八位學生教師數學教學知識的個案研究：以無限概念為例

林恆理

指導教授 ： 金鈐

國立臺灣師範大學/理學院/數學系/碩士(2005年)

摘要

本研究是以國立台灣師範大學數學系教材教法的課程為場景，透過問卷調查法，從其中一班38位修課學生中篩選出8位個案，進行為期一年的個案追蹤研究。本研究採用個案研究法，搭配問卷、訪談與教室觀察的方式，探討相關學生教師在學習教數學的第一年裡，數學教學知識的內涵與發展。透過分析數學層次和教學層次的知識，瞭解學生教師理解無限概念的深度、廣度與完全性，以及教學使用的問話、類比與操作性活動的表徵方式，並以Wenger (1998) 的實務社群(community of practice)理論作為個案數學教學知識發展的脈絡。

研究結果顯示，個案學生教師的模擬教學與理想中的教學有相當的落差，他們有不同層次的迷失概念，可以進一步區分成「數學層次的迷思」、「教學層次的迷思」與「數學與教學層次的迷思」。此外，個案對於自己教學問題的察覺與反思，呈現出「沒察覺沒反思」、「有察覺沒反思」與「有察覺有反思」三種狀態。由於學生教師普遍無法精確掌握無限數學概念的內涵，因此，個人認為「精確性」應該可以作為「對基礎數學有深刻理解」(Ma, 1999)的第四個面向。培育活動中採用的案例教學，也對部分個案產生一定程度的影響，而學習型實務社群的功能雖然有部分的限制，但是仍有助於瞭解個案數學教學知識發展的脈絡。

這個階段的研究結果反應出學生教師在學習教數學時的個別差異，接續的研究可以從瞭解迷思教學概念產生的原因、如何克服與修正這些迷思概念以及如何進一步促使學生教師產生更深層的察覺與反思等幾個方向著手。最終，希望所培育出的中學數學教師，不僅對數學內容有深刻的理解，也能適切地將其轉換成對於不同能力與背景的學生有意義、有效且合適的理解形式(Shulman, 1987)。

關鍵字

學生教師 ； 教師專業發展 ； 教師的數學教學知識 ； 無限概念

並列摘要

The research is one-year case study on the students who participate the method course for secondary mathematics. Eight student cases from the thirty-eight participants were selected in terms of question survey, and followed by interview and classroom observation to understand the content and development of the student teachers’ pedagogical knowledge in mathematics in the process of learning to teach mathematics. By analyzing mathematical and pedagogical knowledge, the author tries to investigate the depth, broad and thoroughness of the concept of infinity of the eight student teachers, and the representation of teacher questioning, analogy and manipulatives used in microteaching. The theoretical context for situating their knowledge development is based on Wenger’s theory of community of practice.

The results show that there were distinct differences between microteaching and ideal teaching among cases. They have misconceptions of mathematic and pedagogy. The eight cases also reflect three states of awareness, including no awareness and no reflection, awareness but no reflection, or awareness and reflection. Due to most of the student teachers could not understand fully the mathematical concept of infinity, the author adds a fourth element “accurateness” to the Ma’s notion of profound understanding of fundamental mathematics. The method course of the teacher education program seems to have potential influences on the cases, while the theory of learning community supports those cases to develop their pedagogical knowledge in mathematics.

The study results reflects also some individual differences among the student cases when they are learning to teach, and the future research is expected to understand the causes and adjustments of the misconceptions, also to stimulate deeply the reflection and awareness of student teachers. Finally, the author also expects that the well–trained secondary mathematics teachers can not only understand profoundly mathematical content but also transform the content knowledge into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by the students (Shulman, 1987)