The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

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

DOI QR Code

A study of representing activities of preservice secondary mathematics teachers in 3D geometric thinking and spatial reasoning

3차원 기하 사고와 공간적 추론에서 예비 중등 수학교사의 표상활동에 관한 연구

  • Received : 2014.01.27
  • Accepted : 2014.05.19
  • Published : 2014.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

This study investigated the types of the 3D geometric thinking and spatial reasoning through the observation of the 2D representing activities for representing the 3D geometrical objects with preservice secondary mathematics teachers. For this purpose, the 43 sophomoric students in college of education were divided into 10 groups and observed their group task performance on the basis of the representation they used. Observed processes were all recorded and the participants were interviewed based on the task. As a result, the role of physical object that becoming the object of geometric thinking and spatial reasoning, and diverse strategies and phenomena of the process that representing the 3D geometric figures in 2D were discovered. Furthermore, these processes of representing were assumed to be influenced by experience and study practice of students, and various forms of representing process were also discovered in the process of small group activities.

Keywords

References

  1. 교육과학기술부 (2011). 2009 개정 수학과 교육과정[별책 8]. 교육과학기술부. Ministry of Education and Science Technology (2011). The mathematics curriculum with 2009 revision[separated edition 8] . Ministry of Education and Science Technology.
  2. 김혜연 (2011). 예비교사들의 학습자 이해 지식과 교수방법 지식에 관한 연구 : 중학교 2학년 기하영역 증명을 중심으로, 석사학위논문, 이화여자대학교. Kim, H. H. (2011). A study on pre-service teachers' knowledge of students' understanding and teaching methods: Based on the geometric proofs for the second graders of middle school, Master's thesis, Ewha Womans University.
  3. 류현아 (2008). 중등 기하문제 해결에서 시각화와 추론과정, 박사학위논문, 건국대학교. Ryu, H. A. (2008). Visualization and reasoning in geometric problem solving among secondary school students, Doctoral dissertation, Konkuk University.
  4. Elena, B., & Leong, D. J. (1998). 정신의 도구:비고츠키 유아교육 (김억환, 박은혜 번역.). 서울:이화여자대학교 출판부. (원본 출판 1996). Elena, B., & Leong, D. J. (1998). Tools of the Mind: the Vygotskian approach to early childhood education (E. H. Kim & E. H. Park Trans.). Seoul: Ewha Womans University Press. (Original work published 1996).
  5. Miles, M. B., & Huberman, A. M. (2009). 질적자료분석론 (박태영, 박소영, 반정호, 성준모, 은선경, 이재령, 이화영, 조성희 번역.). 서울: 학지사. (원본 출판 1994). Miles, M. B., & Huberman, A. M. (2009). Qualitative data analysis. (T. Y. Park, S. Y. Park, J. H. Van, J. M. Sung, S. K. Eun, J. R. Lee, H. Y. Lee, S. H. Cho Trans.). Seoul: Hakjisa. (Original work published 1994).
  6. Baki, A., Kosa, T., & Guven, B. (2011). A comparative study of the effects of using dynamic geometry software and physical manipulatives on the spatial visualisation skills of pre-service mathematics teachers. British Journal of Educational Technology, 42(2), 291-310. https://doi.org/10.1111/j.1467-8535.2009.01012.x
  7. Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester(Ed.), Second handbook of research on mathematics teaching and learning (pp. 843-908). New York: Information Age Publishing.
  8. Ben-Chaim, D., Lappan, G., & Houang, R. (1989). Adolescents' ability to communicate spatial information: Analysing and effecting students' performance. Educational Studies in Mathematics, 20, 121-146. https://doi.org/10.1007/BF00579459
  9. Berthelot, R., & Salin, M. H. (1998). The role of pupils' spatial knowledge in the elementary teaching of geometry. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 71-78). Dordrecht: Kluwer.
  10. Bishop, A. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht: Kluwer.
  11. Brown, D. L., & Wheatley, G. H. (1997). Components of imagery and mathematical understanding. Focus on Learning Problems in Mathematics, 19(1), 45–70.
  12. Burton, L. J., & Fogarty, G. J. (2003). The factor structure of visual imagery and spatial abilities. Intelligence, 31, 289-318. https://doi.org/10.1016/S0160-2896(02)00139-3
  13. Christou, C., & Pittalis, M. (2010). Type of reasoning in 3D geometry thinking and their relation with spatial ability. Educational Studies in Mathematics, 75, 191-212. https://doi.org/10.1007/s10649-010-9251-8
  14. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York: Macmillan Publishing Co.
  15. Cohen, N. (2003). Curved solid nets. In N. Paterman, B. J. Doughery, & J. Zillox (Eds.), Proceedings of the 27th international conference of psychology in mathematics education, Vol. 2 (pp. 229-236). Honolulu, USA.
  16. Colom, R., Contreras, M. J., Botella, J., & Santacreu, J. (2001). Vehicles of spatial ability. Personality and Individual Differences, 32, 903–912.
  17. Coren, S., Ward, L. M., & Enns, J. T. (1994). Sensation and perception. Fort Worth, TX: Harcourt Brace College.
  18. Despina A. S. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76, 265-280. https://doi.org/10.1007/s10649-010-9273-2
  19. Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th conference of the international group for the psychology of mathematics education, Vol. 1 (pp. 33-48). Genova: Universita de Genova.
  20. Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century: An ICMI study. Dordrecht: Kluwer.
  21. Edelman, G. M. (1992). Bright air, brilliant fire. New York, NY: Basic Books.
  22. Ericson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed.). New York: Macmillan Publishing Company.
  23. Gagatsis, A., Christou, C., & Elia, I. (2004). The nature of multiple representations in developing mathematical relationships. Quaderni di Ricerca in Didattica, 14, 150-159.
  24. Greeno, J. G. (1988)."For the study of mathematics epistemology", In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving, Vol. 3 (pp. 23-31). Reston, VA: NCTM.
  25. Gutiérrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry. Structural Topology, 18, 31–48.
  26. Hegarty, M., & Waller, D. A. (2005). Individual differences in spatial abilities. In P. Shah & A. Miyake (Eds.), The cambridge handbook of visuospatial thinking. Cambridge: Cambridge University Press.
  27. Kimura, D. (1999). Sex and cognition. Cambridge: MIT.
  28. Kozhevnikov, M., Motes, M., & Hegarty, M. (2007). Spatial visualization in physics problem solving. Cognitive Science, 31, 549–579. https://doi.org/10.1080/15326900701399897
  29. Laborde, C. (1993). The computer as part of the learning environments: The case of geometry. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (Vol. 121, 48-67). Grenoble Cedex, France: NATOASI Series, Computer and Systems Sciences.
  30. Laborde C. (2002). Integration of technology in the design of geometry tasks with Cabri-Geometry. International Journal of Computers for Mathematical Learning, 6(3), 283-317.
  31. Lohman, D. (1988). Spatial abilities as traits, processes and knowledge. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence, Vol. 40 (pp. 181-248). Hillsdale: LEA.
  32. Ma, H. L., Wu, D., Chen, J. W., & Hsieh, K. J. (2009). Mithelmore's development stages of the right rectangular prisms of elementary school students in Taiwan. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd conference of the international group for the psychology of mathematics education, Vol. 4 (pp. 57-64). Thessaloniki: PME.
  33. Margaret, S., Ami M., & Walter J. W. (2011). Designing spatial visual tasks for research: The case of the filling task. Educational Studies in Mathematics, 78, 135-163. https://doi.org/10.1007/s10649-011-9315-4
  34. Maria, M. D., & Vanessa, S. T. (2012). The role of visual representations for structuring classroom mathematical activity. Educational Studies in Mathematics, 80(3), 413-431. https://doi.org/10.1007/s10649-011-9358-6
  35. Mariotti, M. A. (1989). Mental images: some problems related to the development of solids. In G. Vergnaud, J. Rogalski, & M. Artique (Eds.), Proceedings of the 13rd international conference for the psychology of mathematics education, Vol. 2 (pp. 258-265). Paris, France.
  36. Miyake, A., Friedman, N. P., Rettinger, D. A., Shah, P., & Hegarty, M. (1991). How are visuospatial working memory, executive functioning, and spatial abilities related? A latent-variable analysis. Journal of Experimental Psychology, 130, 621–640.
  37. NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  38. Newman, D., Griffin, P., & Cole, M. (1989). The construction zone: working for cognitive change in school. Cambridge: Cambridge University Press.
  39. Parzysz, B. (1988). "Knowing" vs "Seeing" Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79-92. https://doi.org/10.1007/BF00428386
  40. Potari, D., & Spiliotopoulou, V. (2001). Patterns in children's drawings and actions while constructing the nets of solids: The case of the conical surfaces. Focus on Learning Problems in Mathematics, 23(4), 41-62.
  41. Presmeg, N. C. (1997). Generalization using imagery in mathematics. In L. D. English (Ed.), Mathematical reasoning (pp. 299-312). Mahwah, NJ: Erlbaum.
  42. Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205-236). Rotterdam: Sense.
  43. Saito, N., Akita, M., & Inprasitha, M. (2013). Instructional method for developing students' mathematical creativity. Paper presented at the 6tth East Asia Regional Conference on Mathematics Education (EARCOME 6), Phuket, Thailand.
  44. Salomon, G. (1993). Distributed cognitions: psychological and educational considerations. Cambridge: Cambridge University Press.
  45. Schulze, D., Beauducel, A., & Brocke, B. (2005). Semantically meaningful and abstract figural reasoning in the context of fluid and crystallized intelligence. Intelligence, 33, 143-159. https://doi.org/10.1016/j.intell.2004.07.011
  46. Sevil, A., & Fatma, A. T. (2013). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. International Journal of Science and Mathematics Education, 1-22.
  47. Wheatley, G. H. (1997). Reasoning with images in mathematical activity. In L. D. English (Ed.), Mathematical reasoning (pp. 281-298). Mahwah, NJ: Erlbaum.

Cited by

  1. 원기둥을 이용한 중학생의 공간기하 이해 능력 분석 vol.59, pp.2, 2014, https://doi.org/10.7468/mathedu.2020.59.2.113