한국과학교육학회지 (Journal of The Korean Association For Science Education)

  • 제34권4호
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  • Pages.335-347
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  • 2014
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  • 1226-5187(pISSN)
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  • 2288-8489(eISSN)
한국과학교육학회 (The Korean Association for Science Education)
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척도개념의 이해: 수학적 구조 조사로 과학교과에 나오는 물질의 크기를 표현하는 학생들의 이해도 분석

Student Understanding of Scale: From Additive to Multiplicative Reasoning in the Constriction of Scale Representation by Ordering Objects in a Number Line

  • 투고 : 2014.02.13
  • 심사 : 2014.06.13
  • 발행 : 2014.06.30
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초록

관찰과 측정을 기본으로 하는 과학의 교과에서 "크기(size)"와 그를 나타내는 "척도(scale)"는 물질의 물리적 속성과 과학적 현상을 이해하도록 돕는 중요한 개념이다. 또한, 사물의 수, 크기나 양을 어림잡거나 그것을 정확하게 표현하는 것은 수학에서 수의 개념 형성과 발달, 표현법의 습득, 나아가서는 연산에 관한 사고로의 발전과 관련되어있는 문제라고 볼 수 있어 "크기와 척도" 개념은 수학과 과학의 기본이며 동시에 두 교과를 연결하는 개념이다. 일반적으로 "크기와 척도"는 쉬운 개념이라 생각되지만, 실제 학생들은 물질의 크기를 제대로 이해하지 못하거나 척도로 나타내는 것을 어려워하는 것을 알 수 있다. 이는 단지 물질의 크기를 정확히 알지 못하는 정확성에 관한 오류로만 그치는 것이 아니라 종종 연관된 개념을 추론하거나 개념을 확장해 과학의 현상을 이해하는 과정에서의 어려움으로 이어진다. 이와 관련해 수와 연산에 관한 개념이해와 학습의 어려움에 관한 수학교육분야의 연구는 다양하게 진행되었지만, 과학교육분야에서의 연구는 많지 않았다. 본 연구에서는 "크기와 척도"에 관한 학생들의 사고를 더 잘 이해하고 과학 학습의 어려움에 관한 원인을 분석하기 위해 수학적 구조분석을 적용하였다. 수학교육에서 설명한 수 개념의 발달에 따른 사고유형(덧셈이전의 사고, 덧셈적 사고-additive reasoning, 곱셈적 사고-multiplicative reasoning)을 적용하여 7단계의 수학적 구조를 만들고 이를 이용하여 "크기와 척도"와 관련된 과제를 수행한 학생들의 인터뷰 데이터를 체계적으로 분석하였다. 수학적 구조를 바탕으로 한 개념 틀은 다양한 학생들의 사고를 분석하는 기준이 되었고, 또한 학생들이 겪는 개념이해의 어려움을 해석하는 도구가 되었다. 수 개념의 발달에 맞춘 수학적 사고구조를 적용한 분석은 학생들의 개념 유형의 구분을 명확히 하였고 설명이 모호했던 전환 단계(transition stage) 유형을 밝혀내어 수업에서 고려되어야 할 점들을 구체적으로 드러내었다. 이는 수학과 과학, 두 교과 간의 틈을 줄일 뿐 아니라 연결점을 찾아 학생들의 개념이해와 어려움의 원인을 분석하는데 폭넓은 시각을 제공한다는 점에서 최근 많은 관심을 받고 있는 STEM 혹은 수학과 과학의 융합 수업을 위한 소재로의 가능성을 제시해준다.

Size/scale is a central idea in the science curriculum, providing explanations for various phenomena. However, few studies have been conducted to explore student understanding of this concept and to suggest instructional approaches in scientific contexts. In contrast, there have been more studies in mathematics, regarding the use of number lines to relate the nature of numbers to operation and representation of magnitude. In order to better understand variations in student conceptions of size/scale in scientific contexts and explain learning difficulties including alternative conceptions, this study suggests an approach that links mathematics with the analysis of student conceptions of size/scale, i.e. the analysis of mathematical structure and reasoning for a number line. In addition, data ranging from high school to college students facilitate the interpretation of conceptual complexity in terms of mathematical development of a number line. In this sense, findings from this study better explain the following by mathematical reasoning: (1) varied student conceptions, (2) key aspects of each conception, and (3) potential cognitive dimensions interpreting the size/scale concepts. Results of this study help us to understand the troublesomeness of learning size/scale and provide a direction for developing curriculum and instruction for better understanding.

키워드

참고문헌

  1. Adjiage, R., & Pluvinage, F. (2007). An experiment in teaching ratio and proportion. Educational Studies in Mathematics. 65, 149-175. https://doi.org/10.1007/s10649-006-9049-x
  2. Akatugba, A. H., & Wallace, J. (1999). Sociocultural influences on physics students' use of proportional reasoning in a non-western country. Journal of Research in Science Teaching, 36(3), 305-320. https://doi.org/10.1002/(SICI)1098-2736(199903)36:3<305::AID-TEA5>3.0.CO;2-1
  3. Akatugba, A. H., & Wallace, J. (2009). An integrative perspective on students' proportional reasoning in high school physics in a West African context. International Journal of Science Education, 31(11), 1473-1493. https://doi.org/10.1080/09500690802101968
  4. An, S. (2008). A survey on the proportional reasoning ability of fifth, sixth, and seventh graders. Journal of Educational Research in Mathematics, 18(1), 103-121.
  5. Angell, C., Kind, P. M., Henriksen, E. K., & Guttersrud, O. (2008). An empirical-mathematical modelling approach to upper secondary physics. Physics Education, 43(3), 256-264. https://doi.org/10.1088/0031-9120/43/3/001
  6. Bar. V. (1987). Comparison of the development of ratio concepts in two domains. Science Education, 71(4), 599-613. https://doi.org/10.1002/sce.3730710411
  7. Beland, A., & Mislevy, R. J. (1996). Probability-based inference in a domain of proportional reasoning tasks. Journal of Educational Measurement, 33(1), 3-27. https://doi.org/10.1111/j.1745-3984.1996.tb00476.x
  8. Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7thgrade students with different curricular experiences. Educational Studies in Mathematics, 36(3), 247-273. https://doi.org/10.1023/A:1003235712092
  9. Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 41(6), 189-201.
  10. Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016-1031. https://doi.org/10.1111/j.1467-8624.2008.01173.x
  11. Cheek, K. A. (2010). Why is geologic time troublesome knowledge? In J. H. F. Meyer, R. Land, & C. Baillie (Eds.), Threshold concepts and transformational learning (pp.117-129). Rotterdam: Sense Publishers.
  12. Clark, F., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27(1), 41-51. https://doi.org/10.2307/749196
  13. Dawkins, K. R., Dickerson, D. L., McKinney, S. E., & Butler, S. (2008). Teaching density to middle school students: Preservice science teachers' content knowledge and pedagogical practices. The Clearing House: A Journal of Educational Strategies, Issues, and Ideas, 82(1), 21-26. https://doi.org/10.3200/TCHS.82.1.21-26
  14. De Lozano, S. R., & Cardenas, M. (2002). Some learning problems concerning the use of symbolic language in physics. Science Education, 11, 589-599. https://doi.org/10.1023/A:1019643420896
  15. Dehaene, S. (2011). The number sense: How the mind creates mathematics. (Revised & Expanded Edition), New York, NY: Oxford University Press.
  16. Drane, D., Swarat, S., Light, G., Hersam, M., & Mason, T. (2009). An evaluation of the efficacy and transferability of a nanoscience module. Journal of Nano Education, 1(1), 8-14. https://doi.org/10.1166/jne.2009.001
  17. Erickson, T. (2006). Stealing from physics: modeling with mathematical functions in data-rich contexts. Teaching Mathematics and its Applications, 25(1), 23-32. https://doi.org/10.1093/teamat/hri025
  18. Evans, K. L., Yaron, D., & Leinhardt, G. (2008). Learning stoichiometry: a comparison of text and multimedia formats. Chemistry Education Research and Practice, 9(3), 208-218. https://doi.org/10.1039/B812409B
  19. Fauvel, J. (1995). Revisiting the history of logarithms. Learn from the Masters, 39-48.
  20. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16 (1), 3-17. https://doi.org/10.2307/748969
  21. Guckin, A. M., & Morrison, D. (1991). Math*Logo: A project to develop proportional reasoning in college freshmen. School Science and Mathematics, 91(2), 77-81. https://doi.org/10.1111/j.1949-8594.1991.tb15575.x
  22. Hart, K. M. Brown, M. L., Kuchemann, D. E., Kerslake, D., Ruddock, G., & McCartney, M. (1981). Children's understanding of mathematics: 11-16. London: John Murray.
  23. Hart, K. M. (1988). Ratio and proportion. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp.198-219). Reston, VA: National Council of Teacher of Mathematics.
  24. Hines, E., & McMahon, M. T. (2005). Interpreting middle school students' proportional reasoning strategies: Observations from preservice teachers. School Science and Mathematics, 105(2), 88-105. https://doi.org/10.1111/j.1949-8594.2005.tb18041.x
  25. Hodson, D. (1985). Philosophy of science, science and science education. Studies in Science Education, 12(1), 25-57. https://doi.org/10.1080/03057268508559922
  26. Hofstein, A., & Lunetta, V. N. (1982). The role of the laboratory in science teaching: Neglected aspects of research. Review of Educational Research, 52(2), 201-217. https://doi.org/10.3102/00346543052002201
  27. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books, Inc.
  28. Jang, M., & Park, M. (2006). A study on the multiplicative thinking of 2nd grade elementary students. Communications of Mathematical Education. Series E, 20(3), 443-467.
  29. Jones, M. G., Taylor, A., & Broadwell, B. (2009). Concepts of scale held by students with visual impairment. Journal of Research in Science Teaching, 46(5), 506-519. https://doi.org/10.1002/tea.20277
  30. Jones, M. G., Taylor, A., Minogue, J., Broadwell, B., Wiebe, E., and Carter, G. (2006). Understanding scale: Powers of ten. Journal of Science Education and Technology, 16(2), 191-202.
  31. Kadosh, R., Tzelgov, J., & Henik, A. (2008). A synthetic walk on the mental number line: The size effect. Cognition, 106, 548-557. https://doi.org/10.1016/j.cognition.2006.12.007
  32. Kamii, C., & Livingston, S. (1994). Young children continue to reinvent arithmetic, 3rdgrade. New York: Teachers College Press.
  33. Kim, J., & Bang, J. (2013). An analysis on third graders' multiplicative thinking and proportional reasoning ability. Journal of Educational Research in Mathematics, 23(1), 1-16.
  34. Kohn, A. S. (1993). Preschoolers' reasoning about density: Will it float? Child Development, 64, 1637-1650. https://doi.org/10.2307/1131460
  35. Korea Foundation for the Advancement of Science & Creativity. 2009 -Revised national curriculum.
  36. Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), 41-61. https://doi.org/10.2307/749385
  37. Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. J. Harel, & J, Confrey (Eds.) The development of multiplicative reasoning in the learning of mathematics (pp. 89-122). Albany, NY: State University of New York Press.
  38. Lee, H., Son, D., Kwon, H., Park, K., Han, I., Jeong, H., Lee, S., Oh, H., & Nam, J. (2012). Secondary teachers' perceptions and needs analysis on integrative STEM education. Journal of the Korean Association for Science Education, 32(1), 30-45. https://doi.org/10.14697/jkase.2012.32.1.030
  39. Lesh, R., Post, R., & Behr, M. (1988). Proportional reasoning. In J. M. Hiebert Behr (Ed.), Number concepts and operations in the middle grades (pp. 93-118). Reston, VA: National Council of Teachers of Mathematics.
  40. Lindquist, M. (1989). Results from the fourth mathematics assessment of the national assessment of educational progress. Reston, VA: National Council of Teachers of Mathematics.
  41. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Retrieved from http://www.nctm.org/standards/content.aspx?id=16909
  42. National Research Council (1996). National science education standards. Washington, D. C: National Academy Press.
  43. Noelting, G. (1980). The development of proportional reasoning and the ratio concept Part I-Differentiation of stages. Educational studies in Mathematics, 11(2), 217-253. https://doi.org/10.1007/BF00304357
  44. O'Brien, T., & Casey, S. (1983). Children learning multiplication. School Science and Mathematics. 83, 246-251 https://doi.org/10.1111/j.1949-8594.1983.tb15518.x
  45. Oon, P. T., & Subramaniam, R. (2011). On the declining interest in physics among students-from the perspective of teachers. International Journal of Science Education, 33(5), 727-746. https://doi.org/10.1080/09500693.2010.500338
  46. Park, E-J., & Choi, K. (2010). Analysis of mathematical structure to identify students' understanding of a scientific concept: pH value and scale. Journal of the Korean Association for Science Education, 30(7), 920-932.
  47. Piaget, J. (1987). Possibility and necessity: The role of possibility in cognitive development. Minneapolis, MN: The University of Minnesota Press.
  48. Prain, V., & Waldrip, B. (2006). An exploratory study of teachers' and students' use of multi-modal representations of concepts in primary science. International Journal of Science Education, 28(15), 1843-1866. https://doi.org/10.1080/09500690600718294
  49. Redlich, O. (1970). Intensive and extensive properties. Journal of Chemical Education, 47(2), 154-156
  50. Reys, R. E., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2009). Helping children learn mathematics. Hoboken, NJ: John Wiley & Sons.
  51. Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers' knowledge: concept of division. International Education Journal, 8(2), 377-392.
  52. Sanders, M., Kwon, H., Park, K., & Lee, H. (2011). Integrative STEM education: contemporary trends and issues. Secondary Education Research, 59(3), 729-762. https://doi.org/10.25152/ser.2011.59.3.729
  53. Sheppard, K. (2006). High school students' understanding of titrations and related acid-base phenomena. Chemistry Education Research and Practice, 7(1), 32-45. https://doi.org/10.1039/B5RP90014J
  54. Siegler, R. S., & Opfer, J. (2003). The developmental of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237-243. https://doi.org/10.1111/1467-9280.02438
  55. Siegler, R. S., Thompson, C. A., & Opfer, J. E. (2009). The logarithmic to linear shift: One learning sequence, many tasks, many time scales. Mind, Brain, and Education, 3(3), 143-150. https://doi.org/10.1111/j.1751-228X.2009.01064.x
  56. Siemon, D., Breed, M., & Virgona, J. (2006). From additive to multiplicative thinking-The big challenge of the middle years. www.education.vic.gov.au/studentlearning/teachingresources/maths/
  57. Siemon, D., & Virgona, J. (2001). Road maps to numeracy-Reflections on the middle years numeracy research project. Paper presented at the annual conference of the Australian Association for Research in Education, Fremantle, WA.
  58. Sin, Y., & Han, S. (2011). A study of the elementary school teachers' perception in STEAM Education. Elementary Science Education, 30(4), 514-523.
  59. Singer, J. A., & Resnick, L. B. (1992). Representations of proportional relationships: Are children part-part or part-whole reasoners? Educational Studies in Mathematics, 23(3), 231-246. https://doi.org/10.1007/BF02309531
  60. Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43(3), 271-292. https://doi.org/10.1023/A:1011976904850
  61. Smith, E., & Confrey, J. (1994). Multiplicative structures and the development of logarithms: What was lost by the invention of function? In G. J. Harel, & J, Confrey (Eds.) The development of multiplicative reasoning in the learning of mathematics (pp333-364). Albany, NY: State University of New York Press.
  62. Smith, C., Maclin, D., Grosslight, L., & Davis, H. (1997). Teaching for understanding: a study of students' pre-instruction theories of matter and a comparison of the effectiveness of two approaches to teaching about matter and density. Cognition and Instruction, 15(3). 317-393. https://doi.org/10.1207/s1532690xci1503_2
  63. Stevens, S., Sutherland, L., Schank, P., & Krajcik, J. (2009). The big ideas of nanoscale science & engineering: A guidebook for secondary teachers. NSTA Press.
  64. Streefland, L. (1984). Search for the roots of ratio: Some thoughts on the long term learning process (Towards... a theory). Educational Studies in Mathematics, 15(4), 327-348. https://doi.org/10.1007/BF00311111
  65. Swarat, S., Light, G., Park, E-J., & Drane, D. (2011). A typology of undergraduate students' conceptions of size and scale: Identifying and characterizing conceptual variation. Journal of Research in Science Teaching, 48(5), 512-533. https://doi.org/10.1002/tea.20403
  66. Taber, K. S. (2006). Conceptual integration: a demarcation criterion for science education?. Physics Education, 41(4), 286-287. https://doi.org/10.1088/0031-9120/41/4/F01
  67. Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. The development of multiplicative reasoning in the learning of mathematics (pp179-234). Albany, NY: State University of New York Press.
  68. Trend, R. (2000). Conceptions of geological time among primary teacher trainees, with reference to their engagement with geosciences, history, and science. International Journal of Science Education, 22(5), 539-555. https://doi.org/10.1080/095006900289778
  69. Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16,181-204. https://doi.org/10.1007/BF02400937
  70. Tretter, T. R., Jones, M. G., Andre, T., Negishi, A., & Minogue, J. (2006). Conceptual boundaries and distances: Students' and experts' concepts of the scale of scientific phenomena. Journal of Research in Science Teaching, 43(3), 282-319. https://doi.org/10.1002/tea.20123
  71. Tretter, T. R., Jones, M. G., & Minogue, J. (2006). Accuracy of scale conceptions in science: Mental maneuverings across many orders of spatial magnitude. Journal of Research in Science Teaching, 43(10), 1061-1085. https://doi.org/10.1002/tea.20155
  72. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127-174). New York: Academic Press.
  73. Wagner, E. P. (2001). A study comparing the efficacy of a mole ratio flow chart to dimensional analysis for teaching reaction stoichiometry. School Science and Mathematics, 101(1), 10-22. https://doi.org/10.1111/j.1949-8594.2001.tb18185.x
  74. Weber, K. (2002). Developing students' understanding of exponents and logarithms. ERIC Documents, 471-763.
  75. Zen, E.-A. (2001). What is deep time and why should anyone care? Journal of Geoscience Education, 49(1), 5-9. https://doi.org/10.5408/1089-9995-49.1.5

피인용 문헌

  1. The Students’ Perception on the Real Thing’s Actual Size through the Illustrations in Life Science’s Textbooks vol.46, pp.3, 2018, https://doi.org/10.15717/bioedu.2018.46.3.398