建構數學方法的動態學習環境之研究

Abstract

近年來，電腦網路的普及與資訊設備的提升，帶動課堂上的數學教師將資訊科技融入教學，有助於增強學生學習動機，發展學生主動探索的精神與解決問題的能力，對於學習數學賦予肯定的價值意義。本研究主要針對教學設計加以探討，以「計數」、「遞廻」、「機率」等三種數學方法為例，在PowerPoint平台下搭配Activate Mind Attention的特性，佈置一個動態數學解題情境，教導學生如何在不同的脈絡中應用數學推理，傳達數學方法在實際生活應用之重要性。

本論文分三部分，第三章透過排隊買票問題，將問題條件動態呈現轉化為數學模型，來探索問題中卡特蘭數的計數方法；第四章藉由AMA建構重複的實驗環境，來模擬操作骨牌拼圖遊戲，探索費布納西數列的遞迴性；第五章則從數學史的觀點和個人生活經驗，探索數學期望值的概念，並透過彈珠台的Java程式體驗機率方法的隨機性，來說明期望值如何應用在決策判斷上。期望本研究能啟發數學教育者讓學生體驗數學知識創造的過程，並享受發現的樂趣。

The popularity of Internet and enhancements of information devices enable teachers to integrate information technology into their teaching materials. Based on mathematical methods including counting, recurrence and probabilistic expectations, we will explore course material designing in this research. In particular, dynamic environments based on PowerPoint with the featuring of Activate Mind Attention (AMA) are bulit for instructing how to apply mathematical reasoning in different contexts, followed by conveying the importance of mathematical methods in real life.

The dollar change problem is used to explore the Catalan Numbers in Chapter 3. In Chapter 4, a dynamic environment via AMA is given to simulate the finding the recurrence relations behind the Fibonacci sequence based on the of domino coverings of a 2 by n chess board. In Chapter 5, focusing on the historical deverlopments and personal experiences, the concept of mathematical expectation is analyzed. Through a Java applet in pinball games, the notion of randomness and the role of probabilistic expectation in decision making are illustrated.