以教學反思探究一位高中資深數學教師教學用數學知識的內涵與適應

A Case Study on Content and Adaptation of Experienced High-School Mathematics Teacher’s Mathematical Knowledge for Teaching Through Pedagogical Reflection

本研究旨在探究一位資深教師教學用數學知識（Mathematical Knowledge for Teaching）的內涵與適應。首先針對個案教師進行教學前訪談，以及課室觀 察和錄影；次則透過「高中數學教學討論會」以分析個案教師數學教學之核心 活動，並形成訪談問題；最後則進行個案教師的訪談及教學反思。結果發現： 第一、個案教師的數學和學生知識（Knowledge of Content and Students）是指他 能夠知道學生學習過的數學技能、數學知識與解題策略；數學和教學知識 （Knowledge of Content and Teaching）是指教師能夠將課堂教學的數學內容排 序；數學和課程知識（Knowledge of Content and Curriculum）是指教師能夠指 出不同的數學概念被安排在哪個年級的數學課程之中；共通數學知識（Common Content Knowledge）是指個案教師能夠自行設計並且解出這些數學問題與任 務；專門數學知識（Specialized Content Knowledge， SCK）是指個案教師能夠選 擇和發展合適的表徵與例子以解釋數學概念、由二維類推到三維，以及組織連 結不同概念；眼界數學知識（Horizon Content Knowledge， HCK）是指個案教師 能夠知道哪些數學知識是學生需要卻未編入課本。第二、個案教師在教學反思 後，其SCK 與影片中其他兩位案例教師的SCK 產生適應（adaptation），有些 產生同化，有些則調適為HCK。這樣的教學反思使得教師的SCK 成為下次教 學行動的基礎，也使得教師達到反思即行動（reflection-as-action）的狀態。

The purpose of this study was to explore, through pedagogical reflection, the content and adaptation of an experienced high-school mathematics teacher’s mathematical knowledge for teaching. Firstly, via observation and discussion, the experienced high-school mathematics teacher’s “core activities” of mathematics teaching were identified, and some interview questions were compiled in terms of the mathematical analysis of these core activities. Secondly, the reflections regarding the demonstrations of two teachers were provided. Findings of this study indicated that knowledge of content and students involved the teacher’s knowing and understanding of his students’ learned mathematical skill, mathematical knowledge and strategies for solving mathematical problems; knowledge of content and teaching involved the teacher’s ability of sequencing the teaching topics; knowledge of content and curriculum involved the teacher’s familiarity with the concepts that had been and would be taught in the same subject area during the preceding and subsequent years in school; common content knowledge involved the design of mathematical exercises and tasks; specialized content knowledge (SCK) meant that the teacher would be able to choose an appropriate example and representation to explain the mathematical concepts; horizon content knowledge (HCK) implied that the teacher would be able to know the mathematical contents that students needed but were not included in textbooks. Furthermore, the case teacher’s SCK was adapted to that of the two teachers. Specifically, some was assimilated to the two teachers’ SCK, and some was accommodated to HCK. This kind of pedagogical reflection made the teacher’s SCK the basis of the pedagogical action.