本研究所編排的幾何實驗教材，主要是按照波利亞的解題四步驟為原則而設計，其目的是想了解波利亞的方法是否對教導學生學習幾何證明有所幫助，亦即從了解題意、分析題目、由已知條件找到可供解題的線索、擬定證明的計畫、到實現計畫完整地把題目證明出來。本研究探討的是藉由這樣的教材編排對於學生解幾何證明題的表現是否有所提升。本研究另一個目的是想教導學生用兩欄式撰寫幾何證明寫法，並探討這種方式是否對學生寫幾何證明有幫助。本研究的研究對象為21名國中數學資優生，當中大部分曾代表台灣參加國際性的中小學數學競賽。本研究將這21名學生根據他們在前測的表現分成三組：「高證明能力組」、「中證明能力組」及「低證明能力組」。本研究在資料分析中所使用的評量工具有全等三角形問題(I)及(II)、圓的問題、共邊定理問題、共角定理問題、塞瓦定理問題、訪談問題(I)及(II)等八個問題。
從資料分析看來，在「了解題意」階段，這群學生在本研究教學實驗末了約有85%的學生能從幾何證明題的文字敘述中，掌握到題目的已知條件及所要求證的部分為何，其餘的學生可能了解題意，但對於題目的已知條件及所要求證的部分比較無法清楚地掌握。而全部的學生都能由題目的敘述中把題目的圖形畫出來，並且標上適當的符號。
在「擬定計劃、執行計劃」階段，本研究根據波利亞的解題四步驟為原則而設計的教材編排，對於高證明能力組的學生似乎幫助不太明顯；然而，對於中證明能力組及低證明能力組的學生卻比較有幫助。其原因可能是因為：高證明能力組的學生本身的證明能力就不錯，且也較容易配合教材編排的方式，所以從第二節課開始進行幾何證明時，便有不錯的表現。而中證明能力組及低證明能力組的學生，則需要教師在教學活動期間不時地提醒學生作幾何證明題時可先嘗試擬定計劃，再將所擬定的計劃正確地執行出來，這需要教師不斷地提醒及鼓勵學生，才能使學生不僅知道如何去擬定計劃，進而執行計劃。且教師經常提醒與鼓勵，可使學生自然而然地養成擬定計劃、執行計劃的解題習慣。而且，從資料分析中也發現，這21名學生中，大部分的學生若是依照本研究所設計的波利亞解題四步驟為一般性原則解幾何證明題者，其證明步驟大致上正確；然若沒有依照本研究所設計的波利亞解題四步驟為一般性原則解幾何證明題者，其證明步驟有誤。而其中只有1、2位學生沒有依照本研究所設計的波利亞解題四步驟為一般性原則解幾何證明題者，其證明步驟卻正確。由此看來，若要幫助學生能正確地寫幾何證明，教師需要反覆地提醒學生此一般性的解題原則。
至於「回顧」階段，本研究想要了解學生在經過畫圖、擬定計劃、證明等步驟後，是否能根據題目的敘述，將圖形作適當的延伸。而由學生作答的情形可以發現：在共邊定理的「回顧」中，11位學生具有回顧的能力，其中，只有1位學生不僅把不同於題目的圖形畫出來，並且還詳加證明；而其餘的學生都只是把不同於題目的圖形畫出，證明部分比較簡略甚至沒有證明。在共角定理中，17位具有回顧能力的學生，不僅能畫出與題目不同的圖形，還能加以證明。
從學生第二節課學習寫幾何證明的情形可發現，學生以「兩欄式」撰寫幾何證明題時，所感到的困難，有以下幾點：
1.理由部分不知道該如何寫。
2.引用前面證明步驟所得的結果時，理由部分不知道該如何寫。
3.在寫幾何證明時，對於理由部分的陳述，並不覺得有必要。
而經過本教學實驗後，學生在這幾方面的困難漸漸有改善。同時，這群學生經過本教學實驗後，幾何證明的寫法比較符合數學邏輯且證明步驟的正確性也有所提升。
根據以上的研究發現與心得，本研究建議教師宜在教學過程中，讓學生體會到了解題意、擬定計劃、執行計劃、回顧等階段的重要性。且在課堂上宜讓學生有機會思考，同時教師應適時且不落痕跡地幫助學生，使學生能將波利亞的四個解題步驟的一般性原則運用在平時解幾何證明題中。至於未來的研究方向，可考慮將研究時間延長，以及考慮學習風格、後設認知等因素作更深入的探討。

The geometry teaching material in this study is arranged mainly according to the Polya's 4-step problem solving heuristics with a purpose to understand if Polya's heuristics help guiding students in learning geometry proofs how to start from understanding the question, analyzing it, finding clues to answer from the given conditions, devising a plan to prove and completing the whole proof. This study wants to know if the arrangement of teaching material does upgrade students' performance in proving geometry. Another purpose for this study is to guide students in applying two-column writing as they prove geometric questions and to see if this style help their performance. I have 21 top junior high students in math as my study object. Most of them have represented Taiwan in international math competitions. They are divided, according to their proving competence showed in pre-tests, into 3 groups: high, middle and low. The measuring tools used for data analysis in this study are eight questions regarding the congruent triangles (I) and (II), circles, the coedged theorem, the coangular theorem, the Ceva's theorem and interview (I) and (II).
Gathered from the data analysis, in the stage of "understanding questions," 85% of the students can grasp the given conditions and respond to the core geometric statement. The remaining 15% have no problem in understanding but have troubles in using the given data for proving the key geometric concept. However all of them can draw the diagram expressed in the question statement and mark proper places.
In the stage of devising plan and executing it, the Polya's 4-step problem solving heuristics does not help the high level students very much. However, to the middle and low level students, it gives much help. The high level students are strong in proving and adapt to the arrangement of the teaching materials without much efforts. They have good performance from the second class on. For the middle and low level students, their teacher has to remind them constantly of how to devise a feasible plan and execute it accurately when they proceed to prove a geometric concept. They depend very much on the teacher's frequent encouragement and speaking to form such a problem-solving habit. Besides, from the findings, most of the 21 students who follow Polya's heuristics are correct in their proving steps, while some that do not follow are wrong in their steps. Only one or two students remain correct in their proving steps without following Polya's heuristics. I gathered that if a teacher wants to help students with correct geometry proofs, s/he has to repeatedly remind students of this heuristics.
In the "reflecting" stage, this study intends to see how students extend their diagrams properly according to the question statement, after drawing, devising a plan and proving. From findings on their work, in "reflecting" on the coedged theorem, there are 11 students has such ability and one of them can even draw different figures with sufficient explanations responding to various questions. While the remaining students only mark out the difference with simple, or even little explanations. On co-angular theorem, 17 students have such reflecting power who are able to draw pictures different from the question statement and give proper explanation.
Findings in their second class in learning geometry proving show that students have following difficulties in applying "two column" writing geometry proofs.
1.Don't know how to write the reasons.
2.In quoting the previous proofs, they don't know how to write the reasons.
3.In writing the geometry proofs, they don't feel necessary to write the reasons.
After this teaching experiment, students make much progress in this area. At the same time, their geometry proof writing become more logical in math with correct proving steps than before.
Based on the findings and after-thoughts, this study suggests that teachers make students understand the importance of the stages in understanding the questions, devising a plan, executing the plan and reflecting in their teaching. Students should have time to think in class, while the teacher should help them timely in applying Polya's 4-step problem solving heuristics in their geometry proofs.
For the future study, the program should have longer time frame and take different factors such as various learning styles, meta-cognition etc into consideration for a deep understanding.

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