在本研究中，我們特地針對『如何活用數學觀念』的目標，以國中數學第六冊第三章『數值三角及其應用』為範圍，實際發展出一套可在網路上提供多人同時使用的數學解題系統。本系統以派翠西網路(Petri-Net)理論為基礎，將解題過程對應至派翠西網路結構中的位置(place)、轉變(transition)、流向關係(arc)、記號(token)等觀念，使得系統具有追蹤、記錄、分析、判斷學生解題過程與行為的能力。此套解題系統提供一個解題環境給予學生進行數學解題的行為，並且在學生的解題過程中予以適當的提示與引導。此外，本系統亦可允許多人同時上網使用。當某個學生不會解數學問題時，若系統所給予學生的幫助效果不大，學生亦可尋求網路上其他正在使用本系統的同學幫忙。本系統提供『線上求助』及『討論區』等功能，使學生們可在網路上進行一對一的討論。系統會將求助者所有的學習資訊同步傳送至另一方，讓教導者可以一邊觀察求助者之解題過程，一邊以交談方式提供適當的協助。

The objective of this research is to design and implement an Internet-based learning environment for middle-school-levelmathematics. The kernel of the environment will be a problemsolver which will allow students to learn critical mathematicalconcepts by doing. In this environment the typical learning activity is a learner-centered problem-solving process. Learners may select any problem from a rich set of built-in problems and proceed with the steps of solving the problem. The environment will provideabundant tools to assist learners. Furthermore, the system will be capable of keeping track of the learner's problem-solving steps so it can determine the correctness of the learner's application of concepts in each step and provide hints or guidance when the learner gets stuck somewhere in the process. Furthermore, this Internet-based learning environment will allow simultaneous accesses of the system by multiple users. When the system fails to provide useful assistance to the learner, he/she may seek help from other users with system-provided communication functions.