Analysis of Learning Opportunities Provided in Elapsed Time Instruction: Focusing on Quantitative Objectification

경과시간 수업에서 제공되는 학습기회 분석: 양적 대상화를 중심으로

  • Received : 2021.10.13
  • Accepted : 2021.10.25
  • Published : 2021.10.31


Seeing the elapsed time as a quantity that can be measured is quite challenging for students while making students see it is also challenging for teachers. Tuning on these challenges, this article reports on what learning opportunities elementary teachers provide when they teach elapsed time focusing on quantitative objectification. I observed three mathematics classrooms where the elapsed time was taught by three elementary teachers and did a narrative analysis on the instructions. All three teachers utilized certain tools to support students access to the elapsed time as a quantity. They appropriated various quantitative attributes of the tool. In the case of the analog clock, one teacher tried to quantification the elapsed time with the number of minute hand's turning, while the other teacher indicated the distance of minute hand's moving. One teacher represented the elapsed time with the longitudinal attribute of the time band. Standing on the findings, the didactical implications of various attempts for quantitative objectification of the elapsed time implemented were discussed.

이 연구의 목적은 초등학교 수학교실에서 시행되는 경과시간 수업에서 학생들에게 제공되는 수학 학습기회를 탐색하는 것이다. 경과시간을 도입하는 데에 있어 후속 학년과의 수학적 연결성을 갖도록 경과시간을 양적으로 대상화하려는 교사들의 노력을 문서화하고자 하였다. 이를 위해 세 초등교사의 경과시간 도입 수업을 관찰하고 내러티브 분석을 시행하였다. 그 결과, 교사들은 도구를 사용하여 학생들이 경과시간을 양적으로 접근할 수 있도록 지원하고 있었으며, 같은 도구라도 서로 다른 양적 측면을 강조하였다. 아날로그 시계의 경우 한 교사는 시계바늘의 회전 바퀴 수로 양적 대상화를 시도한 반면, 다른 교사는 시계바늘이 아동한 거리로 경과시간의 양을 표상하였다. 시간띠의 길이 속성으로 경과시간의 양을 표상하는 경우도 있었다. 결과를 바탕으로 경과시간의 수업에서 다양하게 포착된 양적 대상화 사례들의 교수학적 의미를 논의하였다.



  1. 교육부(2015). 수학과 교육과정(교육부 고시 제2015-74호 별책8). 세종: 교육부.
  2. 교육부(2017). 수학 2-2. 서울: (주) 천재교육.
  3. 권미선(2019). 시각과 시간에 대한 초등학교 2 학년 학생들의 이해 실태 조사. 수학교육학연구, 29(4), 741-760.
  4. 한채린(2021a). 시각과 시간에 대한 수학과 교육과정 국제 비교 연구: 한국, 일본, 호주, 미국, 핀란드를 중심으로. 초등수학교육, 24(3), 115-134.
  5. 한채린(2021b). 기표의 구현과 수학적 이해: 경과시간을 중심으로. 수학교육, 60(3), 249-264.
  6. 한채린(2021c). '몇 시 몇 분 전'으로 시각 읽기에 얽힌 수학적 활동의 탐색. 학교수학, 23(3), 457-475.
  7. Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2012). Australian Curriculum: Mathematics. Retrieved from
  8. Barnett, J. E. (1998). Time's pendulum: From sundials to atomic clocks, the fascinating history of timekeeping and how our discoveries changed the world. New York, NY: Plenum Press.
  9. Berliner, D. C. (1979). Tempus educare. In P. L. Peterson & H. J. Walberg (Eds.), Research on teaching: Concepts, findings, and implications (pp. 120-135). Berkeley, CA: McCutchan.
  10. Burny, E., Valcke, M., & Desoete, A. (2009). Towards an agenda for studying learning and instruction focusing on time-related competencies in children. Educational Studies, 35(5), 481-492.
  11. Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., ... & Bakker, A. (2020). Maximizing the quality of learning opportunities for every student. Journal for Research in Mathematics Education, 51(1), 12-25.
  12. Clandinin, D. J., & Connelly, F. M. (2000). Narrative inquiry: Experience and story in qualitative research. San Francisco: Jossy-Bass Publisher.
  13. Cipolla, C. M. (1978). Clocks and culture, 1300-1700. New York, NY: Norton.
  14. Cohen, D. K., & Ball, D. L. (1999). Instruction, capacity, and improvement (CPRE Research Report RR-043). Philadelphia: Consortium for Policy Research in Education.
  15. Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 119-142.
  16. Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches (3rd ed.). Thousand Oaks, CA: Sage Publications.
  17. Dixon, J. K., Larson, M., Burger, E. B., Sandoval-Martinez, M. E., & Leinwand, S. J. (2015). Go math! grade 4 volume 2: Student edition. Orlando, FL: Houghton Mifflin Harcourt Publishing Company.
  18. Earnest, D. (2017). Clock work: How tools for time mediate problem solving and reveal understanding. Journal for Research in Mathematics Education, 48(2), 191-223.
  19. Earnest, D. (2019). The invisible quantity: time intervals in early algebra/La cantidad invisible: los intervalos de tiempo en el algebra temprana. Infancia y Aprendizaje, 42(3), 664-720.
  20. Earnest, D. (2021). About time: Syntactically-guided reasoning with analog and digital clocks. Mathematical Thinking and Learning,
  21. Earnest, D., & Chandler, J. (2021). Making time: words, narratives, and clocks in elementary mathematics. Journal for Research in Mathematics Education, 52(4), 407-443.
  22. Earnest, D., Gonzales, A. C., & Plant, A. M. (2018). Time as a measure: Elementary students positioning the hands of an analog clock. Journal of Numerical Cognition, 4(1), 188-214.
  23. Ellis, A. B. (2007). Connections between generalizing and justifying: Students' reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194-229.
  24. Finnish National Board of Education (2016). National core curriculum for basic education 2014. Helsinki, Finland: Author.
  25. Friedman, W. J., & Laycock, F. (1989). Children's analog and digital clock knowledge. Child Development, 60(2), 357-371.
  26. Hackenberg, A. J. (2010). Students' reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383-432.
  27. Han, C. (2020). Making sense of time-telling classroom: Interplay of cognition, instruction, and tools. Unpublished doctoral dissertation, Seoul National University.
  28. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.
  29. Husen, T. (Ed.). (1967). International study of achievement in mathematics: A comparison of twelve countries (Vol. I). New York, NY: John Wiley & Sons.
  30. Kamii, C., & Russell, K. A. (2012). Elapsed time: Why is it so difficult to teach? Journal for Research in Mathematics Education, 43(3), 296-315.
  31. Kurz, A. (2011). Access to what should be taught and will be tested: Students' opportunity to learn the intended curriculum. In S. N. Elliott, R. J. Kettler, P. A. Beddow, & A. Kurz (Eds.), The handbook of accessible achievement tests for all students: Bridging the gaps between research, practice, and policy (pp. 99-129). Springer.
  32. Males, L. M., & Earnest, D. (2015, April). Opportunities to learn time measure in elementary curriculum materials. In D. Earnest, L. M. Males, C. Rumsey, & R. Lehrer (Discussants), The measurement of time: Cognition, instruction, and curricula. Symposium conducted at the 2015 Research Conference of the National Council of Teachers of Mathematics, Boston, MA.
  33. Monroe, E. E., Orme, M. P., & Erickson, L. B. (2002). Links to literature: working cotton: Toward an understanding of time. Teaching Children Mathematics, 8(8), 475-479.
  34. Moore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), 225-245. doi:10.1007/s10649-012-9450-6
  35. Moore, K. C., & Carlson, M. P. (2012). Students' images of problem contexts when solving applied problems. The J ournal of Mathematical Behavior, 31, 48-59.
  36. National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CCSSO] (2010). Common core state standards for mathematics. Washington, DC: Authors.
  37. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
  38. Schmidt, W. H., McKnight, C. C., Valverde, G. A., Houang, R. T., & Wiley, D. E. (1997). Many visions, many aims: A cross-national investigation of curricular intentions in school mathematics. Dordrecht: Kluwer Academic Publishers.
  39. Smith, J. P., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In D. W. Carraher, J. J. Kaput, & M. Blanton (Eds.), Algebra in the early grades (pp. 95-132). Mahwah, NJ, USA: Lawrence Erlbaum Associates.
  40. Sowder, L. (1988). Children's solutions of story problems. The Journal of Mathematical Behavior, 7, 227-238.
  41. Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education: Issues in mathematics education (pp. 21-44). Providence, RI, USA: American Mathematical Society.
  42. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), WISDOMe Monographs: Vol. 1. New perspectives and directions for collaborative research in mathematics education (pp. 33-57). Laramie, WY, USA: University of Wyoming.
  43. Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics.
  44. Williams, R. F. (2012). Image schemas in clock-reading: Latent errors and emerging expertise. Journal of the Learning Sciences, 21(2), 216-246.
  45. Yerushalmy, M., & Shternberg, B. (2005). Chapter 3: Epistemological and cognitive aspects of time: A tool perspective. Journal for Research in Mathematics Education. Monograph XIII. Medium and meaning: Video papers in mathematics education research.
  46. Yin, R. K. (2014). Case Study Research Design and Methods (5th ed.). Thousand Oaks, CA: Sage.
  47. 文部科学省. (2017). 小学校学習指導要領. 文部科学省.