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

척도개념의 이해: 수학적 구조 조사로 과학교과에 나오는 물질의 크기를 표현하는 학생들의 이해도 분석

  • Received : 2014.02.13
  • Accepted : 2014.06.13
  • Published : 2014.06.30

Abstract

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.

Keywords

size and scale;number;multiplicative reasoning;operation

Acknowledgement

Supported by : 이화여자대학교

References

  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. Kohn, A. S. (1993). Preschoolers' reasoning about density: Will it float? Child Development, 64, 1637-1650. https://doi.org/10.2307/1131460
  3. Korea Foundation for the Advancement of Science & Creativity. 2009 -Revised national curriculum.
  4. 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
  5. 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.
  6. 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
  7. 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.
  8. Lindquist, M. (1989). Results from the fourth mathematics assessment of the national assessment of educational progress. Reston, VA: National Council of Teachers of Mathematics.
  9. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Retrieved from http://www.nctm.org/standards/content.aspx?id=16909
  10. National Research Council (1996). National science education standards. Washington, D. C: National Academy Press.
  11. 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
  12. 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
  13. 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
  14. 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.
  15. 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
  16. 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
  17. Fauvel, J. (1995). Revisiting the history of logarithms. Learn from the Masters, 39-48.
  18. 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
  19. 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
  20. 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.
  21. 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.
  22. 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
  23. Hodson, D. (1985). Philosophy of science, science and science education. Studies in Science Education, 12(1), 25-57. https://doi.org/10.1080/03057268508559922
  24. 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
  25. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books, Inc.
  26. 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.
  27. 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
  28. 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.
  29. 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
  30. Kamii, C., & Livingston, S. (1994). Young children continue to reinvent arithmetic, 3rdgrade. New York: Teachers College Press.
  31. 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.
  32. 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
  33. 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.
  34. 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
  35. 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
  36. 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
  37. 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
  38. Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 41(6), 189-201.
  39. 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
  40. 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.
  41. 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
  42. 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
  43. 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
  44. 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
  45. Dehaene, S. (2011). The number sense: How the mind creates mathematics. (Revised & Expanded Edition), New York, NY: Oxford University Press.
  46. 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
  47. 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
  48. Stevens, S., Sutherland, L., Schank, P., & Krajcik, J. (2009). The big ideas of nanoscale science & engineering: A guidebook for secondary teachers. NSTA Press.
  49. 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
  50. 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
  51. 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
  52. 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.
  53. 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
  54. 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
  55. 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
  56. 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
  57. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127-174). New York: Academic Press.
  58. 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
  59. Weber, K. (2002). Developing students' understanding of exponents and logarithms. ERIC Documents, 471-763.
  60. 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
  61. Piaget, J. (1987). Possibility and necessity: The role of possibility in cognitive development. Minneapolis, MN: The University of Minnesota Press.
  62. 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
  63. Redlich, O. (1970). Intensive and extensive properties. Journal of Chemical Education, 47(2), 154-156
  64. Reys, R. E., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2009). Helping children learn mathematics. Hoboken, NJ: John Wiley & Sons.
  65. Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers' knowledge: concept of division. International Education Journal, 8(2), 377-392.
  66. 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
  67. 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
  68. 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
  69. 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
  70. 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/
  71. 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.
  72. Sin, Y., & Han, S. (2011). A study of the elementary school teachers' perception in STEAM Education. Elementary Science Education, 30(4), 514-523.
  73. 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
  74. 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
  75. 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.