本研究採用個案研究法，分析並比較兩位資深高中數學教師的教學專業知識。在Ball等（2008）所提出的教學用的數學知識（mathematical knowledge for teaching）的理論架構下，收集並分析兩個案教師各三階段的實徵資料。教學單元包括四領域的數學主題：李師的微積分、邱師的機率、李師和邱師的矩陣（進度教學單元）、李師和邱師的排列組合（複習教學單元）。

資料收集的主要方法是參與觀察和訪談，其中，教學影片和訪談錄音的內容都轉譯成逐字稿。為了在質性取向的研究裡加入量化分析，本研究使用教學觀察系統來分析參與觀察的教學影片資料，它主要是修改自LMT計畫（2007）所設計的「教學中的數學品質（mathematical quality of instruction）」的影片編碼詞彙表。作者在詮釋參與觀察和訪談的內容之前，會參考一些文本證據以提升本研究的品質。

　　作者將參與觀察的研究結果視為教師的顯性知識，而訪談的研究結果則視為教師的隱性知識。在不同的研究階段裡，兩位個案教師顯性知識和隱性知識有許多異同；在教師的顯性知識方面，雖然質性部分有許多相異之處，但是量化部分卻是非常相似的；而在教師的隱性知識方面，穩定的教學表象背後具有複雜多變的教學脈絡。

　　研究結果指出，專門的內容知識（specialized content knowledge）在高中數學教師教學專業知識裡佔有舉足輕重的地位，尤其是對教師專業發展而言。對於以教學工作為志業的高中數學教師來說，本研究在教學實務裡所描述的教學專業知識有許多發人省思之處。對於國內數學教師知識的研究而言，本研究結果指出了數學教學專業知識在不同面向的各種觀點。

This study applies case study research to analyze and compare the professional knowledge of teaching of two high school experienced mathematics teachers. On the
theory of mathematical knowledge for teaching presented by Ball et al.(2008), the author collected and analyzed empirical data in the three phases of each teacher.
These mathematical topics cover four areas: calculus by Mr. Li, probability by Mr.
Chiu, matrix by Mr. Li and Mr. Chiu, and permutation and combination by Mr. Li and
Mr. Chiu.
Major methods of the data collection were participant observation and interview,
and the content of instruction films and interview recording was translated into a
transcript. In order to acquire quantitative analysis in the qualitative-oriented study,
this study used observation system to analyze the data of instruction films of
participant observation. The observation system was modified from mathematical
quality of instruction video coding glossary, which was designed by LMT project
(2007). Before the author interpreted the content of participant observation and
interview, she referred to some text for evidence in order to enhance the quality of this
study.
While the research result of participant observation is regarded as explicit
knowledge of the teacher, the research result of interview is regarded as implicit
knowledge of the teacher. In the various research phases, there are many similarities
and dissimilarities between explicit knowledge and implicit knowledge of the teachers.
In the dimension of the teachers’ explicit knowledge, there are many differences on
the qualitative part, but they are similar on the quantitative part. However, in the
dimension of the teachers’ implicit knowledge, there are complex and changeful
contexts of instruction behind the presentation of stable instruction.
The research result implies that specialized content knowledge plays a vital role
in the professional knowledge of teaching of high school mathematics teachers,
especially in the professional development of teachers. For high school mathematics
teachers who have ambition to engage in teaching, there are many points to stimulate
deep thought in the professional knowledge of teaching described in the teaching
practice of our study. For investigating domestic mathematic teachers’ knowledge, the
result of this study points out the various views of the professional knowledge of
teaching mathematics in different aspects.

一、中文文獻
1.吳明隆（2010）。論文寫作與量化研究。臺北市：五南。
2.周祝瑛（2003）。誰捉弄了臺灣教改。臺北市：心理。
3.林清山（1976）。科學教育的心理學基礎。科學教育月刊（1，頁次：27-36；2，頁次：15-20）。臺北市 : 國立師範大學教育硏究中心。
4.林清山（1977）。數學課程設計和數學教學的理論基礎。科學教育月刊（11，頁次：10-20；12，頁次：4-10）。臺北市 : 國立師範大學教育硏究中心。
5.邱皓政（2008）。量化研究法(一)：研究設計與資料處理。臺北市：雙葉書廊。
6.邱皓政（2010）。量化研究與統計分析。臺北市：三民。
7.范良火（2002）。教師教學知識發展研究。上海：華東師範大學出版社。
8.陳亭瑋（2011）。資深高中數學教師教學知識與教學構思的個案研究。國立臺灣師範大學數學研究所碩士論文，臺北市。
9.曾名秀（2011）。資深高中數學教師教學相關知識的個案研究。國立臺灣師範大學數學研究所碩士論文，臺北市。
10.潘淑滿（2003）。質性研究：理論與應用。臺北市：心理。
11.Aron, A., Aron, E. N., & Coups, E. J. (2006)。黃瓊蓉、蘇文賢、江吟梓（譯）（2009）。心理與教育統計學（Statistics for psychology）。臺北市：學富文化。
12.Bogdan, R. C., & Biklen, S. K. (1998)。黃光雄（主譯）（2001）。質性教育研究（Qualitative research for education）。嘉義市：濤石文化。
13.Creswell, J. W. ( 2004)。張宇樑、吳樎椒（譯）（2011）。研究設計（Research design）。臺北市：學富文化。
14.Creswell, J. W., & Clark, V. L. P. (2007)。謝志偉、王慧玉（譯）（2010）。混合方法研究導論（Designing and conducting mixed methods research）。臺北市：心理。
15.Jorgensen, D. L. (1989)。王昭正、朱瑞淵（譯）（1999）。參與觀察法（Participant observation）。臺北市：弘智文化。
16.Kilpatrick, J. (1985)。黃敏晃（譯）（1988）。數學解題的教學：近25年來的回顧。數學傳播，12(2)，26-43。
17.Kvale, S. ( 2007)。陳育含（譯）（2010）。訪談研究法(Doing interviews)。臺北縣：韋伯文化。
18.Seidman, I. (2007)。李政賢（譯）（2009）。訪談研究法（Interviewing as qualitative research）。台北市：五南。
19.Strauss, A., & Corbin, J. ( 1998)。吳芝儀、廖梅花（譯）（2001）。紮根理論研究方法（Basics of qualitative research）。嘉義市：濤石文化。
20.Yin, R. K. ( 1994)。尚榮安（譯）（2001）。個案研究法（Case study research）。臺北市：弘智文化。

二、英文文獻
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(2), 145-172.
2.Andrews, P. (2011). The cultural location of teachers’ mathematical knowledge: another hidden variable in mathematics education research? In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 99-118). New York: Springer.
3.Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. In J. Boaler (Ed.), Multiple Perspectives on Mathematics Teaching and Learning (pp. 83-104). Westport, CT: Ablex.
4.Ball, D. L., Charalambous, C.Y., Thames, M., & Lewis, J.M. (2009). Teacher knowledge and teaching: viewing a complex relationship from three perspectives. Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 121-125). Thessaloniki, Greece: PME.
5.Ball, D. L., & Forzani, F. (2009). The work of teaching and the challenge for teacher education. Journal of Teacher Education, 60(5), 497-511.
6.Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: the unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (4th ed.) (pp. 433-455). New York: Macmillan.
7.Ball, D. L., Thames, M. H., Bass, H., Sleep, L., Lewis, J., ＆Phelps, G. (2009). A practice-based theory of mathematical knowledge for teaching. Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 95-98). Thessaloniki, Greece: PME.
8.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.
9.Bednarz, N., & Proulx, J. (2009). Knowing and using mathematics in teaching: conceptual and epistemological clarifications. For the Learning of Mathematics, 29(3), 11-17.
10.Bell, C. A., Wilson, S. M., Higgins, T., & McCoach, D. B. (2010). Measuring the effects of professional development on teacher knowledge: the case of developing mathematical ideas. Journal for Research in Mathematics Education, 41(5), 479-512.
11.Chinnappan, M., & Lawson, M. J. (2005). A framework for analysis of teachers’ geometric content knowledge and geometric knowledge for teaching. Journal of Mathematics Teacher Education, 8(3), 197-221.
12.Cockbum, A. D. (2008). Assessment of mathematical knowledge of prospective teachers. In P. Sullivan & T. Wood (Eds.), The International Handbook of Mathematics Teacher Education (Vol. 1) (pp. 247-272). Rotterdam, The Netherlands: Sense.
13.Davis, J. D. (2009). Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12(5), 365-389.
14.Davis, B. & Renert, M. (2009). Mathematics-for-teaching as shared dynamic participation. For the Learning of Mathematics, 29(3), 37-43.
15.Davis, B. & Simmt, E. (2006). Mathematics-for-teaching: an ongoing investigation of the mathematics that teachers (need to) know. Education Studies in Mathematics, 61(3), 293-319.
16.Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 11(3), 171-197.
17.Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Education Studies in Mathematics, 21(6), 521-544.
18.Even, R., & Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter. Education Studies in Mathematics, 29(1), 1-20.
19.Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 147-164). New York: Macmillan.
20.Figueiras, L., Ribeiro, M., Carrillo, J., Fernández, S., & Deulofeu, J. (2011). Teachers’ advanced mathematical knowledge for solving mathematics teaching challenges: a response to Zazkis and Mamolo. For the Learning of Mathematics, 31(3), 26-28.
21.Foster, C. (2011). Peripheral mathematical knowledge. For the Learning of Mathematics, 31(3), 24-26.
22.Frick, T., & Semmel, M. I. (1978). Observer agreement and reliabilities of classroom observational measures. Review of educational research, 48(1), 157-184.
23.Gellert, U. (2000). Mathematics instruction in safe space: prospective elementary teachers’ views of mathematics education. Journal of Mathematics Teacher Education, 3(3), 251-270.
24.Graeber, A., & Tirosh, D. (2008). Pedagogical content knowledge: useful concept or elusive notion. In P. Sullivan & T. Wood (Eds.), The International Handbook of Mathematics Teacher Education (Vol. 1) (pp. 117-132). Rotterdam, The Netherlands: Sense.
25.Hill, H. C. (2010). The nature and predictors of elementary teachers’ mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41(5), 513-545.
26.Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: results from California’s mathematics professional development institutes. Journal for Research in Mathematics Education, 35(5), 330-351.
27.Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: conceptualizing and measuring teachers’ topic-specific knowledge of student. Journal for Research in Mathematics Education, 39 (4), 372-400.
28.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. Cognition and Instruction, 26, 430-511.
29.Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers' mathematical knowledge: what knowledge matters and what evidence counts? In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 111-155). Charlotte, NC: Information Age Publishing.
30.Howson, G. (1996). Mathematics and common sense. In C. Asina, J. M. Alvarez, B. Hodgson, C. Laborde, & A. Perezi (Eds.), Proceedings of the Eighth International Congress on Mathematics Education (pp. 257-269). Sevilla, Spain.
31.Huillet, D. (2009). Mathematics for teaching: an anthropological approach and its use in teacher training. For the Learning of Mathematics, 29(3), 4-10.
32.Isiksa, M., & Cakiroglu, E. (2011). The nature of prospective mathematics teachers’ pedagogical content knowledge: the case of multiplication of fractions. Journal of Mathematics Teacher Education, 14(3), 213-230.
33.Kilpatrick, J. (1987). What constructivism might be in mathematics education. Paper present in the 11th annual meeting of the International Group for the Psychology of Mathematics Education. Montreal, Canada: PME.
34.Learning Mathematics for Teaching (LMT) Project (2006). A coding rubric for measuring the mathematical quality of instruction. Retrieved September 17, 2011, from http://sitemaker.umich.edu/lmt/files/lmt-mqi_description_of_codes.pdf
35.Learning Mathematics for Teaching (LMT) Project (2007). Mathematical quality of instruction video coding glossary. Retrieved September 17, 2011, from http://sitemaker.umich.edu/lmt/files/lmt-mqi_glossary_1.pdf
36.Learning Mathematics for Teaching (LMT) Project (2011). Measuring the mathematical quality of instruction. Journal of Mathematics Teacher Education, 14(1), 25-47.
37.Ma, L. (1999). Knowing and Teaching Elementary Mathematics. New Jersey: Lawrence Erlbaum Associates.
38.Petrou, M., & Goulding, M. (2011). Conceptualizing teachers’ mathematical knowledge in teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 9-26). New York: Springer.
39.Piaget, J. (1975). Comments on mathematical education. Contemporary Education, 47(1), 5-10.
40.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.
41.Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: the knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.
42.Ruthven, K. (2011). Conceptualizing mathematical knowledge in teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 83-96). New York: Springer.
43.Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
44.Shulman, L. S. (1987). Knowledge and teaching: foundations of the new reform. Harvard Educational Review, 57(1), 1-22.
45.Sim, J., & Wright, C. C. (2005). The kappa statistic in reliability studies: use, interpretation, and sample size requirements. Physical therapy, 85(3), 257-268.
46.Simon, M. A. (2006). Key developmental understandings in mathematics: a direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359-371.
47.Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307-332.
48.Sveiby, K. E. (1997). The new organizational wealth. Retrieved April 6, 2012, from http://ptarpp2.uitm.edu.my/silibus/neworgan.pdf
49.Tamir, P. (1991). Professional and personal knowledge of teacher and teacher education. Teaching and Teacher education, 7(3), 263-268.
50.Turner, F., & Rowland, T. (2011). The knowledge quartet as an organizing framework for developing and deepening teachers’ mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical Knowledge in Teaching (pp. 195-212). New York: Springer.
51.Vale, C., McAndrew, A., & Krishnan, S. (2011). Connecting with the horizon: developing teachers’ appreciation of mathematical structure. Journal of Mathematics Teacher Education, 14(3), 193-212.
52.Watkins, M. W., & Pacheco, M. (2000). Interobserver agreement in behavioral research: importance and calculation. Journal of behavioral education, 10(4), 205-212.
53.Wise, A. E. (2005). Establishing teaching as profession: the essential role of professional accreditation. Journal of Teacher Education, 56(4), 318-331.
54.Zazkis, R., & Mamolo, A. (2011). Reconceptualizing knowledge at the mathematical horizon. For the Learning of Mathematics, 31(2), 8-13.
55.Zazkis, R. & Zazkis, D. (2011). The significance of mathematical knowledge in teaching elementary methods courses: perspectives of mathematics teacher educators. Education Studies in Mathematics, 76(3), 247-263.