本研究是以國立台灣師範大學數學系教材教法的課程為場景，透過問卷調查法，從其中一班38位修課學生中篩選出8位個案，進行為期一年的個案追蹤研究。本研究採用個案研究法，搭配問卷、訪談與教室觀察的方式，探討相關學生教師在學習教數學的第一年裡，數學教學知識的內涵與發展。透過分析數學層次和教學層次的知識，瞭解學生教師理解無限概念的深度、廣度與完全性，以及教學使用的問話、類比與操作性活動的表徵方式，並以Wenger (1998) 的實務社群(community of practice)理論作為個案數學教學知識發展的脈絡。

研究結果顯示，個案學生教師的模擬教學與理想中的教學有相當的落差，他們有不同層次的迷失概念，可以進一步區分成「數學層次的迷思」、「教學層次的迷思」與「數學與教學層次的迷思」。此外，個案對於自己教學問題的察覺與反思，呈現出「沒察覺沒反思」、「有察覺沒反思」與「有察覺有反思」三種狀態。由於學生教師普遍無法精確掌握無限數學概念的內涵，因此，個人認為「精確性」應該可以作為「對基礎數學有深刻理解」(Ma, 1999)的第四個面向。培育活動中採用的案例教學，也對部分個案產生一定程度的影響，而學習型實務社群的功能雖然有部分的限制，但是仍有助於瞭解個案數學教學知識發展的脈絡。

這個階段的研究結果反應出學生教師在學習教數學時的個別差異，接續的研究可以從瞭解迷思教學概念產生的原因、如何克服與修正這些迷思概念以及如何進一步促使學生教師產生更深層的察覺與反思等幾個方向著手。最終，希望所培育出的中學數學教師，不僅對數學內容有深刻的理解，也能適切地將其轉換成對於不同能力與背景的學生有意義、有效且合適的理解形式(Shulman, 1987)。

The research is one-year case study on the students who participate the method course for secondary mathematics. Eight student cases from the thirty-eight participants were selected in terms of question survey, and followed by interview and classroom observation to understand the content and development of the student teachers’ pedagogical knowledge in mathematics in the process of learning to teach mathematics. By analyzing mathematical and pedagogical knowledge, the author tries to investigate the depth, broad and thoroughness of the concept of infinity of the eight student teachers, and the representation of teacher questioning, analogy and manipulatives used in microteaching. The theoretical context for situating their knowledge development is based on Wenger’s theory of community of practice.

The results show that there were distinct differences between microteaching and ideal teaching among cases. They have misconceptions of mathematic and pedagogy. The eight cases also reflect three states of awareness, including no awareness and no reflection, awareness but no reflection, or awareness and reflection. Due to most of the student teachers could not understand fully the mathematical concept of infinity, the author adds a fourth element “accurateness” to the Ma’s notion of profound understanding of fundamental mathematics. The method course of the teacher education program seems to have potential influences on the cases, while the theory of learning community supports those cases to develop their pedagogical knowledge in mathematics.

The study results reflects also some individual differences among the student cases when they are learning to teach, and the future research is expected to understand the causes and adjustments of the misconceptions, also to stimulate deeply the reflection and awareness of student teachers. Finally, the author also expects that the well–trained secondary mathematics teachers can not only understand profoundly mathematical content but also transform the content knowledge into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by the students (Shulman, 1987)

一、中文部分

王幼君和金之明(2000)。著名的數學家和他的一個重大發現。新竹市：凡異。
王安蘭(2005)。一個重構高中生機率概念的行動研究。國立台灣師範大學碩士論文(未出版)。
王惠中(2003)。青少年無限概念發展研究(2/2) 。行政院國家科學委員會補助專題研究計劃成果報告。國立台灣師範大學數學系。
朱綺鴻和譚克平(2000)。現職教師對數教導數學歸納法意見初探。科學教育月刊，232，2-15。
何福田和羅瑞玉(1992)。教育改革與教師專業化。載於中華民國師範教育主編：教育專業(pp. 1-30)。台北市：師大書苑。
李源順(1999)。數學教師在校內互動促進自我專業發展的個案研究。國立台灣師範大學博士論文(未出版)。
沈亞梵(1995)。師資培育多元化與教師品管之研究。台北市：師大書苑。
林恆理(2003)。中學數學科實習教師的教學知識研究。行政院國家科學委員會補助大專學生專題研究計劃成果報告。國立台灣師範大學數學系。
林碧珍(2001)。發展國小教師之學生數學認知知識：理論結合實務研究取向的教師專業發展。台北市︰師大書苑。
林福來(1998)。教師思維的發展：整合數學教學知識的教材教法。行政院國家科學委員會補助專題研究計劃成果報告。國立台灣師範大學數學系。
林福來、郭汾派和林光賢(1995)。比例推理與錯誤診斷補救。行政院國家科學委員會補助專題研究計劃成果報告。國立台灣師範大學數學系。
林曉雯(1994)。國中生物教師教學表徵的詮釋性研究。國立台灣師範大學博士論文(未出版)。
林靜雯和邱美虹(2005)。整合類比與多重表徵研究取向探索多重類比設計對兒童電學概念學習之影響。科學教育學刊，13(3)，317-345。
金鈐(2005)。職前中學數學教師教學信念和價值的評量研究(2/3)。行政院國家科學委員會補助專題研究計劃期中進度報告。國立台灣師範大學數學系。
范良火(2003)。教師教學知識發展研究。上海市：華東師範大學。
張民杰(2001)。案例教學法：理論與實務。台北市︰五南。
張玉成(1993)。思考技巧與教學。台北市：心理。
張春興(1996)。教育心理學(二版)。台北市：東華書局。
張繼元(2005)。六位學生教師教學認同的個案研究。國立台灣師範大學碩士論文(未出版)。
教育部(2003)。國民中小學九年一貫課程綱要數學學習領域。台北市：教育部。
教育部(2003)。教師法。台北市：教育部。
教育部(2005)。師資培育法。台北市：教育部。
教育部(2004)。普通高級中學課程暫行綱要。台北市：教育部。
教育部國語推行委員會(1998)。國語辭典。台北市：教育部。
曹亮吉(1996)。阿草的葫蘆(下)─文化活動中的數學。台北市：遠哲基金會。
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(未出版)。
陳松靖(2002)。三位學生教師數學教學概念轉變歷程的個案研究。國立台灣師範大學碩士論文(未出版)。
陳長城(2003)。數學史話。台北市：台灣東華。
陳恆迪(1993)。國中學生物理概念類比學習之研究。國立彰化師範大學碩士論文(未出版)。
蔡崑霖(2000)。牛津當代大辭典(四版)。台北市：旺文社。
鄭英豪(2000)。實習教師數學教學概念的學習：以「概念啟蒙例」的教學概念為例。國立台灣師範大學博士論文(未出版)。
鄭智馨(2000)。個案研究─影響教師類比選用之因素以及教師的類比對學生學習之影響。國立台灣師範大學碩士論文(未出版)。
盧增緒(1973)。教育人員。載於田培林：教育學新論(pp. 349-383)。台北市：文景。
繆龍驥(1992)。微積分中極限概念之診斷教學。科學教育學術研討會。高雄市：國立高雄師範大學。
饒見維(2003)。教師專業發展─理論與實務(二版)。台北市：五南圖書。
Berklinski, D. (1997). A tour of the calculus. New York, Random House. [陳雅茜譯(2000)。微積分之旅。台北市：天下遠見。]
Blatner, D. (1997). The joy ofπ. New York: Walker and company. [潘恩典譯(1999)。神奇的π。台北市：商業周刊]
Bogdan, R. C., & Biklen S. K. (1998). Qualitative research for education: An introduction to theory and method (3rd ed.). Boston: Allyn & Bacon. [黃光雄主譯(2003)。質性教育研究理論與方法。嘉義市：濤石文化。]
Ginsburg, H. P. (1997). The clinical interview in psychological research and practice. Cambridge: Cambridge University Press. [謝如山譯(2004)。進入兒童心中的世界。台北市：五南圖書。]
Maxwell, J. A. (1996). Qualitative research design: An interactive approach. Thousand Oaks: Sage. [高熏芳、林盈助和王向葵合譯(2001)。質化研究設計：一種互動取向的方法。台北市：心理。]
Robert, K. Y. (1994). Case study research design and methods. Thousand Oaks: Sage. [尚榮安譯(2001)。個案研究法。台北市：弘智文化。]
Skemp, R. R. (1987). The psychology of learning mathematics. Hillsdale, Lawrence Erlbaum Associates. [陳澤民譯(1995)。數學學習心理學。台北市：九章。]
Skemp, R. R. (1989). Mathematics in the primary school. London: Routledge. [許國輝譯(1995)。小學數學教育─智性學習。香港：公開進修學院。]
Strauss, A., & Corbin, J. (1998). Basis of qualitative research: techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks: Sage. [吳芝儀和廖梅花合譯(2003)。質性研究入門─紮根理論研究方法。嘉義市：濤石文化。]
Wenger, E., McDermott, R., & Snyder, W. M. (2002).Cultivating communities of practice: A guide to managing knowledge. Boston: Harvard Business School Press. [黃維譯(2003)。實踐社群：推動學習型組織之輪。台北市：天下遠見。]
Zippin, L. (1962). Uses of infinity. New York: Random House. [應隆安譯(1999)。無限的用處。新竹市：凡異。]

二、英文部分

Artzt, A., & Thomas, E. (1999). A cognitive model for examining teachers’ instructional practice in mathematics: A guide for facilitating teacher reflection. Educational Studies in Mathematics, 40, 211-235.
Ball, D. (1989). Teaching mathematics for understanding: What do teachers need to know about the subject matter. East Lansing: National Center for Research on Teacher Education.
Ball, D. L. (1988). Unlearning to teach mathematics (Issue Paper 88-1). Ease Lansing: Michigan State University, National Center for Research on Teacher Education.
Ball, D. L. (1991). Research on teaching mathematics: Making subject matter knowledge part of the equation. In J. E. Brophy (Ed.), Ad¬vances in research on teaching: Teachers' subject matter knowledge and classroom instruction (Vol. 2, pp. 1-48). Greenwich: JAI Press.
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). London: Ablex Publishing.
Ball, D. L., Lubienski S. T., & Mewborn D. S. (2001). Research on teaching mathematics: The unsolved problem of teachings' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.) (pp. 433-455). Washington D.C.: AERA.
Berliner, D. (1988). The development of expertise in pedagogy. (ERIC Document Reproduction Service No. ED 298122)
Boaler, J. (1998). Open and closed mathematics: Student experiences and understands. Journal for Research in Mathematics Education, 29(1), 41-62.
Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40, 259-281.
Bromme, R. (1994). Beyond subject matter: A psychological topology of teachers’ professional knowledge. In R. Biehler, R. Scholz, R. Strasser ,& B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 73-88). Dordrecht: Kluwer Academic Publishers.
Brousseau, G. (1986). Fondements et metheods de la didactique des mathematiques. Recherches en didactique des mathematiques, 7, 33-115.
Brown, C. A., & Borko, H. (1992). Becoming a mathematics teacher. In A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 209-239). New York: Macmillan.
California Department of Education. (2000). Mathematics framework for California public schools, kindergarten though grade twelve. California: Author.
Cobb, P., Perlwitz, M., & Underwood, D. (1996). Constructivism and activity theory: A consideration of their similarities and differences as they relate to mathematic education. In C. S. Mansfield, N. A. Peteman, & N. Bednarz (Eds.), Mathematics for tomorrow’s young children (pp. 10-58). Dordrecht: Kluwer Academic Publishers.
Cooney, T. J. (1994). Teacher education as an exercise in adaptation. In D. B. Aichele & A. F. Coxford (Eds.), Professional development for teachers of mathematics (pp. 9-22). Reston: NCTM.
Cooney, T. J. (2001). Considering the paradoxes, perils, and purposes of conceptualizing teacher development. In F. L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 9-31). Dordrecht: Kluwer Academic Publishers.
Cooney, T. J., & Wiegel, H. G. (2003). Examining the mathematics in mathematics teacher education. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), Second international handbooks of mathematics education (pp. 795-828). Dordrecht: Kluwer Academic Publishers.
Cooney, T. J., Wilson, P. S., Albright, M., & Chauvot, J. (1998, April). Conceptualizing the professonal development of secondary preservice mathematics teachers. Paper presented at the American Educational Research Association annual meeting , San Diego, CA.
Cooney, T. J., Shealy, B. E., & Arvold, B. (1998). Conceptualizing belief structures of pre-service secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306-333.
Dale, E. A. (1969). Audiovisual method in teaching. New York: Holt Rinehart and Winston.
Department of Education and Employment. (1998). Initial teacher training national curriculum for secondary mathematics. London: Author.
Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Lexington: D.C. Heath.
Duit, R. (1991). On the role of analogies and metaphors in learning science. Science Education, 75(6), 649-672.
Ernest, P. (1984). Mathematical induction: A pedagogical discussion. Educational Studies in Mathematics, 15, 173-189.
Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). New York: Macmillan.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: Reidel.
Fischbein, E., Tirosh, D., & Hess, P. (1979). The infinity. Educational Studies in Mathematics, 10, 3-40.
Fischbein, E., Tirosh, D., & Melamed, U. (1981). Is it possible to measure the intuitive acceptance of a mathematical statement? Educational Studies in Mathematics, 12, 491-512.
Foss, D. H., & Kleinsasser, R. C. (1996). Preservice elementary teachers’ views of pedagogical and mathematics content knowledge. Teaching and Teacher Education, 12(4), 429-442.
Gardner, H. (1991). The unschooled mind: How children think and how schools should teach. New York: Basicbooks.
Gentner, D. (1989). The mechanisms of analogical learning. In S. Vosoniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 199-241). London: Cambridge University Press.
Goffree, F., & Dolk, M. (1995). Standards for primary mathematics teacher education. Utrecht: SLO/NVORWO.
Graven, M. (2003). Teacher learning as changing meaning, practice, community, identity and confidence: The story of Ivan. For the Learning of Mathematics, 23(2), 42-47.
Green, T. F. (1971). The activities of teaching. New York: McGraw-Hill.
Grossman, P., Wilson, S., & Shulman, L. (1989). Teacher of substance: Subject matter of knowledge for teaching. In M. C. Reynolds (Ed.), Knowledge base for the beginning teacher (pp. 23-34). Oxford: Pergamon Press.
Gudmundsdottir, S. (1990). Values in pedagogical content knowledge. Journal of Teacher Education, 41(3), 44-52.
Guskey, T. R. (1986). Staff development and the process of teacher change. Educational Researcher, 15(2), 5-12.
Hashweh, M. (1986). Toward and explanation of conceptual change. European Journal of Science Education, 8(3), 229-249.
Herriott, R. E., & Firestone. W. A. (1983). Multisite qualitative policy research: Optimizing description and generalizability. Educational Researcher, 12, 14-19.
Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81-116.
Hyman, R. T. (1979). Strategic questioning. Englewood Cliffs: Prentice-Hall.
Jaworski, B., & Waston, A. (1994). Mentoring in mathematics teaching: Mentoring co-mentoring, and the inner mentor. London: The Falmer Press.
Lerman, S. (1983). Problem solving of knowledge centered: The influence of philosophy on mathematics teaching. International Journal of Mathematical education in Science and Technology, 14(1), 59-61.
Lerman, S. (2001). A review of research perspectives on mathematics teacher education. In F. L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 33-52). Dordrecht: Kluwer Academic Publishers.
Lesh, R., Post, T., & Behr M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale: Lawrence Erlbaum Associates.
Leung, F.K.S. (2001). In search of an East Asian identity in mathematics education. Educational Studies in Mathematics, 47, 35-51.
Lin, P. J. (2005). Using research-based video-cases to help preservice primary teachers conceptualize a contemporary view of mathematics teaching. International Journal of Science and Mathematics Education, 3, 351-377.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Lawrence Erlbaum Publishers.
Maor, E. (1991). To infinity and beyond: A cultural history of the infinite. Princeton, Princeton University Press.
Marsh, L. G., & Cooke, N. L. (1996). The effects of using manipulatives in teaching math problem solving to students with learning disabilities. Learning Disabilities Research and Practice, 11(1), 58-65.
Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243-267
Mastrilli, T. M. (1997). Instructional analogies used by Biology teachers: Implications for practice and teacher preparation. Journal of Science Teacher Education, 8(3), 187~204.
McIntyre, D. (1993). Theory, theorizing and reflection in initial teacher education. In J. Calderhead & P. Gates (Eds.), Conceptual reflection in teacher development (pp. 39-52). London: Flamer Press.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47(2), 175-197.
Moyer, P. S., & Jones, M. G. (2004). Controlling choice: Teachers, students, and manipulatives in mathematics classrooms. School Science and Mathematics, 104(1), 16-31.
National Council of Teachers Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston: Author.
National Council of Teachers Mathematics. (1991). Professional standards for teaching mathematics. Reston: Author.
National Research Council (2001). Adding it up: Helping children learn mathematics. 2006年5月12日，取自http://darwin.nap.edu/books/0309069955/html/5.html
Niss, M. (2002). Mathematical competencies and the learning of mathematics: The Danish kom project. 2006年5月12日，取自http://www7.nationalacademies.org/mseb/Mathematical_Competencies_and_the_Learning_of_Mathematics.pdf
Noddings, N. (1992). Professionalization and mathematics teaching. In D. A. Grovws (Ed.), Handbook of research on mathematics teaching and learning (pp. 209-239). New York: Macmillan.
Novak, J. D. (1988). Learning science and the science of learning. Studies in Science Education, 15, 77-101.
Schön, D. A. (1983). The reflective practitioners: How professionals think in action. New York: Basic Books.
Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. J. Wilkof (Eds.), Science, curriculum, and liberal education. Chicago: University of Chicago Press.
Senger, E. S. (1999). Reflective reform in mathematics: The recursive nature of teacher change. Educational Studies in Mathematics, 37, 199-221.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Research, 15(2), 4-14.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-23.
Simon, M. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics, 26, 71-94.
Streefland, L. (1991). Realistic mathematics education in primary school. Utrecht: Freudenthal Institute.
Stevens, C., & Wenner, G. (1996). Elementary preservice teachers’ knowledge and beliefs regarding science and mathematics. School Science and Mathematics. 96(1), 2-9.
Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of limit of a function. Journal for Research in Mathematics Education, 31(3), 258-276.
Tall, D. O. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 271-284
Tall, D. O. (1992). The transition to advanced mathematical thinking: functions, limits, infinity and proof. In A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495-511). New York: Macmillan.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In A. Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 127-146). New York: Macmillan.
Tirosh, D. (1991). The role of students’ intuitions of infinity in teaching the Cantorian theory. In D. Tall (Ed), Adnvanced mathematical thinking (pp. 199-214). Dordrecht: Kluwer Academic Publishers.
Tirosh, D., & Graeber, A. O. (2003). Challenging and Changing Mathematics Teaching Classroom Practices. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), Second international handbooks of mathematics education (pp. 643-687). Dordrecht: Kluwer Academic Publishers.
Tsamir, P., & Tirosh, D. (1994). Comparing infinite sets: Intuition and representations. In Proceedings of the 18th annual conference of the international group for the Psychology of Mathematics Education, 345-352, Portugal, London.
Tzur, R. (2001). Becoming mathematics teacher-educators: Conceptualizing the terrain through self-reflective analysis. Journal of Mathematics Teacher Education, 4, 259-283.
Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge: Cambridge University Press.
Wilson, P. S., Cooney, T. J., & Stinson, D. W. (2005). What constitutes good mathematics teaching and how it develops: Nine high school teachers’ perspectives. Journal of Mathematics Teacher Education, 8(2), 83-111.
Wood, K. D. (1982). The differential effects of interspersed questions on learning form text. Doctoral Dissertation. Athens: University of Georgia.
Zaslavsky, O., Chapman, O., & Leikin, R. (2003). Professional development of mathematics educators: Trends and tasks. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), Second international handbooks of mathematics education (pp. 877-917). Dordrecht: Kluwer Academic Publishers.