在本研究中，探究數學層面一般資賦優異 (G) 和數學學習優異 (EM) 對高中生數學表現的影響和相互作用，透過事件相關電位研究 (ERP) 了解函數圖形與方程式表徵之間轉換的大腦皮質活動。本研究對161名10、11年級的高中生 (16-18歲)進行了抽樣調查，並根據G和EM因子的組合，將學生分成八個實驗組。數據分析將透過兩種方式進行分析，第一種是行為測量，分析G和EM因子在解決函數圖形與方程式表徵轉換問題時，正確率 (Acc) 和答對題目的反應時間 (RT) 的影響。另一種是腦波測量，主要運用ERP成分和腦波平均振幅分析G和EM因子對於大腦不同區域反應差異的影響。
本研究結果依行為測量與腦波測量分別介紹：在行為測量方面，一般資賦優異和數學學習優異學生比非一般資賦優異和數學學習優異學生表現來的好，有更高的正確率與更短的答對題目反應時間。G和EM因子都顯著影響學生解決函數表徵轉換問題的表現。在腦波測量中，學生在閱讀題目階段和解決題目階段，大腦啟動視覺刺激與推理的過程相似，且有相似的腦力活躍程度，但在不同G和EM因子的學生在不同問題解決階段啟動的大腦區域不同，而受到神經傳導效應影響，被認定為一般資賦優異的學生比非一般資賦優異的學生投入了更少的腦力活躍程度。最後，G和EM因子對行為和腦波測量沒有交互影響。透過這些發現，本研究建議進一步透過特徵識別一般資賦優異和數學學習優異的學生，並為他們提供適應性教育以培養和發展他們的數學能力，並設計能激發不同大腦思考區域的教案與課程。透過跨領域的研究，期許透過腦神經科學的面向，能為數學教育有更多面向的探討。

In this study, the effects and interactions of general giftedness (G) and excellence in mathematics (EM) on high school students' performances in mathematics were investigated. The event-related potentials (ERPs) analysis method was conducted to understand the cortical activity when students were solving conversion problems between function graphs and equation representations. This study sampled 161 grade 10-11 high-school students and distributed the students into eight experimental groups by a combination of the G and EM factors. The data were analyzed in two measurements. The behavioral measurement analyzed the effects of the G and EM factors on accuracy (Acc) and reaction time (RT) in solving function graph and equation representation conversion problems. The other is the electrophysiological measurement, which focused on the effects of the G and EM factors on the ERP components and the mean amplitude of brainwaves from different brain regions that caused differences in responses.
The results revealed several dimensions. From behavioral measures, the giftedness and excellence in mathematics students had a superior performance by showing higher accuracy and shorter reaction time compares to non-giftedness and excellence in mathematics students. Both G and EM factors significantly influenced students' performance in solving the function representation transformation problem. Similar visual stimulation and reasoning processes and levels of brain activity were discovered in students' cortical activity during the question-reading and problem-solving stages. However, students from different G and EM factors groups generated different brain regions at different problem-solving stages. The neuro-efficiency effect decoded that the students who were considered to be gifted invested less brain activity than students who were not gifted. Finally, the G and EM factors did not have an interactive effect on behavioral and brainwave measures. This study suggests using characteristics to identify general giftedness and excellence in mathematics students. Adaptive education such as design lesson plans and curricula that help stimulate different brain thinking areas should provide to students to nurture and further develop their mathematical abilities. Also, expectations on exploring more aspects of mathematics education from the brain neuroscience perspective were made through this cross-disciplinary research.

摘要..................................................................................................................................... i
Abstract .............................................................................................................................. ii
致謝.................................................................................................................................. iii
I Introduction ..................................................................................................................... 1
1.1 Background .......................................................................................................... 1
1.2 Purpose ................................................................................................................. 3
1.3 Terminology ......................................................................................................... 3
1.4 Hypothesis............................................................................................................ 5
II Literature Review ........................................................................................................... 7
2.1 Studying Functions in High School ..................................................................... 7
2.1.1 Definition and Importance of Functions ................................................... 7
2.1.2 Functions Representation and Transformation ......................................... 8
2.1.3 Study of Functions in Textbooks ............................................................ 12
2.2 Neurocognitive Research ................................................................................... 15
2.2.1 Physiological Signal................................................................................ 15
2.2.2 Electroencephalogram............................................................................. 18
III Methodology ............................................................................................................... 30
3.1 Research Design................................................................................................. 30
3.2 Research Tools ................................................................................................... 30
3.2.1 Paper-and-Pencil Tests ............................................................................ 30
3.2.2 Computerized Brainwave Tests .............................................................. 32
3.3 Participant .......................................................................................................... 37
3.3.1 Study Groups .......................................................................................... 37
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3.4 Data Collection .................................................................................................. 38
3.4.1 Before the Experiment ............................................................................ 39
3.4.2 During the Experiment ............................................................................ 46
3.4.3 After the Experiment ............................................................................... 48
3.5 Data Analysis ..................................................................................................... 49
3.5.1 Behavioral Measures ............................................................................... 49
3.5.2 Electrophysiological Measures ............................................................... 50
3.6 Statistical Analysis ............................................................................................. 53
IV Result and Discussion ................................................................................................. 54
4.1 Behavioral Measures .......................................................................................... 54
4.1.1 Basic Information.................................................................................... 54
4.1.2 Function in Accuracy and Reaction Time ............................................... 55
4.1.3 Simple and Complex Function in Accuracy and Reaction Time ............ 58
4.2 Electrophysiological Measures .......................................................................... 66
4.2.1 Early components.................................................................................... 67
4.2.2 Late potential components ...................................................................... 74
4.2.3 Early components in G factor and EM factor ......................................... 77
4.2.4 Late potential components in G factor and EM factor ............................ 84
V Discussion .................................................................................................................... 90
5.1 Conclusion ......................................................................................................... 90
5.2 Implication ......................................................................................................... 94
5.3 Limitation ........................................................................................................... 95
VI Reference .................................................................................................................... 96

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