電腦輔助學習系統之解題器設計( II )

Abstract

本研究以派特里網路(Petri net)追蹤與記錄數學解題路徑，並以現今高中三角函數單元做為數學知識範圍，發展一套名為MathCAL-2 之數學解題練習系統。我們將數學題目中的已知條件對應到派特里網路中的位置(place)，基本定理對應到轉變(transition)，解題過程於是等同於動態地觸發一序列的轉變，而系統所記錄的轉變觸發順序也就反應了使用者解題時的思考路徑。MathCAL-2 對於使用者解題過程的逐步掌握使其得以在使用者遭遇解題困難時判斷其困難所在並給予適當的提示。MathCAL-2 的解題運算核心以及整個系統架構的設計均依循物件導向理念，賦予位置與轉變統一的格式與溝通機制，使得龐雜的數學知識得以對應至單純的程式架構，亦使得解題核心可以彈性地加入新的轉變與位置，並因而能不斷擴增MathCAL-2 所能處理的題型。我們在後續的研究中將進行實證研究，以了解本系統之適用性以及它對於增進學生解題能力的成效。

In this research we have developed an Internet-based computer-assisted-learning system, called MathCAL-2, which allows learners to practice mathematical problem solving. Petri nets are used by the system as its knowledge representation model. Content knowledge of high school trigonometry is pre-analyzed to derive a set of macro functions for use in solving problems. Each macro function typically embodies a math concept or rule which may be applied to transform a math problem from a state to another. The macro functions map onto the transitions in a Petri net graph, while the conditions for triggering those functions map onto the places. With Petri net graphs, the system can store experts' static problem-solving knowledge as well as learners' dynamic problem-solving procedures. The graphs may then be used effectively for diagnosing learners' problem-solving difficulties and for offering guidance. Users may add new problems to the problem bank using system-provided interface. New places and transitions can also be easily incorporated into MathCAL-2's system kernel because of the uniform structures that are used to represent all places and all transitions respectively and also because of the well-defined communication mechanism that handles interactions between places and transitions. Experimental research will be carried out in the remainder of this research to investigate MathCAL-2's usability and its effectiveness in enhancing students' mathematical problem-solving skills.