The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

Semiotic mediation through technology: The case of fraction reasoning

초등학생들의 측정으로서 분수에 대한 이해 : 공학도구를 활용한 기호적 중재

  • Received : 2020.10.15
  • Accepted : 2020.11.21
  • Published : 2021.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

This study investigates students' conceptions of fractions from a measurement approach while providing a technological environment designed to support students' understanding of the relationships between quantities and adjustable units. 13 third-graders participated in this study and they were involved in a series of measurement tasks through task-based interviews. The tasks were devised to investigate the relationship between units and quantity through manipulations. Screencasting videos were collected including verbal explanations and manipulations. Drawing upon the theory of semiotic mediation, students' constructed concepts during interviews were coded as mathematical words and visual mediators to identify conceptual profiles using a fine-grained analysis. Two students changed their strategies to solve the tasks were selected as a representative case of the two profiles: from guessing to recursive partitioning; from using random units to making a relation to the given unit. Dragging mathematical objects plays a critical role to mediate and formulate fraction understandings such as unitizing and partitioning. In addition, static and dynamic representations influence the development of unit concepts in measurement situations. The findings will contribute to the field's understanding of how students come to understand the concept of fraction as measure and the role of technology, which result in a theory-driven, empirically-tested set of tasks that can be used to introduce fractions as an alternative way.

본 연구는 초등학생들이 공학도구를 활용하여 측정으로서의 분수의 과제를 해결하는 과정을 분석하고 해결전략의 변화 과정에 대해서 논의하였다. 초등학생 13명을 대상으로 과제 중심의 임상면담을 실시하였고, 특히 분수를 처음 학습한 3학년 학생들의 측정 문제 해결 전략을 심층분석하였다. 그 결과, 추측하기에서 반복적인 분할하기, 임의의 단위 사용에서 주어진 단위 사용과 같은 두 가지 프로파일이 발견되었다. 각 프로파일의 대표적인 사례를 바탕으로, 공학도구의 활용이 역동적인 단위 개념을 형성하는데 기여하고 또한 분수와 관련된 의미형성과정에 드래깅과 같은 수학적 조작 활동이 영향을 줄 수 있음을 알 수 있었다. 본 연구의 결과가 분수의 다양한 의미를 탐구하고 학습하는 후속 연구를 위한 밑거름이 되길 기대한다.

Keywords

References

  1. Alajmi, A. H. (2012). How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait. Educational Studies in Mathematics, 79(2), 239-261. https://doi.org/10.1007/s10649-011-9342-1
  2. Arzarello, F. (2006). Semiosis as a multimodal process. RLIME-Revista Latino americana de Investigacionen Matematica Educativa, 9(1), 267-299.
  3. Arzarello, F., & Robutti, O. (2008). Framing the embodied mind approach within a multimodal paradigm. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 720-749, 2nd ed.). Mahwah, NJ: Erlbaum.
  4. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: artefacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 720-749, 2nd ed.). Mahwah, NJ: Erlbaum.
  5. Battista, M. T. (2008). Representations and cognitive objects in modern school geometry. In G. W. Blume, & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Cases and perspectives (pp. 341-362). Charlotte, NC: Information Age Publishing, Inc.
  6. Behr, M. J., Harel, G., Post, Th. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296-332). New York, NY: Macmilian Publishing Company.
  7. Common Core State Standards Initiative (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers, Retrieved from http://www.corestandards.org/wp-content/uploads/MathStandards.pdf
  8. Davydov, V. V., & Tsvetkovich, Z. H. (1991). The object sources of the concept of fraction. In V. V. Davydov (Soviet Edition Editor) & L. P. Steffe (English Language Editor), Soviet studies in mathematics education: Psychological abilities of primary school children in learning mathematics (pp. 86-147). Reston, VA: National Council of Teachers of Mathematics.
  9. Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to support reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.
  10. Dougherty, B. J. (2008). Measure up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389-412). New York, NY: Lawrence Erlbaum Associates.
  11. Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283-342. https://doi.org/10.1207/S1532690XCI1703_3
  12. Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2006). Fractions as the coordination of multiplicatively related quantities: Across-sectional study of children's thinking. Educational Studies in Mathematics, 63(1), 1-28. https://doi.org/10.1007/s10649-005-9000-6
  13. Empson, S. B., & Levi, L. (2011). Extending Children's Mathematics: Fractions and Decimals: [innovations in Cognitively Guided Instruction]. Portsmouth, NH: Heinemann.
  14. Ginsburg, H. (1997). Entering the child's mind: The clinical interview in psychological research and practice. UK: Cambridge University Press.
  15. Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164-192.
  16. Hunting, R. P., Davis, G., & Pearn, C. A. (1996). Engaging whole-number knowledge for rational-number learning using a computer-based tool. Journal for Research in Mathematics Education, 27(3), 354-379. https://doi.org/10.2307/749369
  17. Kang, H. K., & Ko, J. H. (2003). The educational significance of the method of teaching natural and fractional numbers by measurement of quantity. School Mathematics, 5(3), 385-399.
  18. Kaur, H. (2015). Two aspects of young children's thinking about different types of dynamic triangles: Prototypicality and inclusion. ZDM, 47(3), 407-420. https://doi.org/10.1007/s11858-014-0658-z
  19. Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. J. Behr (Eds.), Number concepts andoperations in the middle grades (pp. 162-181). Hillsdale, NJ: Erlba
  20. Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning, 12(3), 217-230. https://doi.org/10.1007/s10758-007-9123-1
  21. Laborde, J. M. (2016). Technology-enhanced teaching/learning at a new type with dynamic mathematics as implemented in the new Cabri. In M. Bates & Z. Usiskin (Eds.). Digital curricula in school mathematics (pp. 53-74). Charlotte, NC: Information Age Publishing, Inc.
  22. Lee, J., & Pang, J. (2014). Sixth grade students' understanding on unit as a foundation of multiple interpretations of fractions. Journal of Educational Research in Mathematics, 24(1). 83-102.
  23. Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295. https://doi.org/10.2307/749828
  24. Ministry of Education (2015). Mathematics curriculum. [Supplement 8]. Statute Notice of Ministryof Education (No. 2015-74). Seoul, South Korea: Ministry of Education.
  25. Moreno-Armella, L., Hegedus, S. J., & Kaput, J. J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational Studies in Mathematics, 68(2), 99-111. https://doi.org/10.1007/s10649-008-9116-6
  26. Morris, A. K. (2000). A teaching experiment: Introducing fourth graders to fractions from the viewpoint of measuring quantities using Davydov's mathematics curriculum. Focus on Learning Problems in Mathematics, 22(2), 33-84.
  27. Mortimer, E. F., & El-Hani, C. N. (2014). Conceptual profiles: A theory of teaching and learning scientific concepts. New York, NY: Springer.
  28. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
  29. Ng, O. L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. ZDM, 47(3),421-434. https://doi.org/10.1007/s11858-014-0660-5
  30. Noh, J., Lee, K., & Moon, S. (2019). A case study on the learning of the properties of quadrilaterals through semiotic mediation: Focusing on reasoning about the relationships between the properties. School Mathematics, 21(1), 197-214. https://doi.org/10.29275/sm.2019.03.21.1.197
  31. Olive, J., & Lobato, J. (2008). The learning of rational number concepts using technology. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Research syntheses (pp.1-54). Charlotte, NC: Information Age and the National Council of Teachers of Mathematics.
  32. Saxe, G. B., Diakow, R., & Gearhart, M. (2013). Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principal representational context. ZDM, 45(3), 343-364. https://doi.org/10.1007/s11858-012-0466-2
  33. Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of VV Davydov. The Mathematics Educator, 8(1), 60-87.
  34. Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13-25. https://doi.org/10.3102/0013189X031001013
  35. Schoenfeld, A. H., Smith, J. P., & Arcavi, A. A. (1993). Learning: The microgenetic analysis of one student's evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in Instructional Psychology (Volume 4) (pp. 55-175). Hillsdale, NJ: Erlbaum.
  36. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York, NY: Cambridge University Press.
  37. Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the Learning Through Activity research program. The Journal of Mathematical Behavior, 52, 122-133. https://doi.org/10.1016/j.jmathb.2018.03.004
  38. Son, T., Hwang, S., & Yeo, S. (2020). An analysis of the 2015 revised curriculum addition and subtraction of fractionsin elementary mathematics textbooks. School Mathematics, 22(3), 489-508. https://doi.org/10.29275/sm.2020.09.22.3.489
  39. Steffe, L. P., & Olive, J. (2002). Design and use of computer tools for interactive mathematical activity (TIMA). Journal of Educational Computing Research, 27(1), 55-76. https://doi.org/10.2190/GXQ4-60W8-CJY4-GKE1
  40. Streefland, L. (1991). Fractions in realistic mathematics education. Boston, MA: Kluwer.
  41. Streefland, L. (1993). Fractions: A realistic approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 289-325). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
  42. Suh, J., Moyer, P. S., & Heo, H. J. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1-21.
  43. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. Gary Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95-113). Reston, VA: The National Council of Teachers of Mathematics.
  44. Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.
  45. Webel, C., Krupa, E., & McManus, J. (2016). Using representations of fraction multiplication. Teaching Children Mathematics, 22(6), 366-373. https://doi.org/10.5951/teacchilmath.22.6.0366
  46. Yang, E., & Shin, J. (2014). Students' mathematical reasoning emerging through dragging activities in open-ended geometry problems. Journal of Educational Research in Mathematics, 24(1), 1-27.
  47. Yeo, S. (2019). Investigating children's informal thinking: The case of fraction division. Journal of KSME Series D: Research in Mathematics Education, 22(4), 283-304.
  48. Yeo, S. (2020). Integrating digital technology into elementary mathematics: Three theoretical perspectives. Journal of KSME Series D: Research in Mathematics Education, 23(3), 165-179.
  49. Yin, R. K. (2014). Case study research: Design and methods (5th ed.). Thousand Oaks, CA: Sage.