfield, J. (2004). Reasoning about sampling distributions. In D. Ben-Zvi, & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 295-324). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  22. Chiesi, F., & Primi, C. (2010). Cognitive and non-cognitive factors related to students' statistics achievement. Statistics Education Research Journal, 9(1), 6-26.
  23. Cobb, G. W., & Moore, D. S. (1997). Mathematics, statistics, and teaching. The American Mathematical Monthly, 104(9), 801-823. https://doi.org/10.2307/2975286
  24. delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi, & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 79-95). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  25. Denzin, N. K., & Lincoln, Y. S. (1994). Handbook of qualitative research. Thousand Oaks, CA: Sage.
  26. Department for Education. (1999). Mathematics: The National Curriculum for England. Wellington, New Zealand: Author. [http://publications.education.gov.uk/eOrderingDownload/ QCA-99-460.pdf]
  27. Dreyfus, T., & Tsamir, P. (2004). Ben's consolidation of knowledge structures about infinite sets. Journal of Mathematical Behavior, 23(3), 271-300. https://doi.org/10.1016/j.jmathb.2004.06.002
  28. Fleiss, J. L. (1981). Statistical methods for rates and proportion. New York: Wiley.
  29. Greenes, C. (1981). Identifying the gifted student in mathematics. Arithmetic Teacher, 28(6), 14-17.
  30. Goetz, J. P., & LeCompte, M. D. (1984). Ethnography and qualitative design in educational research. Orlando, FL: Academic Press.
  31. Howley, C. B., Pendarvis, E. D., & Howley, A. (1986). Teaching gifted children, principles and strategies. Boston Toronto: Little, Brown and Company.
  32. Ko, E. S., & Lee, K. H. (2011). Are mathematically talented elementary students also talented in statistics? In B. Sriraman, & K. H. Lee, The elements of creativity and giftedness in mathematics (pp. 29-43). Rotterdam, The Netherlands: Sense Publishers.
  33. Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. Chicago, IL: The University of Chicago Press.
  34. Lalonde, R. N., & Gardner, R. C. (1993). Statistics as a second language?: A model for predicting performance in psychology students. Canadian Journal of Behavioural Science, 25(1), 108-125. https://doi.org/10.1037/h0078792
  35. Lee, K. (2005). Three types of reasoning and creative informal proofs by mathematically gifted students. Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 3, 241-248. Melbourne, Australia.
  36. Limm, S. (1984). The characteristics approach: Identification and beyond. Gifted Child Quarterly, 28(4), 181-187. https://doi.org/10.1177/001698628402800408
  37. Lipson, K. (2002). The role of computer based technology in developing understanding of the sampling distribution. In B. Phillips (Ed.), Proceedings of the 6th International Conference on Teaching Statistics. [CD-ROM] Voorburg, The Netherlands: International Statistics Institute.
  38. Ministry of Education. (1992). Mathematics in the New Zealand curriculum. Wellington, New Zealand: Author. [http://www.minedu.govt.nz/-/media/MinEdu/Files/EducationSectors/Schools/ MathematicsInTheNZCurriculum.pdf]
  39. Moore, D. S. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants (pp. 95-137). Washington, DC: National Academy Press.
  40. Moore, D. S. (1997). New pedagogy and new content: The case of statistics. International Statistical Review, 65(2), 123-165. https://doi.org/10.1111/j.1751-5823.1997.tb00390.x
  41. Moore, D. S., & Cobb, G. W. (2000). Statistics and mathematics: Tension and cooperation. The American Mathematical Monthly, 107(7), 615-630. https://doi.org/10.2307/2589117
  42. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
  43. Pfannkuch, M. (2008). Building sampling concepts for statistical inference: A case study, paper presented at the ICME 2008 TSG. Monterrey, Mexico. [http://tsg.icme11.org/document/get/476]
  44. Rubin, A., Bruce, B., & Tenney, Y. (1991). Learning about sampling: Trouble at the core of statistics. In D. Vere-Jones (Ed.), Proceedings of the Third International Conference on Teaching Statistics (pp. 314-319). Voorburg, The Netherlands: International Statistical Institute.
  45. Saldanha, L., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51, 257-270. https://doi.org/10.1023/A:1023692604014
  46. Schneider, W. (2000). Giftedness, expertise, and exceptional performance: A developmental perspective. In K. A. Heller, F. J. Monks, R. J. Sternberg, & R. F. Subotnik (Eds.), International handbook of giftedness and talent (pp. 165-178). NY: Elsevier.
  47. 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
  48. Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. Journal of Secondary Gifted Education, 14, 151-165.
  49. Sriraman, B. (2004). Gifted ninth graders' notions of proof: Investigating parallels in approaches of mathematically gifted students and professional mathematicians. Journal for the Education of the Gifted, 27(4), 267-292.
  50. Tsamir, P., & Dreyfus, T. (2002). Comparing infinite sets - a process of abstraction: The case of Ben. Journal of Mathematical Behavior, 21, 1-23. https://doi.org/10.1016/S0732-3123(02)00100-1
  51. Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal of Research in Mathematics Education, 31(1), 44-70. https://doi.org/10.2307/749819
  52. Wheatley, G. H. (1983). Mathematics curriculum for the gifted and talented. In J. VanTassel-Vaska, & S. M. Reis (Eds.), Curriculum for gifted and talented students (pp.137-146). Thousand Oaks, CA: Corwin Press.
  53. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265. https://doi.org/10.1111/j.1751-5823.1999.tb00442.x