本研究採用質性取向的個案研究法，透過課室教學觀察及訪談，探討一位資深高中數學教師的教學知識與教學思考。研究設計分為前導研究、第一階段研究與第二階段研究；研究的教學單元分別為「空間平面和空間直線方程式」、「重複組合」與「數學期望值」；研究資料的主要來源包括，教學影片檔、訪談錄音檔和個案教師自編的數學科講義。本研究所使用的數學教學觀察系統，引自Learning Mathematics for Teaching (2006)所發展的Mathematical Quality of Instruction(MQI)登錄系統。首先，個人修改系統的編碼，以符合個案數學教師實際的教學特質，其次，針對教學影片進行分析，最後，商請另一位獨立的登錄者協助信度的檢驗，以提昇系統的有效性。藉由三個不同的數學教學單元，以及為期一年的研究進程，對個案數學教師的教學知識與教學思考，以及MKT (Mathematical Knowledge for Teaching, Ball等人, 2008)的內涵有更深入的了解。
本研究的結果顯示，個案教師的數學教學大多呈現PUFM (Profound Understanding of Fundamental Mathematics, Ma, 1996)的連通性、多重觀點、一致性和基本概念四項特徵。然而，由於PUFM著重數學學科知識，對於某些PUFM無法涵蓋的數學教學片段，個人則藉助Ball (2008)的MKT架構來進一步分析。據此，個人提出一些比較凸顯MKT特徵的數學教學實例，這也表明，MKT和PUFM似乎有部分的重疊，但是亦有不同。結果進一步顯示，個案教師的數學教學傾向，遠看似PUFM，近看卻有部分MKT的元素。此外，個案數學教師也有一些其他的數學教學特色，例如螺旋式教學。
最後，根據研究的結果，個人進一步指出PUFM和MKT可能的關係，以作為未來接續探究高中數學教師MKT和PUFM內涵與關係之參考。希望本研究的結果，能夠有助於提昇高中數學教學觀察系統的品質與實用性，並用來檢測和發展高中數學教師的數學教學專業知識。

This study applies qualitative case study, which explores the professional knowledge and thinking of an experienced high school mathematics teacher. The study is structured into three research stages including the pilot, initial, and second stage. The teaching units include「plane and line in space」、「combination with repetition」and「mathematical expectation」. The study material comes from videotaped lessons, interviews, and class handouts provided by the participant teacher. The observation system is adapted from coding system of the Mathematical Quality of Instruction (MQI) developed by Learning Mathematics for Teaching (2006). First of all, I modified the system codes to adapt to the actual quality of the teacher’s classroom teaching. Second, I analyzed the video tapes. Last, the coding results were mostly supported from another independent coder to establish the acceptable inter-coder reliability. Using three different observed teaching units within an academic year, I describe the participant teacher’s knowledge and thinking of/about mathematics teaching in terms of both MKT (Mathematical Knowledge for Teaching) (Ball et al., 2008) and PUFM (Profound Understanding of Fundamental Mathematics, Ma, 1996) theoretical frameworks.

The results reveal that, the teaching of the teacher mostly presented PUFM qualities of connectedness, multiple perspectives, unity, and the basics. However, considering of emphasis on subject matter knowledge of PUFM, I took the aid of MKT structure to further analyze some teaching clips which PUFM is unable to cover. According to the instruction above, I brought up some mathematical teaching cases which highlight MKT characteristics, which also indicates that MKT and PUFM seem to have some parts overlapped, but some not neither. The study result in addition presents the teaching tendency of the teacher was much alike PUFM as a whole, yet also reflecting some MKT elements inside. Furthermore, the teacher also showed sort of spiral features of mathematics teaching.

Finally, the present study described the possible relationship between PUFM and MKT, which may provide some insights into the future study of both the tension and relationship between high school mathematics teacher’s PUFM and MKT. Hopefully, my study results may be used to increase the quality and usefulness of the MKT observational system at high school level, as well as to examine and develop professional knowledge of high school mathematics teachers.

目次………………………………………………………………………I
附錄目次………………………………………………………………III
圖目次……………………………………………………………………IV
表目次…………………………………………………………………….V

目次
第一章 緒論 …………………………………………………………1
第一節 研究背景和動機………………………………………………1
第二節 研究問題和研究目的…………………………………………5
第二章 文獻探討………………………………………………………7
第一節 數學教師的專業………………………………………………7
第二節 數學教師的教學相關知識…………………………………12
第三章 研究方法……………………………………………………35
第一節 研究的場域和參與者………………………………………35
第二節 質性取向的個案研究法……………………………………37
第三節 研究的設計…………………………………………………45
第四節 研究可能的限制……………………………………………72
第四章 研究結果……………………………………………………79
第一節 李師數學教學的觀點………………………………………79
第二節 前導階段研究………………………………………………81
第三節 第一階段研究………………………………………………106
第四節 第二階段研究………………………………………………122
第五節 跨階段綜合分析……………………………………………136
第五章 討論和建議…………………………………………………159
第一節 李師的MKT特徵…………………………………………159
第二節 接續研究的建議……………………………………………162

參考文獻……………………………………………………………….167

附錄目次
附錄一︰數學教學觀察系統…………………………………………………173
附錄一(1)︰LMT (2006)的MQI 系統....................................................................173
附錄一(2)︰本研究的數學教學觀察系統...............................................................181
附錄一(3)︰數學教學觀察系統登錄單...................................................................182
附錄一(4)︰數學教學觀察系統登錄單劃記範例...................................................185
附錄一(5)︰2009 年11 年6 日登錄結果..............................................................188
附錄一(6)︰2010 年4 月27 日登錄結果..............................................................191
附錄一(7)︰2010 年6 月2 日登錄結果................................................................194
附錄一(8)︰前導階段研究登錄結果總表...............................................................197
附錄一(9)︰第一階段研究登錄結果總表...............................................................199
附錄一(10)︰第二階段研究登錄結果總表.............................................................200
附錄二︰數學教學影片與訪談轉譯稿…………………………………….201
附錄二(1)︰2009 年10 月27 日教學影片轉譯稿................................................201
附錄二(2)︰2009 年11 月6 日教學影片轉譯稿..................................................210
附錄二(3)︰2010 年4 月27 日教學影片轉譯稿..................................................217
附錄二(4)︰2010 年6 月2 日教學影片轉譯稿....................................................225
附錄二(5)︰2009 年10 月29 日與11 月10 日課後訪談轉譯稿.......................233
附錄二(6)︰前導階段研究總結性訪談轉譯稿.......................................................236
附錄二(7)︰第一階段研究總結性訪談轉譯稿.......................................................242
附錄二(8)︰第二階段研究總結性訪談轉譯稿.......................................................247

中文部分︰
1. 林清山(1992)。心理與教育統計學。台北：東華書局。
2. 范良火(2003)。教師教學知識發展研究。上海市：華東師範大學出版社。
3. 金鈐(2009)。資深高中數學教師MKT的初探研究。台北市：國科會。
4. 許秀聰(2005)。一位資深高中數學教師重構教學概念的行動研究。國立台灣師範大學碩士論文，台北市。
5. 黃凱旻(2002)。一個輔導中學數學實習教師教學概念轉變的行動研究。國立台灣師範大學碩士論文，台北市。
6. 崔懷芝。量表信度的測量︰kappa統計量之簡介。查詢日期︰99年10月1日，檢自http://www2.cmu.edu.tw/~biostat/online/teaching_corner_011.pdf。
7. 饒見維(1996)。教師專業發展－理論與實際。台北市：五南。
8. Bogdan, R. C., ＆Biklen, S. K. (2001)。質性教育研究理論與方法(黃光雄主譯)。嘉義市︰濤石文化。(1998)
9. Strauss, A., ＆Corbin, J. (2001)。紮根理論研究方法(吳芝儀、廖梅花譯)。嘉義市︰濤石文化。(1998)
10. Yin, R. K. (2001)。個案研究法(尚榮安譯)。台北市︰弘智文化。(1994)

英文部分︰
1. An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in china and the U.S. Journal of Mathematics Teacher Education, 7, 145-172.
2. Ball, D. L. (1988). Knowledge And Reasoning in Mathematical Pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, Michigan. (博士論文)
3. Ball, D. L. (1989). Teaching mathematics for understanding: What do teachers need to know about the subject matter. National Center for Research on Teacher Education. East Lansing.
4. Ball, D. L. (1990a). The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal, 90(4), 449-466.
5. Ball, D. L., ＆Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teaching: Knowing and using mathematics. Multiple Perspectives on Mathematics Teaching and Learning (pp. 83-104). London：Ablex Publishing.
6. Ball, D. L., ＆Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical future. Paper presented on a keynote address at the 43rd Jahrestagung für Didaktik der Mathematik held in Oldenburg, Germany, March 1-4, 2009.
7. Ball, D. L., Lubienski, S. T., ＆Mewborn, D. S. (2001). Research on Teaching Mathematics: The Unsolved Problem of Teachers’ Mathematical Knowledge. In Virginia, Richardson (Ed.), Handbook of Research on Teaching Edition (4th ed.) ( pp. 433-456). Washington D.C. : American Educational Research Association.
8. Ball, D. L., Thames, M. H., Bass, H., Sleep, L., Lewis, J., ＆Phelps, G. (2009). A practice-based theory of mathematical knowledge for teaching. Psychology of Mathematics Education, 33, 1-98.
9. Ball, D. L., Thames, M. H., ＆Phelps, G. (2008). Content knowledge for teaching. What makes it special? Journal of Teacher Education, 59(5), 389-407.
10. Barnes, C. (2002). Standards reform in high-poverty schools: Managing conflict and building capacity. New York: Teachers College Press.
11. Begle, E. G. (1979). Critical variables in mathematics education: Findings from a survey of the empirical literature. Washington, DC: Mathematical Association of America and National Council of Teachers of Mathematics.
12. Brophy, J. E. (1991). Conclusion to advances in research on teaching, Vol. 2: Teachers’ knowledge of subject matter as it relates to their teaching practice. In J. E. Brophy (Ed.), Advances in research on teaching: Teachers’ subject matter knowledge and classroom instruction (pp. 347-362). Greenwich, CT: JAI Press.
13. Brown, J. S., Collins, A., ＆Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-42.
14. Carpenter, T. P., Fennema, E., Peterson, P. L., ＆Carey,D. A. (1988).Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 29-37.
15. Cochran, K. E., DeRuiter, J. A., ＆King, R. A. (1993). Pedagogical content knowledge: An integrated model for teacher preparation, Journal of Teacher Education, 44, 263-272.
16. Cohen, D. K., ＆Ball, D. L. (1999). Instruction, capacity, and improvement. CPRE Research Report Series (RR-043). Philadelpfia: Consortium for Policy Research in Education.
17. Cooney, T. J. (1994). Teacher education as an exercise in adaptation. In D. B. Aichele, ＆A. F. Coxford (Eds.), Profession development: 1994 yearbook (pp. 9-22). Reston, VA：National Council of Teachers of Mathematics.
18. Dewey, J. (1997). Democracy and education. New York: The Free Press. (Original work published 1916)
19. Dunkin, M. J., ＆Biddle, B. J. (1974). The Study of Teaching. New York: Holt, Rinehart and Winston, Inc.
20. Eisenberg, T. A. (1977). Begle revisited: Teacher knowledge and student achievement in algebra. Journal for Research in Mathematics Education, 8, 216-222.
21. Elbaz, F. (1983). Teacher thinking: A study of practical knowledge. New York: Nichols.
22. Etzioni, A. (Ed.). (1969). The semi-professions and their organization: Teachers, nurses, and social workers. New York: Free Press.
23. Even, R., ＆Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter. Educational Studies in Mathematics, 29, 1-20.
24. Fennema, E., ＆Franke, M. L. (1992). Teachers’ knowledge and its impact. Handbook of research on mathematics teaching and learning (pp. 147-164). New York：Macmillan.
25. Frick, T., ＆Semmel, M. I. (1978). Observer agreement and reliabilities of classroom observational measures. Review of Educational Research, 48(1), 157-184.
26. Gilbert, J. K., Boulter, C. J., ＆Elmer, R. (2000). Positioning models in science education and in design and technology education. In J. K. Gilbert, ＆C. J. Boulter (Eds.), Developing Models in Science Education (pp. 3-17). Dordrecht, The Netherlands: Kluwer Academic Publisher.
27. Gilbert, W., Hirst, L., ＆Clary, E. (1987). The NCA Workshop’s taxonomy of professional knowledge. In, Jones, D. W. (Ed.). Professional Knowledge Base: NCATE Approval. Fortieth Annual Report of the North Central Association Teacher Education Workshop (pp. 38-57). Flagstaff, AZ: University of North Arizona.
28. Goodland, J. I. (1984). A place called school. New York: McGraw-Hill.
29. Hansen, D. T. (2001). Teaching as a moral activity. In W. R. Houston (Ed.), Handbook of Research on Teacher Edition (4th ed.) (pp. 826-857). New York: Macmillan.
30. Herriott, R. E., ＆Firestone, W. A. (1983). Multisite qualitative policy research: Optimizing description and generalizability. Educational Research, 12, 14-19.
31. Hill, H. C., ＆Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s Mathematics Professional Development Institutes. Journal of Research in Mathematics Education, 35, 330-351.
32. Hill, H. C., Ball, D. L., ＆Schilling, S. G. (2008).Unpacking pedagogical content knowledge: conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education,39(4),372-400.
33. Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., ＆Ball, D. L. (2008).Mathematical knowledge for teaching and the mathematical quality of instruction: an exploratory study. Taylor ＆ Francies Group, 26, 430-511.
34. Hill, H. C., Rowan, B., ＆Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal ,42, 371-406.
35. Holmes Group, (1986). Tomorrow’s teachers. East Lansing: Author.
36. Lampert, M., ＆Ball, D. L. (1999). Aligning teacher education with contemporary K-12 reform visions. In G. Sykes, ＆L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 33-53). San Francisco: Jossey Bass.
37. Lappan, G., ＆Theule-Lubienski, S. (1994). Training teachers or educating professionals? What are the issues and how are they being resolved? In: Robitaille, D. F., Wheeler, D. H., ＆Kieran, C. (Eds.). Selected Lectures from the 7th International Congress on Mathematical Education. Sainte-Foy, Quebec: Les Presses de ’Universite Laval, 249-261.
38. Learning Mathematics for Teaching (LMT) Project, Learning Mathematics for Teaching (LMT) Project. Retrieved October 20, 2009 from http://sitemaker.
umich.edu/lmt/faq_about_video_codes
39. Leinhardt, G., Putnam, R. T., Stein, M. K., ＆Baxter, J. (1991). Where subject knowledge matters. In J. E. Brophy (Ed.), Advances in research on teaching: Teachers’ subject matter knowledge and classroom instruction (2), pp. 87-113. Greenwich, CT: JAI Press.
40. Leinhardt, G., ＆Smith, D. A. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77, 247-271.
41. Ma, L. (1996). Profound Understanding of Fundamental Mathematics: What is it ,why is it important, and how is it attained? Unpublished doctoral dissertation, Stanford University, Stanford. (博士論文)
42. Ma, L. (1999). Knowing and teaching elementary mathematics. New Jersey: Lawrence Erlbaum Associates.
43. Maccoby, E., ＆Maccoby, N. (1954). The interview: A tool of social science. In G. Lindzey (Ed.), Handbook of social psychology (vol. 1)(pp. 449-487). Cambridge, MA: Addison-Wesley.
44. McDiarmid, G. W., ＆Clevenger-Bright, M. (2008). Rethinking teacher capacity. Handbook of Research on Teacher Education Edition (3), pp.134-156.
45. McEwan, H., ＆Bull, B. (1991). The pedagogic nature of subject matter knowledge. American Educational Research Journal, 28, 316-334.
46. McIntyre, D. I. (1980). Systematic observation of classroom activities. Educational Analysis, 2(2), 3-30.
47. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
48. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
49. Noddings, N. (1992). Professionalization and mathematics teaching. In D. A. Grovws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 197-208). New York: Macmillan.
50. Patton, M. Q. (1987). How to use qualitative methods in evaluation. Newbury Park, CA: Sage.
51. Peterson, P. L. (1988). Teachers’ and students’ cognitional knowledge for classroom teaching and learning. Educational Researcher, 17(5), 5-14.
52. Quinton, A. (1967). Knowledge and belief. The Encyclopedia of Philosophy, 4, 345-352.
53. Rowland, T. (2008). Researching teachers’ mathematics disciplinary knowledge. In P. Sullivan, &T. Wood (Eds.), The international handbook of mathematics teacher education Vol. 1 (pp. 273-298). Rotterdam, The Netherlands: Sense Publishers.
54. Schwab, J. J. (1978). Educational and the structure of the disciplines. In I. Westbury, ＆N. Wilkof (Eds.), Science, curriculum, and liberal education (pp. 167-183). Chicago: University of Chicago. (Original work published 1961)
55. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
56. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.
57. Shulman, L. S. (2005). Signature pedagogies. Retrieved October 1, 2010, from http://www.hub.mspnet.org/index.cfm/11172.
58. Shulman, L. S. (2005).Teacher education does not exist. Retrieved October 1, 2010, from http://www.ed.stanford.edu/suse/news-bureau/educator-newsletter.html.
59. Smith, D. E. (1987). The everyday world as problematic: A feminist sociology. Boston: Northeastern University Press.
60. Sockett, H. T. (1987). Has Shulman got the strategy right? Harvard Education Review, 57, 208-219.
61. Steinberg, T., Haymore, J., ＆Marks, R. (1985). Teachers’ knowledge and structuring content in mathematics. Paper presented at the annual meeting of the American Educational Research Association, Chicago.
62. Stevens, C., & Wenner, G. (1996). Elementary preservice teachers’ knowledge and beliefs regarding science and mathematics. School Science and Mathematics. 96(1), 2-9.
63. Znaniecki, F. (1965). The social role of the man of knowledge. New York: Octagon Books, Inc.