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

Abstract

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.