資深高中數學教師教學相關知識的個案研究

Abstract

本研究採用個案研究法與教室觀察法，描述一位資深高中數學教師「教學用的數學知識(mathematical knowledge for teaching，簡稱MKT)」內涵。依照個案教師所教授的高二數學課程，研究的單元包括幾何、離散與統計三個主題，區分為三個研究階段，為期一年。前導階段研究的教學單元為「空間中的直線與平面」，第一階段的教學單元為「重複組合」，第二階段則是「數學期望值」。個人利用教室觀察與訪談資料，並藉助Ball, Thames與Phelps (2008)的MKT架構，分析個案教師的數學教學相關知識及其樣貌。 研究結果顯示，個案教師MKT的六領域知識中有四項較為突顯，其餘二項則因文獻較少而難以區辨。在教學過程中，個案數學教師的MKT兼具外顯與內隱的特質，而且，隨著當下數學教學所需，各領域知識之間的轉換相當的快速，他為了因應教學的不確定性，展現出流動與彈性的特質。另外，各領域知識之間也會產生交互作用，特別是對學生理解的知識，時常影響其他領域的知識。此外，個案教師的數學教學信念以及課堂的情境脈絡，亦會影響各領域知識的呈現。最後，本研究個案教師的數學教學中，同時兼有「基礎數學的深刻理解(profound understanding of fundamental mathematics)，簡稱PUFM」的部分特質。 本研究的結果，可以用來協助高中在職數學教師，進一步了解自己在教學中用到的各類數學教學知識的內涵以及影響的因素，以幫助教師做教學決策，進而引動數學教學專業的發展。此外，對數學師資培育者而言，了解資深高中數學教師教學時所呈現的各領域數學教學知識，可以用來幫助中學數學科實習學生，發展各領域的知識，以提升其教數學的功力。This study uses case study and classroom observation to explore “mathematical knowledge for teaching (MKT)” of an experienced and purposeful selected senior high school mathematics teacher. This one year qualitative case study was divided into pilot, first and second stage, each explores the selected topics from geometry, discrete and statistics including 「plane and line in space」、「combination with repetition」 and 「mathematical expectation」 units. With the aid of Ball, Thames and Phelps’s (2008) MKT theoretical framework, the author used classroom observations and interview data to analyze participant teacher’s knowledge used in teaching those selected topics.

The results revealed that four domains of MKT of the teacher were highlighted, and the remaining two were not easy to distinguish. The MKT of the participant teacher had both explicit and implicit characteristics in presenting his knowledge. In the process of teaching, the four MKT domains transformed quietly and smoothly. Besides, the quick shifting showed the characteristics of mobility and flexibility in response to the uncertainties of classroom mathematics teaching. Few domains also interacted with each other. Especially, the knowledge of content and students often impacted others. The teacher's mathematics teaching belief and the context of the classroom also affected the use of the knowledge. Finally, the teacher's mathematics teaching showed some distinctive properties of “profound understanding of fundamental mathematics (PUFM)”.

The results of this research might help senior high school mathematics teachers to know and be aware of their MKT in order to make proper decisions, and facilitate their professional development of mathematical teaching. In addition, for mathematics teacher educators, understanding the knowledge in practice of the experienced senior high school mathematics teachers can help student teachers to develop MKT and enhance their pedagogical power.