한국수학교육학회지시리즈D:수학교육연구 (Research in Mathematical Education)

  • 제21권1호
  • /
  • Pages.1-13
  • /
  • 2018
  • /
  • 1226-6191(pISSN)
  • /
  • 2287-9943(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

Tinkering with Number Lines

  • 투고 : 2018.11.21
  • 심사 : 2018.12.09
  • 발행 : 2018.12.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

While the utility of the number line is considerable, articulating its conceptual foundation is often neglected in school mathematics. We suggest that it is important to build up strong conceptual foundations in the earlier grades so that number lines can be used in a more meaningful way and that any misconceptions associated with the number line can be prevented or intervened. This paper addresses unit, direction, and origin as the key elements of number lines and presents activities from Davydov's curriculum for early grades that promote exploration of those key elements and may resolve some students' misconceptions. As shown in sample activities from Davydov's curriculum, this paper suggests that students can broaden their perspectives on the number line and use it versatilely in various areas of mathematics learning when they deeply engage in the construction of a number line and have flexibility in interpreting the relationships between key number line elements.

키워드

SHGHEN_2018_v21n1_1_f0001.png 이미지

Figure 1. An example of a structured number line

SHGHEN_2018_v21n1_1_f0002.png 이미지

Figure 2. The number line for the natural numbers (Tapia et al. as cited in Herbst, 1997)

SHGHEN_2018_v21n1_1_f0003.png 이미지

Figure 3. Example of incorrect identification of the unit

SHGHEN_2018_v21n1_1_f0005.png 이미지

Figure 4. Example of unjustifiable identification of the unit and direction

SHGHEN_2018_v21n1_1_f0006.png 이미지

Figure 5. A number line for sorting even and odd numbers

SHGHEN_2018_v21n1_1_f0007.png 이미지

Figure 6. A curved number line showing that 24 rounded to the nearest ten is 20

SHGHEN_2018_v21n1_1_f0008.png 이미지

Figure 7. Coin lines

Table 1. Examples of misconceptions

SHGHEN_2018_v21n1_1_t0001.png 이미지

Activity 1. Place the given numbers on the number line.

SHGHEN_2018_v21n1_1_t0002.png 이미지

Activity 2. Mark the direction and the beginning on the number line

SHGHEN_2018_v21n1_1_t0003.png 이미지

Activity 3. Place the given numbers on the number line.

SHGHEN_2018_v21n1_1_t0004.png 이미지

참고문헌

  1. Bartolini Bussi, M. G. (2015). The number line: A "western" teaching aid. In X. Sun, B. Kaur, & J. Novotna (Eds.), Conference proceedings of the ICMI Study 23: Primary mathematics study on whole numbers (pp. 298-306). Macau, Chaina: University of Macau.
  2. Bright, G., Behr, M., Post, T., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19(3), 215-232. https://doi.org/10.2307/749066
  3. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669-705). Charlotte, NC: Information Age.
  4. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics instruction. Journal for Research in Mathematics Education, 37(2), 87-115.
  5. Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235-272). New York, NY: Lawrence Erlbaum.
  6. Cramer, K., Ahrendt, S., Monson, D., Wyberg, T., & Miller, C. (2016). Making sense of thirdgrade students' misunderstandings of number line. Investigations in Mathematics Learning, 9(1), 19-37.
  7. Davydov, V. V., Gorbov, S. F., Mikulina, G. G., & Saveleva, O. V. (1999). Mathematics: Class 1, Binghamton: State University of New York.
  8. Frykholm, J. (2010). Learning to think mathematically with the number line: A resource for teachers, a tool for young children. Salem, OR: Math Learning Center.
  9. Gullberg, J. (1997). Mathematics from the birth of numbers. New York: Norton and Company.
  10. Herbst, P. (1997). The number line metaphor in the discourse of a textbook series. For the Learning of Mathematics, 17(3), 36-45.
  11. Kanopka, K. (2016). Beyond the number line: Coordinate systems and vector arithmetic. Yale National Initiative. Retrieved from http://teachers.yale.edu/curriculum/viewer/initiative16.05.04_u.
  12. Lee, J. (2002). An analysis of difficulties encountered in teaching Davydov's mathematics curriculum to students in a US setting and measures found to be effective in addressing them. Doctoral Dissertation, State University of New York at Binghamton.
  13. Lee, J. (2006). Teaching algebraic expressions to young students: The three-day journey of 'a+2'. School Science and Mathematics, 106 (2), 98-104. https://doi.org/10.1111/j.1949-8594.2006.tb18139.x
  14. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: NGA & CCSSO.
  15. Schmittau, J. (2003). Cultural historical theory and mathematics education. In A. Kozulin, B. Gindis, V. Ageryev, & S. Miller (Eds.), Vygotsky's educational theory in cultural context (pp. 225-245). Cambridge, UK: Cambridge University Press.
  16. Schmittau, J. (2010). The relevance of Russian elementary mathematics education. In A. Karp, & B. R. Vogeli (Eds.), Russian mathematics education: History and world significance (pp. 253-278). Hackensack, NJ: World Scientific.
  17. Suh, B., Shin, H., & Na, J. (2005). An analytic study on the figure of number line. Journal of Educational Research in Mathematics, 23(2), 135-152.
  18. van de Walle, J., Karp, K., & Bay-Williams, J. M. (2013). Elementary and middle School mathematics: Teaching developmentally (8th ed.). NJ: Pearson.