MathCAL-II：應用派特里網路理論所建置之數學解題輔助系統

Abstract

數學解題是當今數學教育重要的課題。解題訓練依據坡里亞定義，可區分為瞭解問題、擬定計畫、實施計畫、回顧解答四個步驟。然而以講述為主的傳統教學型態受限於人力及時間，常無法引導學生完成解題步驟並達到適性化教學目標，應用電腦科技做為輔助工具，應可彌補傳統教學的某些不足。民國84年林美娟教授及莊志洋教授提出將派特里網路應用至數學解題，並完成MathCAL數學解題練習系統。此構想利用派特里網路中權杖的傳遞，系統可將解題步驟以文字或圖形的方式呈現，並可分析使用者的解題步驟以提供必要的引導。本研究修正歷年MathCAL系統核心、診斷與矯正機制等部份細節，在發展過程中邀請數學教師和高中學生參與測試與建議，並且將教師的教學知識融入系統發展中，完成MathCAL-II。系統完成後針對高中生進行實驗研究，分析實驗結果發現，MathCAL-II可有效地輔助學生解題，並且達到顯著成效。實驗中的問卷調查結果亦顯示MathCAL-II所提供的功能皆符合使用者解題的需求，並且能夠加強學生對於數學概念的瞭解，對學生的學習態度與學習興趣皆有正向影響。MathCAL-II is a piece of software for students to practice solving mathematical word problems in trigonometry. At the core of the system is a database that stores experts’ problem-solving paths. These solution paths are stored as Petri net graphs internally. MathCAL-II also dynamically records a learner’s solution path during a practicing session. Thus it can compare a learner’s partial solution path against all correct solution paths to identify a best match, which can then be used to provide guidance to a learner in case s/he requests help. In a series of experiments that we conducted in a local high school, 58 students who participated voluntarily were divided randomly into the treatment group and the control group. The treatment group used MathCAL-II to practice solving trigonometric word problems, whereas the control group used the traditional pencil-and-paper approach. Results revealed that the students in the treatment group made significantly more progress than those in the control group. A questionnaire survey also showed that MathCAL-II not only improved students’ understanding of mathematical concepts in trigonometry but had positive influence on students’ attitudes toward learning trigonometry.