本研究採用質性取向的個案研究法，透過課室教學觀察及訪談，探討一位資深高中數學教師的教學知識與教學思考。研究設計分為前導研究、第一階段研究與第二階段研究；研究的教學單元分別為「空間平面和空間直線方程式」、「重複組合」與「數學期望值」；研究資料的主要來源包括，教學影片檔、訪談錄音檔和個案教師自編的數學科講義。本研究所使用的數學教學觀察系統，引自Learning Mathematics for Teaching (2006)所發展的Mathematical Quality of Instruction(MQI)登錄系統。首先，個人修改系統的編碼，以符合個案數學教師實際的教學特質，其次，針對教學影片進行分析，最後，商請另一位獨立的登錄者協助信度的檢驗，以提昇系統的有效性。藉由三個不同的數學教學單元，以及為期一年的研究進程，對個案數學教師的教學知識與教學思考，以及MKT (Mathematical Knowledge for Teaching, Ball等人, 2008)的內涵有更深入的了解。
本研究的結果顯示，個案教師的數學教學大多呈現PUFM (Profound Understanding of Fundamental Mathematics, Ma, 1996)的連通性、多重觀點、一致性和基本概念四項特徵。然而，由於PUFM著重數學學科知識，對於某些PUFM無法涵蓋的數學教學片段，個人則藉助Ball (2008)的MKT架構來進一步分析。據此，個人提出一些比較凸顯MKT特徵的數學教學實例，這也表明，MKT和PUFM似乎有部分的重疊，但是亦有不同。結果進一步顯示，個案教師的數學教學傾向，遠看似PUFM，近看卻有部分MKT的元素。此外，個案數學教師也有一些其他的數學教學特色，例如螺旋式教學。
最後，根據研究的結果，個人進一步指出PUFM和MKT可能的關係，以作為未來接續探究高中數學教師MKT和PUFM內涵與關係之參考。希望本研究的結果，能夠有助於提昇高中數學教學觀察系統的品質與實用性，並用來檢測和發展高中數學教師的數學教學專業知識。

This study applies qualitative case study, which explores the professional knowledge and thinking of an experienced high school mathematics teacher. The study is structured into three research stages including the pilot, initial, and second stage. The teaching units include「plane and line in space」、「combination with repetition」and「mathematical expectation」. The study material comes from videotaped lessons, interviews, and class handouts provided by the participant teacher. The observation system is adapted from coding system of the Mathematical Quality of Instruction (MQI) developed by Learning Mathematics for Teaching (2006). First of all, I modified the system codes to adapt to the actual quality of the teacher’s classroom teaching. Second, I analyzed the video tapes. Last, the coding results were mostly supported from another independent coder to establish the acceptable inter-coder reliability. Using three different observed teaching units within an academic year, I describe the participant teacher’s knowledge and thinking of/about mathematics teaching in terms of both MKT (Mathematical Knowledge for Teaching) (Ball et al., 2008) and PUFM (Profound Understanding of Fundamental Mathematics, Ma, 1996) theoretical frameworks.

The results reveal that, the teaching of the teacher mostly presented PUFM qualities of connectedness, multiple perspectives, unity, and the basics. However, considering of emphasis on subject matter knowledge of PUFM, I took the aid of MKT structure to further analyze some teaching clips which PUFM is unable to cover. According to the instruction above, I brought up some mathematical teaching cases which highlight MKT characteristics, which also indicates that MKT and PUFM seem to have some parts overlapped, but some not neither. The study result in addition presents the teaching tendency of the teacher was much alike PUFM as a whole, yet also reflecting some MKT elements inside. Furthermore, the teacher also showed sort of spiral features of mathematics teaching.

Finally, the present study described the possible relationship between PUFM and MKT, which may provide some insights into the future study of both the tension and relationship between high school mathematics teacher’s PUFM and MKT. Hopefully, my study results may be used to increase the quality and usefulness of the MKT observational system at high school level, as well as to examine and develop professional knowledge of high school mathematics teachers.

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