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研究生: 卓益安
Cho, Yi-An
論文名稱: 臺灣高中數學教師專門數學知識與眼界數學知識的個案研究
The Case Studies of Taiwanese High-school Mathematics Teachers’ Specialized Content Knowledge and Horizon Content Knowledge
指導教授: 金鈐
Chin, Chien
楊文金
Yang, Wen-Gin
學位類別: 博士
Doctor
系所名稱: 科學教育研究所
Graduate Institute of Science Education
論文出版年: 2016
畢業學年度: 104
語文別: 中文
論文頁數: 227
中文關鍵詞: 教學用數學知識教學用數學任務專門數學知識眼界數學知識
英文關鍵詞: mathematical knowledge for teaching, mathematical task for teaching, specialized content knowledge, horizon content knowledge
DOI URL: https://doi.org/10.6345/NTNU202203745
論文種類: 學術論文
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  • 本研究採用詮釋性個案研究法,探索三位高中典型數學教師專門數學知識(Specialized Content Knowledge,簡稱SCK)與眼界數學知識(Horizon Content Knowledge,簡稱HCK)的特徵。依據Ball、Thames與Phelps(2008)提出的教學用數學知識(mathematical knowledge for teaching,簡稱MKT)的理論架構下,收集和分析三位個案教師在拉格朗日插值多項式與數學歸納法兩個教學單元的實徵資料。
    資料收集的範圍包括參與觀察的場域筆記、上課教材、課堂錄影和訪談。研究者一方面依據文獻資料歸納個案教師課堂使用的數學任務(簡稱教學用數學任務),另一方面修改LMT (2007)的「教學的數學品質(mathematical quality of instruction,簡稱MQI)」編碼詞彙表,建立課堂教學錄影的分析工具。修改分析工具的原則有兩個:一是基於文獻資料的輔助,將編碼歸類至SCK與HCK兩個操作型定義;二是依照個案教師實際的課堂活動,刪減、增加與修訂部分編碼。
    主要的研究發現有二:第一是,三位個案教師的SCK在提供數學解釋上展現,例如,教師解釋數學名詞、特殊主題的數學想法、解題歷程等。教學用數學任務的數學內涵更深也更廣,展現了數學知識在數學課堂教學中扮演的關鍵角色,也更能夠使研究者理解個案教師在不同數學主題展現的SCK特徵。第二是,個案教師在不同單元顯現出的HCK中眼界(Horizontal)、周邊(Peripheral)與從入門透視進階(Elementary-on-Advanced)三個面向的特徵都不太一樣。HCK幫助教師察覺數學主題的關鍵核心概念,也幫助學生看到特定主題的數學結構。

    This study combines interpretive case study data with the quantitative video analysis to explore three exemplar high-school mathematics teachers’ characteristics of specialized content knowledge (SCK) and horizon content knowledge (HCK). Firstly, the researcher focused on these two mathematical topics of Lagrange interpolation polynomial and mathematical induction. Based on the theoretical framework of mathematical knowledge for teaching (MKT), the researcher collected and analyzed three case teachers’ emperical data including field notes, teaching material, videos of classroom teaching and follow-up interviews. Secondly, the researcher modified the coding system of Mathematical Quality of Instruction (MQI) developed by Learning Mathematics to Teaching (2007), and categorized the codes into two operational definition of SCK and HCK. The two major priciples of modification include the suggestions of literature and the three case teachers’ actual classroom teaching.
    There are two major findings. One is that three teachers’ SCK was founded in the mathematical explanation of mathematical definions and noun, key ideas of specific mathematics topics and problem-solving. The content of these mathematical tasks for teaching was more borader and deeper, which played the critical role in the classroom teaching. And these mathematical tasks for teaching help the researcher to understand the characteristics of SCK in different mathematical topics. The other is that HCK has three different characteristics, including horizontal, peripheral and elementary-on-advanced dimension. The HCK not only helps teachers be aware of the key and core concept of mathematics topic, but also helps students see the mathematical sturcture of the specific topics.

    目次………………………………………………………………………………….Ⅰ 附錄目次…………………………………………………………………………….Ⅴ 表目次……………………………………………………………………………….Ⅵ 圖目次……………………………………………………………………………….Ⅷ 第壹章 緒論 1 第一節 研究背景與動機 1 一、數學教學的內涵 2 二、數學教學的知識需求 4 第二節 研究目的與研究問題 10 第三節 名詞釋義 12 一、高中典型數學教師 12 二、教學用數學任務 12 三、專門數學知識 13 四、眼界數學知識 13 第四節 研究限制 14 第貳章 文獻探討 15 第一節 數學教師知識五個重要的觀點 15 一、Shulman的觀點 16 二、Ball的觀點 24 三、Rowland的觀點 27 四、Davis的觀點 29 五、Zazkis的觀點 32 六、Ruthven對數學教師知識理論的分類 34 七、MKT後續的實徵研究 36 八、小結 43 第二節 教學用數學任務 43 一、有價值的數學任務 44 二、不同認知需求的數學任務 45 三、教學的數學任務 47 四、小結 48 第三節 如何研究數學教師知識 49 一、Lampert與Ball的自我教學研究 49 二、專家與生手的教學研究 52 三、日本的課室研究 53 四、數學教導學研究 55 五、小結 57 第參章 研究方法 59 第一節 研究架構與流程 59 一、個案選取 60 二、研究對象與研究場域 61 第二節 資料收集與整理 63 一、資料收集 63 二、資料整理 65 第三節 資料分析與呈現 66 一、錄影分析工具的選用 67 二、資料編碼程序 68 三、編碼系統的簡介 69 四、編碼的信度檢測 74 五、資料呈現方式 77 第四節 研究可能的限制 78 一、 關於研究場域 78 二、 關於研究者與被研究者 79 三、 關於研究方法 79 四、 關於研究結果的可類推性 80 第肆章 研究結果 81 第一節 個案教師在插值多項式SCK與HCK的特徵 81 一、個案教師在回顧時展現的SCK與HCK的特徵與其異同 82 二、個案教師在引進主要工作與概念時展現的SCK與HCK的特徵與其異同 92 三、個案教師在教師示範例題、學生練習與檢討學生練習時展現的SCK與HCK的特徵與其異同 101 四、個案教師在總結、延伸與探索時展現的SCK與HCK的特徵與其異同 111 五、個案教師在插值多項式中展現的SCK的特徵與其異同 122 六、個案教師在插值多項式中展現的HCK的特徵與其異同 123 第二節 個案教師在數學歸納法SCK與HCK的特徵 126 一、個案教師在回顧和引進主要工作與概念時展現的SCK與HCK的特徵與其異同 127 二、個案教師在教師示範例題、學生練習與檢討學生練習時展現的SCK與HCK的特徵與其異同 145 三、個案教師在總結、延伸與探索時展現的SCK與HCK特徵與其異同 151 四、個案教師在數學歸納法中展現的SCK的特徵與其異同 161 五、個案教師在數學歸納法中展現的HCK的特徵與其異同 162 第三節 結論 164 一、三位個案教師的SCK特徵 164 二、三位個案教師的HCK特徵 165 第伍章 討論與建議 169 第一節 討論 169 第二節 接續研究的建議 171 參考文獻 175   附錄目次 附錄1:個案教師課堂教學影片的轉譯(筆者整理後) 183 附錄2:MQI教室觀察系統 197 附錄3:MQI教室觀察系統的中文對照 202 附錄4:MQI的編碼和本研究教學觀察系統的修訂對照表 205 附錄5:個案教師的訪談逐字稿(筆者整理後) 208 附錄6:丙師分組討論時使用的數學問題 221 附錄7:個案教師在教師示範例題與學生練習時提供的題目 225   表目次 表2-1 詮釋CCK、SCK、KCS、KCT與KCC四種知識內涵的實徵研究 37 表2-2 教學的數學任務 47 表3-1 高中數學教師教室觀察系統編碼表 69 表3-3 項的K值表 75 表3-3 教學進行方式的K值結果 76 表3-4 SCK與HCK操作型定義的K值結果 77 表4-1 個案教師在插值多項式的「教學進行方式」編碼統計表 82 表4-2 個案教師在「回顧」時SCK與HCK編碼發生的次數與頻率 83 表4-3 甲師多項式唯一性定理的內容與證明 85 表4-4 丙師在回顧時提出的例子 88 表4-5 丙師提問的數學問題 89 表4-6 個案教師在「引進主要工作與概念」時SCK與HCK編碼發生的次數與頻率 93 表4-7 甲師提問的問題 97 表4-8 乙師對於插值多項式的講解步驟與內容 98 表4-9 丙師在中國剩餘定理與插值多項式的起始例 100 表4-10 個案教師在「教師示範例題、學生練習以及檢討學生練習」SCK與HCK編碼發生的次數與頻率 102 表4-11 乙師在回顧時提出的數學問題 107 表4-12 個案教師在「總結和延伸與探索」SCK與HCK編碼發生的次數與頻率 112 表4-13 甲師在同餘概念提供講解與練習的數學問題 114 表4-14 甲師在中國剩餘定理提供講解的起始例 116 表4-15乙師中國剩餘定理的解題步驟與數學內涵 117 表4-16 個案教師在插值多項式的「教學進行方式」編碼統計表 127 表4-17 個案教師在「回顧以及引進主要工作與概念」SCK與HCK編碼發生的次數與頻率 128 表4-18 乙師在教學回顧講解的數學問題 129 表4-19 個案教師在「教師示範例題、學生練習以及檢討學生練習」SCK與HCK編碼發生的次數與頻率 146 表4-20 個案教師在「總結和延伸與探索」SCK與HCK編碼發生的次數與頻率 152 表4-21 甲師在教學回顧提供學生練習的例題 159   圖目次 圖1-1 MKT的領域圖 25 圖2-1 教學用數學的巢狀結構圖 30 圖2-2 乘法概念的實現化與其K-12的對應圖 32 圖2-3 乘法的方格表徵及其延伸運用 32 圖2-4 Zazkis和Mamolo (2011)的眼界數學知識 39 圖2-5 周邊數學知識輔助HCK 40 圖2-6 HCK形朔MKT以及它的本質 41 圖2-7 臺灣高中數學教師HCK的特徵與內涵 43 圖2-8 課室研究教學改善架構圖 55 圖3-1 研究架構圖 60 圖3-2 本研究的資料整理程序 65 圖3-3 本研究的分析架構以及研究結果的呈現模式 67 圖3-4 高中數學教師教室觀察系統的建立流程 68 圖4-1 整數與多項式的除法原理 85 圖4-2 餘式的一般形式假設法 91 圖4-3 餘式的牛頓形式假設法 91 圖4-4 餘式的大雜燴假設法 94 圖4-5 連環套假設法與大雜燴假設法的比對 94 圖4-6 Lagrange插值多項式的數學形式 95 圖4-7 Lagrange插值多項式的數學意涵 97 圖4-8 餘式的連環套假設法 108 圖4-9 餘式唯一性的證明 110 圖4-10 中國剩餘定理與「連環套」的連接 115 圖4-11 丙師插值多項式的討論文本(一) 119 圖4-12 丙師插值多項式的討論文本(二) 120 圖4-13 丙師插值多項式的討論文本(三) 121 圖4-14 甲師與乙師在插值多項式單元教學中展現的HCK 124 圖4-15 丙師在中國剩餘定理與插值多項式兩單元教學中展現的HCK 125 圖4-16 等差級數固定項的求法 129 圖4-17 數學歸納法的初始書寫程序 131 圖4-18 河內塔的遞迴關係式 135 圖4-19 丙師數學歸納法的討論文本 138 圖4-20 數學歸納法的標準書寫(一) 144 圖4-21 數學歸納法的標準書寫(二) 144 圖4-22 費波那西數列一般項的推演歷程 158 圖4-23 甲師在數學歸納法單元教學中展現的HCK 162 圖4-24 乙師在數學歸納法單元教學中展現的HCK 163 圖4-25 丙師在數學歸納法單元教學中展現的HCK 164

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