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How Dense Are Rational Numbers?: An Inclusive Materialist Case Study to Digital Technology

초등학생의 디지털 테크놀로지를 이용한 유리수 조밀성 탐구 사례 분석: 포괄적 유물론에서의 접근

  • Received : 2018.08.21
  • Accepted : 2018.09.04
  • Published : 2018.10.31

Abstract

This study examines the influence of the bodily interaction with digital technology on meaning-making process in a mathematical activity. Increasing interest in the use of multi-touch dynamic digital technology has brought the movement of the body to the center of research focus in recent mathematics education literature. Thereby, we investigate the process in which the meaning of the density of rational numbers emerges around the bodily interaction on the multi-touch dynamic digital technology. We analyze a case of a small group of primary school students with microethnography. In the result, the students formed the higher level of meaning of the density, where the finger movement of zooming in-and-out played a crucial role throughout the meaning-maknig process.

본 연구는 수학적 활동에서 디지털 테크놀로지와의 신체적 상호작용이 수학적 의미 형성 과정에 주는 영향을 탐구하는 것을 연구 목표로 삼는다. 최근의 수학교육 연구는 역동적 환경과 멀티터치 입력의 결합에 주목하기 시작했는데, 이에 따라 신체적 움직임이 연구에서 적극적으로 고려해야 할 대상이 되었다. 이에 따라 이 연구는 유리수 조밀성의 수학적 의미가 형성되는 과정을 역동적 멀티터치 테크놀로지와 상호작용하는 신체의 움직임과 관련하여 살펴보고자 한다. 이를 위해 소규모 집단의 초등학생에 대한 교수실험 사례를 분석하며 미시 문화기술적 방법을 사용한다. 연구 결과, 주어진 활동을 통해 조밀성에 대한 더 높은 수준의 수학적 의미가 형성된 것을 확인할 수 있었는데, 이러한 의미 형성 과정 전반에 걸쳐 학습자가 터치패드를 이용하여 화면을 확대하고 축소하는 손가락 움직임이 결정적인 역할을 하는 것으로 드러났다.

Keywords

References

  1. Ko, H., Kim, E., Yang, S., Kwon, S., Kwon, S. Jung, N., Jang, I., Lim, Y., Choi, S., Lee, S., Roh, S., Baek, H., & Hong, C. (2013). Middle school mathematics 3. Seoul: Kyohaksa.
  2. Ministry of Education, Science and Technology (2011). Mathematics curriculum. Ministry of Education, Science and Technology Notice 2011-361 [Supplement 8] .
  3. Kim, M. H. & Kim, H. J. (2015). Internet of things and hyper-connected society: conceptual foundation and possibilities of the technique humanities. Visual Culture, 27, 215-238.
  4. Gim, C.-C. & Bae, J.-H. (2016). Duleuze and education: Pedagogy of difference-making. Seoul: Hageeshisub.
  5. Lew. H., Lew, S., Lee, K.-H., Kim, C., Kang, S., Yoon, O., Kim, M., Cho, S., Chun, T., & Kim, C. (2013). Middle school mathematics 1. Seoul: Chunjae.
  6. Moon, S. J. & Lee, K.-H. (2017). The function of signs and attention in teaching-learning of mathematics. School Mathematics, 19(1), 189-208.
  7. Woo, J., Park, K., Lee, J., Park, K., KIm, N., Lim, J., Nam, J., Kwon, S., Kim, J., Kang, H., Cho, C., Huh, S., Jeon, J., Koh, H., Lee, J., Choi, E., & Kim, J.. (2013). Middle school mathematics 1. Seoul: Doosan Donga.
  8. Cha, D. & Jin, Y. (2015). The future of hyperconnected society, shared economy and internet of things. Seoul: Hans Media.
  9. Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70(2), 97-109. https://doi.org/10.1007/s10649-008-9163-z
  10. Baccaglini-Frank, A., & Maracci, M. (2015). Multi-touch technology and preschoolers' development of number-sense. Digital Experiences in Mathematics Education, 1(1), 7-27. https://doi.org/10.1007/s40751-015-0002-4
  11. Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Durham: Duke University Press.
  12. Borba, M. C., Askar, P., Engelbrecht, J., Gadanidis, G., Llinares, S., & Aguilar, M. S. (2016). Blended learning, e-learning and mobile learning in mathematics education. ZDM, 48(5), 589-610. doi:10.1007/s11858-016-0798-4
  13. Chatelet, G. (2000). Figuring space: Philosophy, mathematics, and physics (S. Robert & Z. Muriel, Trans.). Dordrecht: Springer.
  14. Coole, D. H., & Frost, S. (2010). Introducing the new materialisms. In D. H. Coole & S. Frost (Eds.), New materialisms: Ontology, agency, and politics (pp. 1-43). Durham, NC: Duke University Press.
  15. de Freitas, E., & Palmer, A. (2016). How scientific concepts come to matter in early childhood curriculum: rethinking the concept of force. Cultural Studies of Science Education, 11(4), 1201-1222. doi:10.1007/s11422-014-9652-6
  16. de Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education: The body in/of mathematics. Educational Studies in Mathematics, 83(3), 453-470. https://doi.org/10.1007/s10649-012-9465-z
  17. de Freitas, E., & Sinclair, N. (2014). Mathematics and the body: Material entanglements in the classroom. NY: Cambridge University Press.
  18. Deleuze, G. (1994). Difference and repetition (P. Patton, Trans.). London: Athlone.
  19. Deleuze, G., & Guattari, F. l. (1987). A thousand plateaus: Capitalism and schizophrenia. Minneapolis: University of Minnesota Press.
  20. Drijvers, P., & Ferrara, F. (2018). Instruments and the body. In E. Bergqvist, M. Osterholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 193-194). Umea, Sweden: PME.
  21. Ferrara, F., Faggiano, E., & Montone, A. (2017). Introduction: Innovative spaces for mathematics education with technology. In E. Faggiano, F. Ferrara, & A. Montone (Eds.), Innovation and technology enhancing mathematics education: Perspectives in the digital era (pp. 1-5). Cham: Springer.
  22. Ferrara, F., & Ferrari, G. (2017). Agency and assemblage in pattern generalisation. Educational Studies in Mathematics, 94, 21-36. doi:10.1007/s10649-016-9708-5
  23. Gucler, B., Hegedus, S., Robidoux, R., & Jackiw, N. (2013). Investigating the mathematical discourse of young learners involved in multi-modal mathematical investigations: The case of haptic technologies. In D. Martinovic, V. Freiman, & Z. Karadog (Eds.), Visual mathematics and cyberlearning (pp. 97-118). Dordrecht: Springer.
  24. Haus, J. M. (2018). Performative intra-action of a paper plane and a child: Exploring scientific concepts as agentic playmates. Research in Science Education. doi:10.1007/s11165-018-9733-8
  25. Hegedus, S. J., & Tall, D. O. (2016). Foundations for the future: The potential of multimodal technologies for learning mathematics. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 556-575). New York, NY: Routledge.
  26. Hoyles, C., & Lagrange, J.-B. (2010). Introduction. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and techonlogy: Rethinking the terrain (pp. 1-11). New York: Springer.
  27. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning.
  28. Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. Journal for Research in Mathematics Education, 44(2), 372-415. https://doi.org/10.5951/jresematheduc.44.2.0372
  29. Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111-126. https://doi.org/10.1007/s10649-008-9127-3
  30. Roschelle, J., Noss, R., Blikstein, P., & Jackiw, N. (2017). Technology for learning mathematics. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 853-872). Reston, VA: NCTM.
  31. Roth, W.-M. (2016). Growing-making mathematics: A dynamic perspective on people, materials, and movement in classrooms. Educational Studies in Mathematics, 93(1), 87-103. https://doi.org/10.1007/s10649-016-9695-6
  32. Rotman, B. (2008). Becoming beside ourselves : the alphabet, ghosts, and distributed human being. Durham: Duke University Press.
  33. Sinclair, N., Arzarello, F., Gaisman, M. T., & Lozano, M. D. (2010). Implementing digital technologies at a national scale. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology: Rethinking the terrain (pp. 61-78). New York: Springer.
  34. Sinclair, N., & de Freitas, E. (2014). The haptic nature of gesture: Rethinking gesture with new multitouch digital technologies. Gesture, 14(3), 351-374. https://doi.org/10.1075/gest.14.3.04sin
  35. Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM, 45(2), 239-252. https://doi.org/10.1007/s11858-012-0465-3
  36. Sinclair, N., & Pimm, D. (2015). Mathematics using multiple senses: Developing finger gnosis with three-and four-year-olds in an era of multi-touch technologies. Asia-Pacific Journal of Research in Early Childhood Education, 9(3), 99-109. https://doi.org/10.17206/apjrece.2015.9.3.99
  37. Streeck, J., & Mehus, S. (2005). Microethnography: The study of practices. In K. L. Fitch & R. E. Sanders (Eds.), Handbook of language and social interaction (pp. 381-404). Mahwah, NJ: Lawrence Erlbaum Associates.
  38. Trouche, L. (2014). Instrumentation in mathematics education. In S. Lerman (Ed.), Encycolopedia of Mathematics Education (pp. 307-313). Dordrecht: Springer.
  39. Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students' understanding of density. Learning and Instruction, 21(5), 676-685. https://doi.org/10.1016/j.learninstruc.2011.03.005
  40. Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453-467. https://doi.org/10.1016/j.learninstruc.2004.06.013
  41. Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational numbers interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 265-282). Amsterdam: Elsevier.
  42. Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students' understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181-209. https://doi.org/10.1080/07370001003676603
  43. Walter, D. (2018). How children using counting strategies represent quantities on the virtual and physical 'twenty frame'. In L. Ball, P. Drijvers, S. Ladel, H.-S. Siller, M. Tabach, & C. Vale(Eds.), Uses of technology in primary and secondary mathematics edcuation: Tools, topics and trends (pp.119-143). Cham: Springer.
  44. Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. P. (2007). Research on technology in mathematics education: The perspective of constructs. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1169-1207). Charlotte, NC: Information Age Publishing.