結合不同學習策略的工作例對理解幾何證明之影響研究

The Effects of Worked-out Examples with Different Strategies on Comprehending Geometry Proof

李健恆(Lei, Kin Hang)

指導教授 ： 左台益

國立臺灣師範大學/理學院/數學系/碩士(2012年)

摘要

幾何證明是發展數學思維和學習演繹推理的重要工具，卻也是學生數學學習的難點之一。工作例是展示數學思維的基本方式，因此尋找合適的學習策略結合工作例來理解幾何證明的內容是值得探討的議題。我們以平行線截比例線段證明做為工作例的內容，在電腦環境下閱讀相關證明後，配合練習或後設認知問題所形成的閱讀學習模組，以檢驗對學生理解幾何證明的影響。本研究選取254位尚未學習幾何演繹證明的八年級學生，使用理解測驗問卷和認知負荷感受量表，分別檢測學生能否理解相關的內容和其學習成效的保留情況，以及學生的認知負荷感受。從學生回答問題的策略檔案中，進一步分析學生的學習過程與理解幾何證明之間的關係。研究結果顯示，使用類似結構的練習策略有助於學生在當下的理解，但卻容易受工作例所產生的原型影響，僅使用模仿改編策略來回答問題；回答後設認知問題對學生來說是較困難的學習任務，但卻能反映學生真正的理解程度且產生較好的保留成效。因此，後設認知問題可以作為幫助學生反思的理想工具，適當搭配練習題的優點相信能有助於學生理解幾何演繹證明的內容。

關鍵字

幾何證明 ； 工作例 ； 後設認知 ； 認知負荷

並列摘要

Geometry proof is an important tool for the development of the mathematical thinking and the learning deductive reasoning; nevertheless, geometry proof is also one of the learning difficulties for students. Worked-out example is a fundamental approach to demonstrate mathematical thinking. Thus, the topic of finding suitable learning strategies in order to understand geometry proof is worth to discuss. The effects of comprehending geometry proof are detected under using different reading learning modes. Proofs are showed with a computer setting. The reading learning modes are formed by worked-out examples with practices or metacognition questions. The intercept theorem (or Thales' theorem) is used as the presenting content of worked-out examples. 254 eighth grade students who have not learned deductive proof are chosen for this research. Reading comprehension test is used to examine students’ understanding and conservation of learning effects. Students also need to fill out the rating-scale measurement of cognitive load. Furthermore, we investigate the relationship between learning process and comprehending geometry proof from students’ writing files of responding questions. The results show that practice with similar structure of worked-out example is helpful for the instant understanding for students; however, students are affected by the prototype of worked-out examples which tend to use copy-and-adapt strategy for doing practices. On the other hand, learning task combined with metacognition questions are more difficult than practices; however, metacognition questions reflect the students’ level of understanding and provide a better conservation. Hence, metacognition question is an ideal tool which is helpful for the reflection in student learning. It is suggested that proper pair of the metacognition question with the advantage of practices may support students to understand the content of deductive proof.

並列關鍵字

geometry proof ； worked-out example ； metacognition ； cognitive load

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