關於建構三角函數動態學習環境之研究

Abstract

三角函數是用來描述現實世界中具有週期現象的重要數學模型，例如潮汐循環、溫度起伏等問題。然而由於三角函數的抽象性，往往也是大多數高中學生感到頭疼與棘手的數學題材。在資訊科技高度發展的現在，許多繁雜的運算可以透過計算機輕鬆完成，如此一來，「公式背誦」以及「正確計算」的能力便不再如以往一般的重要性。因此，是否可以利用資訊科技帶給學生更有效的學習？是否可以讓他們克服對三角函數抽象性的排斥？ 有鑑於此，本論文將透過GSP及MathPS的動態展示功能，來建構動態三角函數的教學與學習環境。一方面，動態的展示可以提供給學生對於三角函數更為直觀的了解；其次，藉由電腦的生動表現，可以彌補書本上只有靜態示意圖的缺憾。我們期望能夠藉由這種方式，將三角函數學習的重點從「公式、定理的推導」轉移到「觀察函數的概念、性質與圖形」。使學生能夠對三角函數有具體而圖形化的了解，進而打破他們對於三角函數的恐懼心理。 本論文也將藉由「數學建模」構想的引入，搭配電腦軟體操作，來解決現實生活中所會面臨到的問題，讓學習者瞭解如何將生活問題轉化成數學問題，進而對該問題進行探究，找出合適的數學模型，最後加以驗證。

Trigonometric functions are basic functions used for describing the periodic patterns in the real world, though the topic of trigonometric functions is one of the difficult subjects for most high school students because of its high degree of abstraction involved, followed by many lengthy formulas and complicate computations. However, complicated computing nowadays can be done easily by computers under well prepared instructions. We are considering if the current information technology can be used to provide students friendly and effective learning environments for studying trigonometric functions.

The purpose of this thesis is to manage friendly and user oriented environments for helping students to catch the insights and the usages of trigonometric functions efficiently and to realize how the mathematical knowledge can be used in treating the problems from the real world as well. In particular, the software packages GSP and MathPS will be used in this thesis for developing environments for dynamic presentations of trigonometric functions. By taking fully advantage of the dynamic presentations, we will have chances to transform the focuses from formulas and proofs to the insights and features of trigonometric functions in the environments supported by GSP and MathPS. Some examples relating to tides and temperatures will also be introduced to show how the real world problems can be transformed to mathematics problems within the frameworks of mathematical modeling.