Journal of Educational Research in Mathematics (대한수학교육학회지:수학교육학연구)

The Korea Society of Educational Studies in Mathematics (대한수학교육학회)
권호 표지

Eye Movements in Understanding Combinatorial Problems

순열 조합 이해 과제에서의 안구 운동 추적 연구

  • Received : 2016.08.09
  • Accepted : 2016.11.11
  • Published : 2016.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Combinatorics, the basis of probabilistic thinking, is an important area of mathematics and closely linked with other subjects such as informatics and STEAM areas. But combinatorics is one of the most difficult units in school mathematics for leaning and teaching. This study, using the designed combinatorial models and executable expression, aims to analyzes the eye movement of graduate students when they translate the written combinatorial problems to the corresponding executable expression, and examines not only the understanding process of the written combinatorial sentences but also the degree of difficulties depending on the combinatorial semantic structures. The result of the study shows that there are two types of solving process the participants take when they solve the problems : one is to choose the right executable expression by comparing the sentence and the executable expression frequently. The other approach is to find the corresponding executable expression after they derive the suitable mental model by translating the combinatorial sentence. We found the cognitive processing patterns of the participants how they pay attention to words and numbers related to the essential informations hidden in the sentence. Also we found that the student's eyes rest upon the essential combinatorial sentences and executable expressions longer and they perform the complicated cognitive handling process such as comparing the written sentence with executable expressions when they try the problems whose meaning structure is rarely used in the school mathematics. The data of eye movement provide meaningful information for analyzing the cognitive process related to the solving process of the participants.

조합(combinatorics)은 확률적 사고의 기초가 되며 정보, 과학 등 타교과와 연계성이 높은 중요한 영역이지만, 학교 수학에서 학생들이 가장 어려워하는 단원 중 하나이다. 본 연구는 순열 조합 문제의 구조를 나타낼 수 있는 표현식을 도입하여, 문제를 표현식으로 변환하는 대학원생의 안구 운동을 분석함으로써 순열 조합 문장제의 이해 과정과 의미 구조에 따른 난이도 차이를 조사하였다. 연구 결과, 연구참여자들의 순열 조합 문장제 이해 전략은 문제에 대한 수학적 모델을 내적으로 직접 표상하는 전략과 보기에 주어진 표현식과 문제를 비교하여 답을 찾는 전략으로 분류할 수 있었다. 전문가 집단인 연구참여자들은 대상들의 구별성, 중복가능성, 의미 구조에 관한 단어나 수치 정보 등 문제의 핵심정보를 빠르게 파악하고 주의를 기울였다. 의미 구조의 변환이 필요한 문제를 풀 때 학생들은 문제의 핵심정보를 더 많이 보고, 보기의 표현식을 더 오래 응시하며, 문제와 보기 사이의 비교를 더 많이 하는 등 복잡한 인지 처리와 연관된 안구운동 지표가 나타났다. 안구 운동 데이터는 문제 이해 과정에서 연구참여자의 수학적 인지를 분석하는데 유의미한 정보를 제공하였다.

Keywords

References

  1. 김선희 (2004). 수학 문장제 해결에 영향을 주는 언어적.인지적 요인 - 혼합물 문제를 중심으로 -. 수학교육학연구 14(3), 267-281.
  2. 우정호 외 (2016). 고등학교 확률과 통계. 서울:동아출판.
  3. 이미진, 이광호 (2015). 시선 추적기를 통해 본, 4학년 학생들의 방정식에 대한 관계적 사고형성. 학교수학 17(3), 391-405.
  4. 이주영, 김서령, 박혜숙, 김완순 (2006). 조합문제 사이의 구조적 동형. 수학교육 45(1), 123-138.
  5. 이지윤 (2015). 3D 입체 변별 과제에서 공간 인지 전략의 유형과 역할 : 체화된 3D 거북 표현식과 전략을 중심으로. 박사학위논문, 서울대학교, 서울.
  6. 이지현, 이정연, 최영기 (2005). 순열 조합 문장제의 문제 변인과 오류 분석. 학교수학 7(2), 123-137.
  7. 조한혁 (2003). 컴퓨터와 수학교육. 수학교육 42(2), 177-191.
  8. 조한혁, 송민호 (2014). 실행식 (Executable expression) 기반 SMART 스토리텔링 수학교육. 수학교육학연구 24(2), 269-283.
  9. Andra, C., Lindstrom, P., Arzarello, F., Holmqvist, K., Robutti, O., & Sabena, C. (2015). Reading mathematics representations: An eye-tracking study. International Journal of Science and Mathematics Education, 13(2), 237-259. https://doi.org/10.1007/s10763-013-9484-y
  10. Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181-199. https://doi.org/10.1023/A:1002954428327
  11. Bates, D., Maechler, M., Bolker, B., Walker, S., Christensen, R. H. B., Singmann, H., ... & Rcpp, L. (2015). Package 'lme4'. convergence, 12, 1.
  12. Campbell, S. R. (2010). Embodied minds and dancing brains: New opportunities for research in mathematics education. In Theories of mathematics education (pp. 309-331). Springer Berlin Heidelberg.
  13. Clifton, C., Staub, A., & Rayner, K. (2007). Eye movements in reading words and sentences. Eye movements: A window on mind and brain, 341-372.
  14. Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive psychology, 20(4), 405-438. https://doi.org/10.1016/0010-0285(88)90011-4
  15. Dambacher, M., & Kliegl, R. (2007). Synchronizing timelines: Relations between fixation durations and N400 amplitudes during sentence reading. Brain research, 1155, 147-162. https://doi.org/10.1016/j.brainres.2007.04.027
  16. DeAngelus, M., & Pelz, J. B. (2009). Top-down control of eye movements: Yarbus revisited. Visual Cognition, 17(6-7), 790-811. https://doi.org/10.1080/13506280902793843
  17. De Smedt, B., & Verschaffel, L. (2010). Traveling down the road: from cognitive neuroscience to mathematics education… and back. ZDM, 42(6), 649-654. https://doi.org/10.1007/s11858-010-0282-5
  18. Dubois, J. G. (1984). A Systematic Study of Simple Combinatorial Configurations. Educational Studies in Mathematics, 15(1), 37-57. https://doi.org/10.1007/BF00380438
  19. Eizenberg, M. M., & Zaslavsky, O. (2004). Students' verification strategies for combinatorial problems. Mathematical Thinking and learning, 6(1), 15-36. https://doi.org/10.1207/s15327833mtl0601_2
  20. English, L. D. (1991). Young children's combinatoric strategies. Educational Studies in Mathematics, 22(5), 451-474. https://doi.org/10.1007/BF00367908
  21. English, L. D. (1993). Children's strategies for solving two-and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 255-273.
  22. English, L. D., & Halford, G. S. (1995). Mathematics education. 고상숙, 고호경, 박만구, 이중권, 정인철, 황우형 역. 서울: 경문사.
  23. Epelboim, J., & Suppes, P. (2001). A model of eye movements and visual working memory during problem solving in geometry. Vision Research, 41(12), 1561-1574. https://doi.org/10.1016/S0042-6989(00)00256-X
  24. Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt fur Didaktik der Mathematik, 5, 193-198.
  25. Gegenfurtner, A., Lehtinen, E., & Saljo, R. (2011). Expertise differences in the comprehension of visualizations: A meta-analysis of eye-tracking research in professional domains. Educational Psychology Review, 23(4), 523-552. https://doi.org/10.1007/s10648-011-9174-7
  26. Goldberg, J. H., & Kotval, X. P. (1999). Computer interface evaluation using eye movements: methods and constructs. International Journal of Industrial Ergonomics, 24(6), 631-645. https://doi.org/10.1016/S0169-8141(98)00068-7
  27. Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving guiding attention guides thought. Psychological Science, 14(5), 462-466. https://doi.org/10.1111/1467-9280.02454
  28. Hegarty, M., Mayer, R. E., & Green, C. E. (1992). Comprehension of arithmetic word problems: Evidence from students' eye fixations. Journal of Educational Psychology, 84(1), 76. https://doi.org/10.1037/0022-0663.84.1.76
  29. Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of educational psychology, 87(1), 18. https://doi.org/10.1037/0022-0663.87.1.18
  30. Holmqvist, K., Nystrom, M., Andersson, R., Dewhurst, R., Jarodzka, H., & Van de Weijer, J. (2011). Eye tracking: A comprehensive guide to methods and measures. OUP Oxford.
  31. Just, M. A., & Carpenter, P. A. (1980). A theory of reading: from eye fixations to comprehension. Psychological review, 87(4), 329. https://doi.org/10.1037/0033-295X.87.4.329
  32. Kapur, J. N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111-127. https://doi.org/10.1007/BF00381598
  33. Knoblich, G., Ohlsson, S., & Raney, G. E. (2001). An eye movement study of insight problem solving. Memory & cognition, 29(7), 1000-1009. https://doi.org/10.3758/BF03195762
  34. Kingsdorf, S., & Krawec, J. (2014). Error analysis of mathematical word problem solving across students with and without learning disabilities. Learning Disabilities Research & Practice, 29(2), 66-74. https://doi.org/10.1111/ldrp.12029
  35. Kintsch, W. (1998). Comprehension: A paradigm for cognition. 김지홍, 문선모 역. 경기도: 나남.
  36. Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81(4), 521. https://doi.org/10.1037/0022-0663.81.4.521
  37. Lin, J. J. H., & Lin, S. S. (2014). Cognitive Load for Configuration Comprehension in Computer-Supported Geometry Problem Solving: AN Eye Movement Perspective. International Journal of Science and Mathematics Education, 12, 605-627. https://doi.org/10.1007/s10763-013-9479-8
  38. Mayer, R. E. (1992). Thinking, problem solving, cognition. WH Freeman/Times Books/Henry Holt & Co.
  39. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc..
  40. Polya, G. (1945). How to solve it; a new aspect of mathematical method. Princeton university press.
  41. Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2-7.
  42. Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological bulletin, 124(3), 372. https://doi.org/10.1037/0033-2909.124.3.372
  43. R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Viena, Austria. URL http://www.R-project.org/.
  44. Roa, R. (2000). Razonamiento combinatorio en estudiantes con preparacion matematica avanzada. (Unpublished Doctoral dissertation). Universidad de Granada, Granada, Spain.
  45. Snowden, R., Thompson, P., & Troscianko, T. (2012). Basic vision: an introduction to visual perception. 오성주 역. 서울: 학지사.
  46. Susac, A., Bubic, A., Kaponja, J., Planinic, M., & Palmovic, M. (2014). Eye Movements Reveal Students' Strategies in Simple Equation Solving. International Journal of Science and Mathematics Education, 12, 555-577. https://doi.org/10.1007/s10763-014-9514-4
  47. Thomas, L. E., & Lleras, A. (2007). Moving eyes and moving thought: On the spatial compatibility between eye movements and cognition. Psychonomic bulletin & review, 14(4), 663-668. https://doi.org/10.3758/BF03196818
  48. Van Dijk, T. A., & Kintsch, W. (1983). Strategies of discourse comprehension. New York: Academic Press.
  49. Yarbus, A. L. (1967). Eye movements during perception of complex objects (pp. 171-211). Springer US.