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研究生:林培棠
研究生(外文):Pai-Tang Lin
論文名稱:兩位資深高中數學教師專門內容知識之嵌入式設計的混合方法研究
論文名稱(外文):The embedded design of mixed-methods research on two experienced senior high school mathematics teachers' specialized content knowledge
指導教授:金鈐金鈐引用關係
指導教授(外文):Chien Chin
學位類別:碩士
校院名稱:國立臺灣師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2011
畢業學年度:100
語文別:中文
論文頁數:305
中文關鍵詞:質性研究個案研究混合方法研究MQIMKTSCK
外文關鍵詞:Qualitative studyCase studyMixed-methods researchMQIMKTSCK
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本研究結合質性取向的個案研究與錄影分析的量化數據,形成一個嵌入式設計的混合研究(embedded design of mixed-methods research),用以探究兩位資深高中數學教師(林師與吳師)專門內容知識(specialized content knowledge)可能的內涵與特質以及它與內容與教學的知識(knowledge of content and teaching,簡稱 KCT)、內容與學生的知識(knowledge of content and student ,簡稱 KCS)間的關係。在為期一年的研究中,作者進入兩位個案教師的教學現場,透過課堂教學觀察與訪談,探索兩位個案教師和學生之間的互動與SCK呈現的情形。在錄影分析系統的部分,則是引用Learning Mathematics to Teaching (2006)所發展的Mathematical Quality of Instruction (MQI)登錄系統。個人首先修改系統的編碼,以符合兩位個案教師實際的數學教學特質,接著,進行教學影片分析,最後,商請另一位獨立登錄者協助信度的檢測。依據蒐集準則的質性與量化資料,並借助Ball, Thames與Phelps (2008)的MKT架構,本文描述兩位資深高中數學教師SCK可能樣貌以及它與KCT、KCS間的關係。
本研究結果顯示,兩位個案教師除了具有MKT原始定義的SCK特性外,也顯現其他SCK的內涵與特質,例如含有近似於HCK的特徵。其次,某些事件中教學的「不確定性(uncertainties)」會喚起林師即興的(improvisational)SCK,它的顯現與林師具有的數學知識相關,也反應了這些教學事件「不確定性」的程度。此外,兩位個案教師的SCK也會影響其教學的安排與教學的評價(亦即KCT),是影響他們教學決策的原因之一,而KCS也會影響兩位個案教師SCK呈現的方式與時機。
最後,根據研究結果,本文指出即興SCK與「不確定性」的關係,可以作為未來進一步探究高中數學教學中的「不確定性」。希望,本研究的結果可以用來幫助在職高中數學教師,進一步了解自己在教學中所需數學知識的內涵與影響的因素,以發展高中數學教師的SCK。

This study combines qualitative case study data with quantitative video analysis as an embedded design of mixed-methods research to explore two experienced senior high school mathematics teachers’ specialized content knowledge (SCK). Using systematic classroom observations and the follow-up interviews, this research explores the explicit and implicit aspects of the SCK that the two teacher cases reveal. For encoding the videos, I used Mathematical Quality of Instruction (MQI) developed by Learning Mathematics to Teaching (2006). First, I modified the MQI coding system to adapt to the classroom teaching of two cases. Second, I analyzed the video tapes by using the adapted codes. Last, the coding results were mostly supported from another independent coder to establish acceptable inter-coder reliability. The study results properly describe two teacher cases’ SCK as well as its relationship with their KCT and KCS.
The results also show that the two teachers not only have the characteristics of SCK of the original definition in MKT study, but also show other SCK types, for example it also embodied some characteristics similar to HCK. Moreover, "uncertainties” in teaching will evoke some improvisational aspects of SCK. In addition, the two teachers’ SCK clearly affected the arrangement of their teaching and evaluation of teaching (i.e. KCT). KCS also affected the manner and timing of their SCK.
As a whole, the research results of the present study clearly point out the inherent relationship between the "uncertainties" of classroom teaching and the improvisational aspects of SCK. It is assumed that the results of this study might be used to help in-service high school mathematics teachers to understand more about the required mathematical knowledge in teaching and to develop their own SCK.

目次......................................................I
附錄目次……………………………………………………………III
圖目次……………………………………………………………………V
表目次……………………………………………………………………VII
第一章 緒論…………………………………………………………1

第一節 研究背景與動機……………………………………………………………..1
第二節 研究問題與目的……………………………………………………………..6

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…………………………………………………..9

第三章 研究方法………………………………………………………29
第一節 研究的參與者和場域………………………………………………………29
第二節 嵌入式設計的混合方法研究………………………………………………32
第三節 研究設計……………………………………………………………………44
第四節 研究限制……………………………………………………………………77

第四章 研究結果………………………………………………………81
第一節 林師前導階段的研究結果…………………………………………………81
第二節 林師第一階段的研究結果………………………………………………..112
第三節 林師第二階段的研究結果………………………………………………..130
第四節 吳師三階段的研究結果…………………………………………………..153
第五節 兩個案的整理與對照……………………………………………………..177

第五章 討論與建議…………………………………………………..184
第一節 林師與吳師SCK的討論…………………………………………………184
第二節 接續研究的建議…………………………………………………………..192

附註……………………………………………………………………195
參考文獻………………………………………………………………197

附錄目次

附錄一:教學影片與訪談轉譯稿……………………………………204
附錄一(1):林師2010年10月13日教學影片轉譯…………….…….…………....204
附錄一(2):林師2011年02月19日教學影片轉譯……………….….……………214
附錄一(3):林師2011年05月25日教學影片轉譯…………………..……………224
附錄一(4):吳師2011年02月14日教學影片轉譯………………….….…………236
附錄一(5):林師2011年01月28日前導階段總結性訪談轉譯……….….………245
附錄一(6):林師2011年09月02日第一階段總結性訪談轉譯………….….……248
附錄一(7):林師2011年09月19日第二階段總結性訪談轉譯………….….……251
附錄一(8):吳師2011年08月04日前導階段總結性訪談轉譯………….….……255

附錄二:錄影分析系統資料…………………………………………..260
附錄二(1):LMT(2006)的MQI系統…………………………………….…..…......260
附錄二(2):本研究的錄影分析系統與MQI系統的比較…………….……..……268
附錄二(3):錄影分析系統登錄單………………………..…………….…..………272
附錄二(4):錄影分析系統登錄單劃記範例……………..…………….…..………275
附錄二(5):林師前導階段登錄結果總表……………..……………….…..………278
附錄二(6):林師第一階段登錄結果總表……………..……………….…..………281
附錄二(7):林師第二階段登錄結果總表……………..……………….…..………284
附錄二(8):吳師前導階段登錄結果總表……………..……………….…..………287
附錄二(9):吳師第一階段登錄結果總表……………..……………….…..………290
附錄二(10):吳師第二階段登錄結果總表……………..………….….…..………293

附錄三:相關參考資料影本…………………………………………..296
附錄三(1):Lagrange插值多項式的引入(康熹文化)…………………..…..…......296
附錄三(2):Lagrange插值多項式的引入(龍騰文化)…………………..…..…......299
附錄三(3):Lagrange插值多項式的引入(南一書局)…………………..…..…......301
附錄三(4):後退的數學歸納法(徐道寧,1980)………………………..…..…......304

圖目次

圖2.1:發展於脈絡的教師知識(引自 Fennema &; Franke, 1992, p. 162)……..…..14
圖2.2:對主題概念性理解模式(引自 Ma,1999, p. 25)……………………………16
圖2.3:MKT領域架構圖(引自Ball等人, 2008, p. 403)…………………..……….20
圖2.4:數學目標中MKT的呈現(引自Sleep, 2009, p. 222)………………………25
圖3.1:錄影資訊的編碼循環模型(引自Jacobs et al. ,1999, p. 719)……….……...43
圖3.2:挑選操作物以表徵數學想法……………………….……………………….61
圖3.3:挑選模型以表徵數學想法………………………………………………….62
圖3.4:多重模型……………………………………………..………………………62
圖3.5:圖像、符號間的連結…………………………………………………………63
圖3.6:河內塔(一)………………………………………………………………..….67
圖3.7:河內塔(二)………………………………………………………………..….67
圖4.1:Lagrange插值多項式的起始例………..…….………………………………87
圖4.2:Lagrange插值多項式的講解……………..….………………………………88
圖4.3:Lagrange插值多項式的展現………………………………………………...89
圖4.4:中國剩餘定理與「連環套」的連接……………………………………..…90
圖4.5:唯一性的說明……………………………………….……………………….91
圖4.6:用手比擬係數多項式…………………………………………………….…97
圖4.7:特殊書寫模型……………………………………………………………....109
圖4.8:數學歸納法,實驗與觀察…………………………………….…………..115
圖4.9:數學歸納法,歸納………………………………………………………..115
圖4.10:使用表格觀察數的大小…………………………………………………..119
圖4.11:用圖形表示增長速度的模糊…………………………………………….119
圖4.12:介紹分割原理與樹狀圖…………………………………………………..132
圖4.13:「挑選模型或操作物以表徵數學想法」、「在符號、具體圖像、圖表等物之間做連結」與「多重模型」一起顯現…………….………………….…136
圖4.14:林師推估比率的板書…………………………..……………………..…137
圖4.15:林師的解法(一)……………………………………………………………140
圖4.16:林師的解法(二)…………………………………………………………..141
圖4.17:矩陣乘法的引入……………………………………………………….….161
圖4.18:利用向量內積表示矩陣…………………………………………………..162
圖4.19:高斯消去法與增廣矩陣……………………………………….………….163
圖4.20:吳師使用的矩陣符號………………………………………….…………166
圖4.21:吳師「為數學想法挑選數字、實例或者脈絡」………….………..…..168
圖4.22:樹狀圖的表徵與條件機率乘法原理的連接………………………….....169
圖4.23:使用表格連接矩陣……………………………………………………….170
圖4.24:三階反方陣的公式解(一)………………………………………………..171
圖4.25:三階反方陣的公式解(二)………………………………………………..172
圖4.26:三階反方陣的公式解(三)………………………………………………..172
圖5.1:林師不確定性與SCK關係圖…………………………………………..….189
圖5.2:吳師不確定性與SCK關係圖………………………………………...…..190

表目次

表2.1:數學教學的任務 (引自 Ball等人,2008, p. 400)………………..……….22
表3.1:混合方法設計的類型(修改自Creswell &; Clark, 2007, p. 82)….……..…..35
表3.2:資料項目的代碼……………………………………………………….……51
表3.3:(C_1,20101015,前)課堂登錄結果的一部份……………….……………….55
表3.4:i×i 項 Kappa統計表………………………………………………………71
表3.5:教學活動kappa統計表(引自(A_1,20101015) 課堂登錄的結果)………….73
表3.6:林師「教學的內容與安排」各類別的K值………………………………74
表3.7:吳師「教學的內容與安排」各類別的K值………………………………74
表3.8:吳師「數學解釋」K值(引自( C_1,20110525)課堂登錄的結果)…….……..75
表3.9:林師「教學活動中數學領域的知識」各類別的K值……………………75
表3.10:吳師「教學活動中數學領域的知識」各類別的K值…………………..76
表3.11:林師「偕同學生的數學使用」的K值結果……………………………..77
表3.12:吳師「偕同學生的數學使用」的K值結果……………………………..77
表4.1:林師前導階段「教學活動」編碼統計表……………………………………..82
表4.2:林師前導階段「教學活動中數學領域知識」編碼統計表……………….93
表4.3:林師前導階段「偕同學生的數學使用」編碼統計表…………………….94
表4.4:林師前導階段闡述數學的方式統計表……...…………………………….100
表4.5:林師前導階段引發學生回應中教師回應的分布…………………………106
表4.6:林師前導階段SCK的樣貌……………………………………………….112
表4.7:林師第一階段「教學活動」編碼統計表………………….………………113
表4.8:林師第一階段「教學活動中數學領域知識」編碼統計表………………116
表4.9:林師第一階段「偕同學生的數學使用」編碼統計表……………………117
表4.10:林師前導階段與第一階段出現的SCK樣貌…………………………..130
表4.11:林師第二階段「教學活動」編碼統計表…………….…………….131
表4.12:林師第二階段「教學活動中數學領域的知識」編碼統計表…………135
表4.13:林師第二階段「偕同學生的數學使用」編碼統計表…………………135
表4.14:林師三階段「教學活動」編碼統計表…………………………………..146
表4.15:林師三階段「教學活動中數學領域知識」編碼統計表………………..147
表4.16:林師三階段「偕同學的數學使用」編碼統計表………………………….147
表4.17:林師三階段闡述數學的方式…………………………………………….149
表4.18:林師三階段「支線」中教師回應的比例分配………………………….152
表4.19:林師三階段SCK的樣貌…………………………………………………153
表4.20:吳師三階段「教學活動」編碼統計表…………………………………..154
表4.21:吳師三階段「教學活動中數學領域知識」編碼統計表……………….165
表4.22:吳師三階段「偕同學生的數學使用」編碼統計表…………………….165
表4.23:吳師三階段闡述數學的方式……………………………………………..167
表4.24:吳師三階段SCK的樣貌………………………………………………….177
表4.25:兩個案SCK的整理………………………………………..…………….178
表4.26:「不確定性」相關編碼的比較……………………………..……………..180
表4.27:林師與吳師「闡述數學的方式」的比較…………………………….…181

一、中文部分

中國教育學會與中華民國師範教育學會(2004)。教師專業成長問題研究:理念、問題與革新。臺北市 : 學富文化。
中華民國師範教育學會(2005)。各師資類科教師專業表現之標準訂定計畫。台北市:教育部。
何福田和羅瑞玉(民81)。教育改革與教師專業化。載於中華民國師範教育學會主編,教育專業(1-30頁)。台北市:師大書苑
吳心楷、宋曜廷、簡馨瑩(2010)。錄影分析在教育研究的應用。教育科學研究期刊,55(4),1-37。
宋曜廷和潘佩妤(2010)。混合方法研究在教育研究的應用。教育科學研究期刊,55(4),97-130。
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。
查孟華(譯)(1980)。數學歸納法(原作者:徐道寧)。台北市:凡異出版社
教育部國語推行委員會(民1996)。重編國語辭典修訂本【網路版】。查詢日期:100年12月15日,檢自http://dict.revised.moe.edu.tw/
教育部(2000)。高級中等以下學校及幼稚園教師分級及審定辦法(草案)。
  台北市:教育部
陳美玉(1997)。教師專業學習與發展。台北市:師大書苑。
陳亭瑋(2011)。資深高中數學教師教學相關知識的個案研究。國立臺灣師範大學碩士論文,台北市。
曾名秀(2011)。資深高中數學教師教學知識與構思的個案研究。國立臺灣師範大學碩士論文,台北市。
饒見維(2003)。教師專業發展理論與實務。台北市:五南出版社。
Creswell, J. W. &; Clark, V. L. (2007)。混合方法研究導論(謝志偉、王慧玉譯)。台北市:心理出版社(2010)。
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)
Irving S. (2009)。訪談研究法(李政賢譯)。台北市:五南。(2006)
Robert, K. Y. (2001)。個案研究法(尚榮安譯)。台北市︰弘智文化(1994)
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)

二、英文部分

Ainley, J., &; Luntley, M. (2007). The role of attention in expert classroom practice. Journal Mathematics Teacher Education, 10(1), 3-22.
Argyris, C., &; Schon, D. (1974). Theory in practice: Increasing professional effectiveness. London: Jossey-Bass.
A.M. Carr-Saunders. (1933). The Profession, Oxford:Clarendon Press.
Ball, D. L. (1996). Teacher learning and the mathematics reforms: What do we think we know and what do we need to learn? Phi Delta Kappan, 77(7), 500-508.
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.
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 the teaching and learning of mathematics (pp. 83-104). London :Ablex Publishing
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.
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