國小六年級學生「數學論證評量工具」之建構

The Construction of Measuring Six-Graders' Mathematical Argumentation

林碧珍(Pi-Jen Lin)；鄭俊彥(Chun-Yung Cheng)；蔡寶桂(Bao-Quei Tsai)

六年級 ； 評量工具 ； 數學論證 ； 驗證性因素分析 ； assessment ； confirmatory factor analysis ； mathematical argumentation ； sixthgrade

測驗學刊

65卷3期（2018 / 09 / 01）

257 - 289

繁體中文

過去有關學生學習成效之評量，其評量內容著重在數學知識而忽視高層次思考層面。本研究旨在建構一個適用於國小學生數學論證之評量工具，建構包含發展與驗證試題的品質。此評量工具透過預試和正式施測兩階段發展完成，兩階段的研究樣本都採用立意抽樣，預試和正式施測階段各從三縣市抽取六年級學生各為456與599人。從理論觀點建構數學論證評量架構的四個向度：「提出猜想」、「效化猜想」、「一般化」及「證明」，以提升評量工具的內容效度和建構效度，並利用統計分析，以項目分析、驗證性因素分析與信度分析，考驗「數學論證評量工具」的信效度。分析結果得到該份試題的難度介於0.30至0.74之間，鑑別度介於0.25至0.69之間，15題的總量表信度係數為.61，效度為專家效度；四個向度「提出猜想」、「效化猜想」、「一般化」及「證明」對數學論證的預測係數分別為.91、.84、.96、.83，顯示本研究中四個一階的預測因素能有效預測二階的數學論證之構念效度。本研究研發的數學論證工具之特色是不僅能對數學論證的四個向度以操作型定義，而且能將高層次思考的數學論證之評量試題以選擇題形式命題，並以論證品質的不同等級為誘答項設計的考量。本研究可提供教師及研究者一個值得信賴的評量工具。

In past, the assessment of student learning focused on mathematical knowledge instead of the high-order thinking. The study was to construct a measure of assessing six-graders' mathematical argumentation through pilot study and main study. The construction of the measure included two aspects: developing and testing. A total of 456 students and 599 students were purposely sampled from three counties for pilot study and main study, respectively. The validity and reliability of the measure to be constructed were based on both theoretical consideration and statistical analysis. From the theoretical consideration, the framework of the measure consisted of two dimensions: mathematical argumentation and mathematical content. Argumentation included four constructs: formulation, validation, generalization, and justification. From statistical analysis perspective, item analysis, reliability analysis, and confirmatory factor analysis were used to test the validity of the measure. The difficulty of the items ranged from 0.30 to 0.74 and the coefficiency of discrimination ranged from 0.25 to 0.69. The reliability coefficient of the total measure of 15 items was .61. We used the validity of expert. The coefficients of predicting the four factors of the mathematical argumentation including formulation, validation, generalization, and justification were .91, .84, .96, .83 respectively. The result indicates that the four first-order predictors had good prediction on the validity of the second-order argumentation. The measure developed in this study with multiple choice of the items to assess students' high-order thinking and mathematical argumentation with operational definition were the two features of the study. This study was able to provide a reliable instrument for teachers or researchers to measure students’ argumentation.

1. 林碧珍(2015)。國小三年級課室以數學臆測活動引發學生論證初探。科學教育學刊，23(1)，83-110。
   連結：
2. 林碧珍、鄭章華、陳姿靜(2016)。數學素養導向的任務設計與教學實踐：以發展學童的數學論證為例。教科書研究，9(1)，109-134。
   連結：
3. 凃金堂(2014)。中學生「數學知識信念量表」之發展與信效度考驗。測驗學刊，61(4)，533-556。
   連結：
4. (2013).The argument of mathematics.Netherlands:Springer.
5. (2014).Encyclopedia of Mathematics Education.
6. Aberdein, A.(Ed.),Dove, I. J.(Ed.)(2013).The argument of mathematics.New York, NY:Springer Dordrecht.
7. Bagozzi, R. P.,Yi, Y.(1988).On the evaluation of structural equation models.Journal of the Academy of Marketing Science,16,74-94.
8. Bloom, B. S.,Engelhart, M. D.,Furst, E. J.,Hill, W. H.,Krathwohl, D. R.(1956).Taxonomy of educational objectives: The classification of educational goals.New York, NY:David McKay Company.
9. Cañadas, M. C.,Castro, E.(2005).A proposal of categorization for analyzing inductive reasoning.Proceedings of the CERME 4 international conference,Catalonia, Spain:
10. Cavagnetto, A. R.(2010).Argument to foster scientific literacy a review of argument interventions in K-12 science contexts.Review of Educational Research,80,336-371.
11. Chen, Y.-C.(2011).Iowa City, IA,University of Iowa.
12. Civil, M.,Hunter, R.(2015).Participation of non-dominant students in argumentation in the mathematics classroom.Intercultural Education,26(4),296-312.
13. Cobb, P.,Wood, T.,Yackel, E.,McNeal, B.(1992).Characteristics of classroom mathematics traditions: An interacitonal analysis.American Educational Research Journal,29(3),573-604.
14. Common Core State Standards Initiative. [CCSSI] (2010). Introduction to the Common Core State Standards.Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/ccssi-introduction.pdf
15. Conner, A.,Singletary, L. M.,Smith, R. C.,Wagner, P. A.,Francisco, R. T.(2014).Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities.Educational Studies in Mathematics,86,401-429.
16. Cox, J. R.(Ed.),Willard, C. A.(Ed.)(1982).Advances in argumentation theory and research.Carbondale, IL:Southern Illinois University Press.
17. Erduran, S.,Simon, S.,Osborne, J.(2004).TAPping into argumentation: Developments in the application of Toulmin's argument pattern for studying science discourse.Science Education,88(6),915-933.
18. Govier, T.(1987).Problems in argument analysis and evaluation.Providence, RI:Foris.
19. Hanna, G.,de Villiers, M.(2008).ICMI Study 19: Proof and proving in mathematics education.ZDM The International Journal on Mathematics Education,40(1),329-336.
20. Inglis, M. J.,Mejia-Ramos, J. P.(2008).Theoretical and methodological implications of a broader perspective on mathematical argumentation.Mediterranean Journal for Research in Mathematics Education,7,107-119.
21. Jackson, D. L.,Gillaspy, J. A.,Purc-Stephenson, R.(2009).Reporting practices in con-firmatory factor analysis: An overview and some recommendations.Psychological Methods,14,6-23.
22. Jöreskog, K. G.,Sörbom, D.(1993).LISREL 8: Structural equation modeling with the SIMPLIS command language.Chicago, IL:Scientific Software International.
23. Kim, S.,Hand, B.(2015).An analysis of argumentation discourse patterns in elementary teachers' science classroom discussions.Journal of Science Teacher Education,26(3),221-236.
24. Kline, R. B.(2005).Principles and practice of structural equation modeling.New York, NY:The Guilford Press.
25. Ko, Y. Y.(2010).Mathematics teachers' conceptions of proof: Implications for educational research.International Journal of Science and Mathematics Education,8,1109-1129.
26. Krummheuer, G.(2007).Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions.Journal of Mathematical Behavior,26(1),60-82.
27. Lester, F. K.(Ed.)(2007).Second handbook of research on mathematics teaching and learning.Greenwich, CT:Information Age.
28. Lin, P. J.(2018).Students' mathematical argumentation through conjecturing in primary classrooms.Educação & Realidade (Education & Reality),43(3),1171-1192.
29. Lin, P. J.(2016).The quality of students' argumentation used in a fourth-grade classroom.Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education,Hungary:
30. Lin, P. J.(2018).The norms of argumentation in a primary classroom.Proceedings of the 8th ICMI-East Asia Regional Conference on Mathematics Education (EARCOME8),Taipei, Taiwan:
31. Lin, P. J.,Tsai, W. H.(2016).Enhancing students' mathematical conjecturing and justification in third-grade classrooms: The sum of even (odd) numbers.Journal of Mathematics Education,9(1),1-15.
32. Makar, K.,Bakker, A.,Ben-Zvi, D.(2015).Scaffolding norms of argumentation-based inquiry in a primary mathematics classroom.ZDM,47(7),1107-1120.
33. Mason, J.,Johnston-Wilder, S.(2004).Fundamental constructs in mathematics education.NY:Routledge.
34. McNeil, K. L.(2011).Elementary students' view of explanation, argumentation, and evi dence, and their abilities to construct arguments over school year.Journal of Research in Science Teaching,48(7),793-823.
35. McNeill, K. L.,González-Howard, M.,Katsh-Singer, R.,Loper, S.(2016).Pedagogical content knowledge of argumentation: Using classroom contexts to assess high-quality PCK rather than pseudo argumentation.Journal of Research in Science Teaching,53(2),261-290.
36. Ministry of Education、 Youth and Sport of the State of Brandenburg, YSSB(2015).Frame curriculum Brandenburg Mathematics 1-10.Berlin, Germany:Science and Technology Publisher.
37. Ministry of Education. [MOE] (2012). Primary mathematics teaching and learning syllabus. Singapore: Ministry of Education.
38. Nussbaum, E. M.(2008).Collaborative discourse, argumentation, and learning: Preface and literature review.Contemporary Educational Psychology,33,345-359.
39. Perloff, R. M.(2003).The dynamics of persuasion.Mahwah, NJ:Lawrence Erlbaum Associates.
40. Polya, G.(1954).Mathematics and plausible reasoning.Princeton, NJ:Princeton University Press.
41. Rijlaarsdam, G.(Ed.),Esperet, E.(Ed.),Andriessen, J. E .B.(Ed.),Coirier, P.(Ed.)(1999).Foundations of argumentative text processing.Amsterdam, The Netherlands:Amsterdam University Press.
42. Schumacker, R. E.,Lomax, R. G.(1996).A beginner's guide to structural equation modeling.Mahwah, NJ:Lawrence Erlbaum Associates.
43. Simon, D.,Holyoak, K. J.(2002).Structural dynamics of cognition: From consistency theories to constraint satisfaction.Personality and Social Psychology,6,283-294.
44. Staples, M.,Newton, J.(2016).Teachers' contextualization of argumentation in the mathematics classroom.Theory Into Practice,55(4),294-301.
45. Stein, M. K.,Grover, B. W.,Henningsen, M.(1996).Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.American Educational Research Journal,33,455-488.
46. Stylianides, G. J.,Stylianides, A. J.,Philippou, G. N.(2007).Preservice teachers' knowledge of proof by mathematical induction.Journal of Mathematics Teachers Education,10,145-166.
47. Toulmin, S. E.(1958).The uses of argument.Cambridge, MA:Cambridge University Press.
48. Yackel, E.(2004).Theoretical perspectives for analyzing explanation, justification and argumentation in mathematics classrooms.Journal of the Korea Society of Mathematics Education Series D: Research in Mathematical Education,8(1),1-18.
49. Yore, L. D.(2003).Quality science and mathematics education research: Considerations of argument, evidence, and generalizability.School Science and Mathematics,103(1),1-7.
50. 余民寧(2006)。潛在變項模式：SIMPLIS的應用。臺北市:高等教育。
51. 徐涵瑟(2013)。碩士論文(碩士論文)。嘉義市，國立嘉義大學。
52. 教育部（2016）。十二年國民基本教育數學領域課程綱要。臺北市：作者。
53. 教育部（2000）。國民中小學九年一貫課程暫行綱要：數學學習領域。臺北市：作者部。
54. 郭生玉(2004)。教育測驗與評量。臺北市:精華書局。
55. 彭淑玲、陳學志、黃博聖(2015)。當數學遇上創造力：數學創造力測量工具的發展。創造學刊，6(1)，83-107。

1. 張廖珮鈺,林碧珍(2020)。數學臆測教學中教師擔任協調者角色之教學行為。臺灣數學教育期刊，7(2)，1-23。