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The Conceptualizing and Practices of Mathematical Classes Based on Students' Thinking

학생 사고기반 수학 수업의 특징과 그 실제

  • Received : 2020.12.15
  • Accepted : 2021.03.18
  • Published : 2021.03.31

Abstract

In this study, the student participation-centered class, which takes students' mathematical thinking as an important issues of the class, is named as student thinking-based math class. The main characteristics of student thinking-based mathematics classes were examined in terms of tasks, students engagement, and the role of teachers. According to the results of analysis of class cases practiced by five secondary mathematics teachers, student thinking-based mathematics classes were conducted in the intersection of the rich mathematics tasks, students' cognitive and social engagement, and the role of teachers' formative facilitator. The results of this study showed that the student's thinking is more important than the activity itself. And it is meaningful in that it examines the influence of the dynamic interaction of the three components of the mathematics class on the direction and outcome of the class.

본 연구는 학생의 수학적 사고를 수업의 중요한 자원으로 삼는 학생 참여형 수업을 학생 사고기반 수학 수업이라 명하고, 학생 사고기반 수학 수업의 주요 특징을 살펴보았다. 문헌 검토를 통해 확인된 학생 사고기반 수학 수업의 중요한 특징은 풍부한 수학 과제, 학생의 인지적 사회적 참여, 그리고 형성적 조력자 역할이다. 수업 사례 분석 결과에 의하면 학생 사고기반 수학 수업은 풍부한 수학 과제, 학생의 인지적·사회적 참여, 그리고 교사의 형성적 조력자 역할의 교집합 속에서 이루어졌다. 연구 결과는 학생 참여형 수업이 활동 자체에서 학생의 사고에 초점을 두었으며, 수업의 세 구성 요소의 상호작용이 수업 방향과 결과에 미치는 영향을 살펴보았다는 점에서 의미가 있다.

Keywords

References

  1. Kim, W. (2019). A discursive approach to the development of the conceptual framework of textbook analysis, and research on teachers' recognition(Doctoral dissertation). Korea University.
  2. Kim, D. Y. & Kim, G. Y. (2014). Secondary mathematics teachers understanding and modification of mathematical tasks in textbooks. School Mathematics, 16(3), 445-469.
  3. Kim, M. H., & Kim, G. Y. (2013). The analysis of matheamtical tasks in the high school mathematics. School Mathematics, 15(1), 37-59.
  4. Kim, H. L., & Lee, K. H. (2016). Pre-service secondary mathematics teachers modification of derivative tasks, School Mathematics, 18(3), 711-731.
  5. Park, Y. E., & Pang, J. S. (2016). Exploring self-study and its application to enhance instructional expertise in mathematics. Journal of educational research in mathematics, 26(3), 467-488
  6. Park, J. M., Park, J, H., & Lee, J, K. (2017). A study on analysis of American CMP textbooks in terms of mathematical connectivity: Focused on equations, inequalities, and functions. Journal of the Korean School Mathematics Society, 20(3), 277-302.
  7. Park, J. H. (2019). Prospective elementary mathematics teachers' difficulties on textbook task modification: focusing on fraction tasks. Journal of Educational Research in Mathematics, 29(4), 551-575. https://doi.org/10.29275/jerm.2019.11.29.4.551
  8. Yu, J. E. & Park, M. H. (2019). Tangram task modification for exploring in elementary mathematics. Educational of Primary School Mathematics, 22(1), 95-111.
  9. Lee, D. G. (2018). A case study on student's thoughts and expressions on various types of geometric series tasks. The Mathematical Education, 57(4), 353-369. https://doi.org/10.7468/MATHEDU.2018.57.4.353
  10. Lee, S. J., & Kim, G. Y. (2019). How middle-school mathematics textbooks of Korea and the US support to develop students' statistical reasoning. Mathematical Education, 58(1), 139-160. https://doi.org/10.7468/MATHEDU.2019.58.1.139
  11. Lee, J. H. (2014). ). How do pre-service teachers disprove 0.99⋯? School Mathematics, 16(3), 491-502.
  12. Lee, J. A. (2017). A qualitative research on middle school teachers' understanding of "student-participatory class"(Master's thesis). Seoul National University.
  13. Yi., Y. B., & Hong, J. K. (2016). Primary students' mathematical thinking analysis of between abstraction of concrete materials and concretization of abstract concepts. School Mathematics, 18(1), 159-173.
  14. Cho, H. H. & Choi, Y. G. (1999). The repreating decimal from the static and dynamic view point. School Mathematics, 1(2), 605-615.
  15. Choi, H. S. (2020). Exploration of Instructional Design Changes of Pre-service Mathematics Teachers by Restructuring the Lesson Plan. Journal of KSMS, 23(1), 159-177.
  16. Han, C. R., Kim, H. J. & Kwon, O. N. (2018). Teacher noticing on students' reasoning of statistical variability. Journal of the Korean School Mathematics, 21(2), 183-206. https://doi.org/10.30807/KSMS.2018.21.2.005
  17. Han, H. J., Kim, Y. G., Kim, B. E. et al. (2018). Analysis of actual conditions to student-participatory class. Sejong Educational Policy Institute, 2018-03.
  18. Hong, C. J., & Kim. G. Y. (2012). Functions in the middle school matheamtics: the cognitive demand of the mathematical tasks. School Mathematics, 14(2), 213-232.
  19. Ahmed, H. O. K. (2016). Flipped learning as a new educational paradigm: an analytical critical study. European Scientific Journal, 21(10), 417-444. doi: 10.19044/esj.2016.v12n10p417
  20. Bahr, D. L. & Bahr, K. (2017). Engaging all students in mathematical discussions. Teaching Children Mathematics, 23(6), 350-359. https://doi.org/10.5951/teacchilmath.23.6.0350
  21. Black, P. & Wiliam, D. (1998). Assessment and classroom learning. Policy & Practice, 5(1), 7-74.
  22. Breen, S. & O.Shea, A. (2010). Mathematical thinking and task design. Irich Mathematical Society Bulletin, 66, 39-49. https://doi.org/10.33232/BIMS.0066.39.49
  23. Chapin, S. H., O'Connor, C., & Anderson, N. C. (2013). Classroom discussions in math: a teacher's guide for using talk moves to support the common core and more, grades K-6. 김진호, 이지은, Jung Colen, 이대현, 김상미 역(2016). 수학교실토론. 서울: 경문사.
  24. Choy, B. H. (2014). Teachers' productive mathematical noticing during lesson preparation. In Nicol, C., Liljedahl, P., Oesterle, S., & Allan, D. (Eds.) Proceedings of the Joint Meeting of PME 38 and PME-NA 36,Vol. 2, pp. 297-304. Vancouver, Canada: PME.
  25. Choy, B. H., Dindyal, J. (2018). An approach to teach with variations: using typical problems. Avances de Investigacion en Educacion Matematica, nΩ 13, 21-38. https://doi.org/10.35763/aiem.v0i13.227
  26. Cohen, D. K., Raudenbush, S. W. & Ball, D. L. (2003). Resources, instruction, and research. Educational evaluation and policy analysis, 25(2), 119-142. https://doi.org/10.3102/01623737025002119
  27. Finn, D. F. (1989). Withdrawing from school. Review of Educational Research, 59(2), 117-142. https://doi.org/10.3102/00346543059002117
  28. Finn, J. D. & Zimmer, K. S. (2012). Student engagement: What is it? Why does it matter? In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of research on student engagement (pp. 97-131). New York, NY: Springer.
  29. Foster, C. (2015). The Convergent-Divergent Model: an opportunity for teacher-learner development through principled task design. Educational Designer, 2(8), 1-25.
  30. Fredricks, J. A., Wang, M. T., Linn, J. S., Hofkens, T. L., Sugn, H., Parr, A. & Allerton, J. (2016). Using qualitative methods to develop a survey measure of math and science engagement. Learning and Instruction, 43, 5-15. https://doi.org/10.1016/j.learninstruc.2016.01.009
  31. Hatch, J. A. (2008). Doing qualitative research in education settings. 진영은 역(2016). 교육적 상황에서 질적 연구 수행하기. 서울: 학지사.
  32. Hattie, J. & Timperley, H. (2007). The power of feedback. Review of educational research, 1, 81-112. https://doi.org/10.3102/003465430298487
  33. Helme, S. & Clarke, D. (2001). Identifying cognitive engagement in the mathematics classroom. Mathematics Education Research Journal, 13, 133-153. https://doi.org/10.1007/BF03217103
  34. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students' learning in second-grade arithmetic. American educational research journal, 30(2), 393-425. https://doi.org/10.3102/00028312030002393
  35. Hiebert, J., Carpenter, T. P., Fennema, E., & Fuson, K. C. (1997). Making sense teaching and learning mathematics with understanding. 김수환, 박영희, 이경화, 한대희 공역(2004). 어떻게 이해하지?. 서울: 경문사.
  36. Jacobs, V. R., Lamb, L. L. C. & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking, Journal for research in mathematics education, 41(2), 169-202. https://doi.org/10.5951/jresematheduc.41.2.0169
  37. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press.
  38. Leon, J., Medina-Garrido, E. & Munez, J. L. (2017). Teaching quality in math class: the development of a scale and the analysis of its relationship with engagement and achievement. Front. Psychol. 8:895. doi: 10.3389/fpsyg.2017.00895.
  39. Li, Y. & Huang, R. (2013). How Chinese teach mathematics and improve teaching. Rouledge, New York, NY 10017
  40. McMahon, B. & Portelli, J. P. (2004). Engagement for what? beyond popular discoursed of student engagement. Leadership and Policy in Schools, 3(1), 59-76. https://doi.org/10.1076/lpos.3.1.59.27841
  41. Martin, D. P. & Rimm-Kaufman, S. E. (2015). Do student self-efficacy and teacher-student interaction quality contribute to emotional and social engagement in fifth grade math?, Journal of school psychology, 53, 359-373. https://doi.org/10.1016/j.jsp.2015.07.001
  42. Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. St. Albans: Tarquin Publications.
  43. Munter, C. (2014). Developing visions of high-quality mathematics instruction. Journal for Research in Mathematics Education, 45(5), 584-635. https://doi.org/10.5951/jresematheduc.45.5.0584
  44. Peterson, K. R., Stockero, S. L. & Zoest, L. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for research in mathematical education, 46(1), 88-124. https://doi.org/10.5951/jresematheduc.46.1.0088
  45. Rimm-Kaufman, S. E., Baroody, A., Larsen, R., Curby, T. W. & Abry, T. (2015). To what extent do teacher-student interaction quality and student gender contribute to fifth graders' engagement in mathematics learning? Journal of Educational Psychology, 107(1), 170-185. https://doi.org/10.1037/a0037252
  46. Rymes, B. (2009). Classroom discussion analysis: a tool for critical reflection by Betsy Rymes. 김종현 역 (2011). 말이 열리는 교실: 교실수업 개선을 위한 담화분석. 서울: 학이시습.
  47. Schoenfeld, A. H. (2010). How to think: a theory of goal-oriented decision making and its educational application. 이경화 역(2013). 수학수업, 설명을 만나다. 서울: 경문사.
  48. Shepard, L. A.(2000). The role of assessment in a learning culture. Educational Researcher, 29(7). 4-14 https://doi.org/10.3102/0013189X029007004
  49. Smith, M. S. & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. 방정숙 역(2013). 서울: 경문사.
  50. Smith, M. S. & Sherin, M. G. (2019). The 5 practices in practice. NY: Corwin mathematics publishing.
  51. Stacey, K. (2006). What is mathematical thinking and why is it important? Retrieved from http://e-archives.criced.tsukuba.ac.jp/data/doc/pdf/2009/02/Kaye_Stacey.pdf
  52. Stein, M. K. & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: from research to practice. Mathematics Teaching in the Middle School, 3, 268-275. https://doi.org/10.5951/MTMS.3.4.0268
  53. Stein, M. K., Smith, M. S. & Silver, S. A. (2009). Implementations standards-based mathematics instruction: a casebook for professional development (2nd ed.). 김남균, 조완영, 권성룡 역(2017). 수학과제가 수학수업을 만났을 때. 서울: 경문사.
  54. Stockero, S. L. & Van Zoest, L. R. (2012). Characterizing pivotal teaching moments in begging matheamtics teachers' practice. Journal of Mathematics Teacher Education, 16, 125-147. https://doi.org/10.1007/s10857-012-9222-3
  55. Sullivan, P., Clarke, D. & Clarke, B. (2013). Teaching with tasks for effective mathematics learning. 이경화, 김동원 역(2016). 수학 수업 이야기: 수학, 과제, 학습의 삼중주. 서울: 경문사.
  56. Swan, M. (2008). Designing a multiple representation learning experience in secondary algebra. Educational Designer, 1(1), 1-17.
  57. Swan, M. & Burkhardt, H. (2012). Designing assessment of performance in mathematics. Educational Designer, 2(5).
  58. van ES, E. A., Hand, V. & Mercado, J. (2017). Making visible the relationship between teachers' noticing for equity and equitable teaching practice. In Schack et al. (Eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks, Research in Mathematics Education, DOI 10.1007/978-3-319-46753-5_15.
  59. Watson, A. & Sullivan, P. (2008). Teachers learning about tasks and lessons. International Handbook of Mathematics Teacher Education, 2, 107-134.
  60. Watson, A. & Thompson, D. R. (2015). Design Issues Related to Text-Based Tasks. In Task Design In Mathematics Education (pp. 143-190). Springer International Publishing.
  61. William, D. & Thompson, M. (2007). Integrating assessment with learning: What will it take to make it work? In: Dwyer, CA, (ed). The Future of Assessment: Shaping Teaching and Learning. (pp. 53-82). New York: Routledge.
  62. Yang, Y. & Rick, T. E. (2013). Chinese lesson study: developing classroom instruction through collaborations in school-based teaching research group activities. In Y. Li & R. Huang (Eds.), How Chinese teach mathematics and improve teaching (pp. 51-65). New York: Routledge.