知識表徵的程序化與概念化：以高二數學的排列原理為例

Proceduralizing and Conceptualizing Representation of Knowledge: An Example in Permutation Theorems

本研究的目的在探討初學一個數學主題時，學習者將所習得的原理知識程序化，或是例題知識概念化的情況。本研究共有54位高中一年級的學生自願參加，他們被隨機分配至原理組、單例題組、及三例題組。參與者首先完成目標學習內容的前測。之後，他們學習相關的先備知識，並且通過先備知識的測驗。接著，參與者學習各組指定的目標內容，然後完成相對應的目標知識測驗。其中原理組學習兩個排列原理然後接受概念測驗，單例題組就各原理學習一個例題然後完成應用測驗，三例題組就各原理學習三個例題然後完成應用測驗。為瞭解原理組將其原理知識轉換為特定程序性知識的效果，在概念測驗之後需完成應用測驗，以便與兩個例題組進行比較。同樣地，為瞭解兩個例題組將其例題知識轉換為一般語意性知識的效果，在應用測驗之後需完成概念測驗。最後，每一組皆完成遷移測驗。結果顯示，在應用測驗上，原理組和單例題組的表現較三例題組為差，而前兩者之間並無差異。在概念測驗上，兩個例題組的表現較原理組為差，而前兩者之間並無差異。在遷移測驗方面，三個組別的表現則不相上下。這些結果意味著，原理組將其一般的語意性知識轉化為特定的程序性知識的效果有限，例題組將其特定的程序性知識轉化為一般的語意性知識的效果亦有限，而三個實驗組在應用其現有的知識以解決較為新穎的問題上亦缺乏彈性。

The purpose of this research was to investigate that， when initialially learning a mathematical topic， how well learners pro- ceduralizes their principle knowledge or conceptualize their example knowledge. Fifty-four tenth grade students of volunteered to par- ticipate in this study. They were randomly assigned to the principle， single-example， or three-example condition. The participants at first finished the pretest on the target content to be learned. Afterwards， they studied material for the prerequisite knowledge and passed the test for that. Next， the participants studied material for the target knowledge according to their assigned conditions and took the corresponding test for it. The principle condition learned two mathematical principles and then took the concept test. The single-example condition learned one example for each principle and then took the application test. Similarly， the three-example condition learned three examples for each principle and then took the application test. In order to understand the effect of the principle condition in transforming the principle knowledge into specific procedural knowledge， the participants had to take the application test afterwards for comparison with the other two conditions. Similarly， in order to understand the effects of two example conditions in transforming specific procedural knowledge into general semantic knowledge， participants took the concept test afterwards. Finally， all conditions completed the transfer test. Results showed that on the application test， the principle and single-example condition had worse performance than the three-example condition， and the former two conditions did not differ. On the concept test， the two example conditions had worse performance than the principle condition， and the former two conditions did not differ. On the transfer test， the three conditions had comparably low performances. These results indicated that the effect of transforming general semantic knowledge into specific procedural knowledge was limited. Similarly， the effect of transforming specific procedural knowledge into general semantic knowledge was limited as well. In addition， all three conditions lacked flexibility in applying their current knowledge to solve novel problems.