臺灣博碩士論文加值系統

電腦繪圖計算機增進了函數學習的效率。學生能快速繪製函數圖形，也能調整特定係數，觀察方程式與函數圖形的表徵轉換，這些行為有助其理解特定係數的功能與意義，進而為真實情境建立合適的數學模型。然而，電腦輔助教學之學習模組設計，需考量性向處理交互作用，以及多媒體設計原則，因應學生的先備知識多寡彈性調整，才能確保學生真正理解教師欲傳達的數學概念。
本研究二次函數圖形為主題，以Desmos電腦繪圖計算機為研究工具，設計示例引導、探索式兩種學習模組，將受測學生分為高先備、低先備知識組，透過相關分析、滯後序列分析與循序樣式探勘，分析前測成績、學習進展、先備知識分組與題組操作行為頻率的相關性，並比較不同先備知識學生，在不同的題組設計中，行為模式是否有差異，嘗試以數學理解層次理論解釋。
行為頻率相關分析結果顯示，在示例引導題組，先備知識越低，多數類別的行為頻率越高，代表受試者需要經歷較長的試誤過程，達到與先備知識較高者相近的數學理解層次；另外，學習進展越高，會有較多的觀察性質行為。而在探索式題組，先備知識較低者則有較頻繁的觀察性質行為和較低的完成答題率；學習進展越高，反而出現較少的觀察性質行為。
行為模式分析結果顯示，相較於示例引導題組，受試者在探索式題組的行為歧異度較高。雖能觀察出高先備組更擅長以特定參數概括問題情境，但不能單純以函數先備知識區分行為差異，需考量受試者的遊戲經驗和力學知識。
另外，分心行為在兩種題組都沒有與任何依變項達顯著相關，但透過滯後序列分析和循序樣式探勘，可以推測受試者出現認知負荷的前因後果，亦即示例引導效應、專家反轉效應、元素互動性效應的存在。
本研究行為頻率相關分析未達顯著高相關，行為模式分析也僅能用以初步推測學生的解題策略，顯示研究樣本數、學習環境、研究工具、題組設計和實驗設計均有未臻完善之處。除了改善研究方法，未來研究應可加強答題回饋功能、增加質化訪談的深度或廣度、使用多元迴歸分析將更多變因納入考量，以拼湊出學生完整的學習歷程。

Computer graphing calculators increase efficiency of learning of functions. With computer graphing calculators, students can generate graphs of functions quickly, adjust specific parameters, and then observe the transformation of representations. These behaviors enable students to understand features and meanings of specific parameters, and to build suitable mathematical models for real-life situations.
On the topic of quadratic functions, the researcher designed worked-example and discovery learning modules with Desmos, an online graphing calculator. The ninth-grade subjects are divided into two groups based on prior knowledge. In the correlation analysis, the researcher analyzed relationships between behavior frequencies and three variables: pretest score, learning progress, and grouping of prior knowledge. In the sequential analysis and the sequential pattern mining, the researcher compared differences of behavior patterns between high and low prior knowledge (PK) group in two different learning modules, and gave further explanations with the dynamical theory for the growth of mathematical understanding.
The results of correlation analysis show that in the worked-example module, less PK leads to more behaviors in most behavior categories. It means that subjects of lower PK experienced longer trial and error process to reach same levels of mathematical understanding as subjects of higher PK. In addition, more learning progress leads to more property-observing behaviors. Comparatively in the discovery learning module, less PK leads to more property-observing behaviors and lower completion rate.
On the other hand, results of behavior analysis show that the subjects have more diverse behaviors in the discovery learning module than those in the worked-example module. The high-PK group is expert in generalizing problem situation, but we cannot distinguish differences of behaviors without considering subjects’ gaming experience and knowledge of Mechanics.
Less but not the least, distraction doesn’t significantly correlate to any dependent variables. However, it is helpful for us to conjecture cause and effect of subjects’ cognitive overload, in other words the existence of worked-example effect, expertise reversal effect, and interactivity effect.
The correlation analysis fails to prove any significantly correlated behaviors. Also, the behavior analysis can only be used to preliminarily speculate on students' problem-solving strategies. Those results reveal that there is room for improvement for number of subjects, learning environment, research tools, module design and experimental design in this research. In addition to improvement of research methods, we can enhance feedback mechanisms of learning modules, increase the depth or breadth of qualitative interviews, and use multiple regression analysis to take more variables into account. Hopefully we can piece together the complete learning process of students in the future.

第一章 緒論 - 1 -
1.1 研究背景與動機 - 1 -
1.2 研究目的與問題 - 5 -
第二章 文獻探討 - 7 -
2.1 二次函數課程內涵與地位 - 7 -
2.1.1 函數課程的內涵 - 7 -
2.1.2 我國二次函數學習能力指標 - 8 -
2.1.3 多元表徵轉換 - 11 -
2.2 電腦輔助函數學習 - 13 -
2.2.1 電腦輔助代數學習 - 13 -
2.2.2 CAS融入二次函數教學文獻回顧 - 15 -
2.2.3 模擬導向教學（simulation-based learning） - 18 -
2.3 先備知識 - 21 -
2.3.1 先備知識與學習表現 - 21 -
2.3.2 專家與新手的先備知識 - 22 -
2.3.3 先備知識與電腦輔助學習 - 23 -
2.3.4 先備知識與多媒體設計原則 - 25 -
2.4 數學理解層次 - 30 -
2.4.1 數學理解的定義 - 30 -
2.4.2 數學理解成長模型 - 32 -
2.4.3 二次函數的數學理解層次 - 34 -
第三章 研究方法 - 37 -
3.1 研究流程 - 37 -
3.2 研究設計 - 38 -
3.2.1 研究法的選擇 - 38 -
3.2.2 研究架構 - 38 -
3.2.3 研究變項 - 39 -
3.2.4 實驗流程 - 42 -
3.3 研究對象 - 44 -
3.3.1 預試樣本 - 44 -
3.3.2 正式施測樣本 - 44 -
3.4 研究工具 - 48 -
3.4.1 學習模組設計 - 48 -
3.4.2 前後測試卷製作 - 55 -
3.4.3 行為模式分析編碼表 - 58 -
3.5 資料分析 - 64 -
3.5.1 SPSS Statistics 25.0 - 64 -
3.5.2 GSEQ 51 (Generalized Sequential Querier 51) - 65 -
3.5.3 PrefixSpan循序樣式探勘演算法 - 65 -
3.5.4 受試者訪談 - 67 -
第四章 實驗結果與分析 - 68 -
4.1 示例引導題組 - 69 -
4.1.1 行為頻率相關分析 - 69 -
4.1.2 滯後序列分析 - 75 -
4.1.3 循序樣式探勘 - 84 -
4.2 探索式題組 - 92 -
4.2.1 行為頻率相關分析 - 92 -
4.2.2 滯後序列分析 - 96 -
4.2.3 循序樣式探勘 - 104 -
第五章 結論 - 115 -
5.1 研究結果 - 115 -
5.1.1 行為頻率相關性 - 115 -
5.1.2 題組操作行為模式 - 117 -
5.1.3 數學理解層次 - 118 -
5.2 研究限制 - 120 -
5.3 未來展望 - 122 -
附錄一 二次函數理解測驗前測 - 131 -
附錄二 二次函數理解測驗後測 - 134 -
附錄三 示例引導題組與答題範例 - 137 -
附錄四 示例引導題組循序樣式探勘結果 - 141 -
附錄五 探索式題組與答題範例 - 150 -
附錄六 探索式題組循序樣式探勘結果 - 156 -

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