直線形幾何之解題策略

The strategies of solving linear classical geometric problems

張景軒(Ching-Hsuan Chang)

指導教授 ： 李金城 ； 何淳雪

中原大學/理學院/應用數學研究所/碩士(2016年)

https://doi.org/10.6840/cycu201600462

摘要

在中學的數學課程中，教師最難教授的就是「證明」，這也是現在學生最缺乏的，和其他主題相比，證明顯得枯燥乏味，然而證明不單能訓練思考邏輯，也是培養觀察力與想像力的好方法。
本研究所討論的題目適合國中九年級平面幾何。研究主題為：解直線形幾何證明題，有哪些策略可以幫助思考？
根據本研究所討論的題目，整理出的策略為：中線延長、作對稱圖、圖形性質、輔助圓、大角中取小角或大邊上取小邊。藉由這些策略可以推導出證明，然而作任何的證明題，必須要有完善的先備知識，解決方法不只一種，經驗越是豐富，想到的策略就越多，最重要的是教師需要觀察學生的想法與反應來給予適當的提示，幫助學生思考與成長，進而擁有更加寬廣的視野，而不受侷限。
這些策略只針對直線形幾何證明題，所以無法概括所有題型，但也藉由這些策略，讓學生理解九年一貫數學領域課程的「幾何」主題上的目標：「能認識證明的意義。」

關鍵字

直線形幾何 ； 證明 ； 解題策略

並列摘要

“Proof question” is the most difficult topic for teachers to teach in high school mathematics classes; and this is proved the most insufficient training for students nowadays. To compare to other topics, "proof question” is less interesting; however, students can be trained effectively to think logically and to inspire their observation skills and imagination by solving proof questions.
The topic of this study is based on the plane geometry of ninth grade in junior high schools and takes the proof of plane geometry into consideration. The topic of this study: What are the strategies for solving proof questions of linear classical geometric?
According to the topic of this study, there are several ways to solve the question are: “extensive midline, graphics features, auxiliary circle, graphic cut edges and corners.” With these strategies, the proof questions can be solved step by step. A decent knowledge is a prerequisite for solving any proof questions; in addition, proof questions usually have various solutions. The more solutions can be thought of when students have more experience. Importantly, teachers are also required to observe students’ ideas and reaction and give appropriate hints for students to think and learn. By having a broader vision given by the teachers, students’ way of thinking and development will be limited less.
Overall, these strategies and techniques are specifically for straight geometric proofs, not applicable for any other kinds of questions. With these strategies skills, students can reach the objective of the requirement of nine-year integrated mathematics courses on the topic of Geometric: To understand the significance of proof.

並列關鍵字

Linear Classical Geometric Problems

參考文獻

翁聖閔(2014)。在平面幾何證明題中使用輔助線的策略。中原大學應用數學系碩士學位論文，未出版。

徐永成(2014)。圓內接四邊形的解題策略。中原大學應用數學系碩士學位論文，未出版。

洪萬生（2004）。教改爭議聲中，證明所為何事？台灣師大學報（科學教育類），49（1），pp.1-14。

Duval, R. (1995). Geometrical Pictures:Kinds of Representation and Specific Processings. In R. Sutherland & J.Mason (Eds.), Exploiting Mental Imagery with computers in Matehmatics Education(pp.142-127). Berlin: pringer (NATO ASI Series n138).

Little Heath, Thomas.(1956). Thirteen Books of Euclid’s Elements. New York: DoverPublications, INC.