本研究以探究九年級學生在三類型幾何題目的解題表現差異，及學生對幾何性質的對自我概念了解、概念理解與解題表現間的交叉分析，為主要研究方向。
透過問卷調查法，採「方便樣本－簡單隨機分派」方式，進行「幾何性質－認知理解」及「幾何代數、幾何計算、幾何證明」問卷施測。
此次施測對象為台北市、新北市、桃園市、新竹縣等七所國民中學之九年級學生，共計378位學生。施測結果以描述性統計、ANOVA-單因子變異、類別變項-關聯分析、卡方檢定等統計分析法，就施測結果進行分析與研究。
研究結果顯示，學生在三類型題目的解題表現中，以「幾何計算」類型題目的解題表現最佳，顯著優於「幾何代數」與「幾何證明」。
在「對自我概念了解」與「概念理解」的解題表現中，學生認為「對稱角性質」及「對應角性質」為較容易被理解之幾何性質，而「圓切線段性質」、「三角形外角定理」與「平行線內錯角性質」則為較困難理解之幾何性質。對幾何性質內容進行描述時，學生在「對稱角性質」及「對應角性質」上有較好的解題表現，在「三角形外角定理」、「圓切線段性質」與「平行線內錯角性質」上，表現較為不理想。在「對自我概念了解」和「概念理解」的交叉分析中顯示，對自我概念了解表現不同之學生，對性質內容的描述表現上，有顯著差異。
在「概念理解」與「解題表現」關聯分析中，學生對關鍵性質的概念理解程度不同，在「幾何代數」與「幾何證明」兩類型題目的解題表現上，呈顯著差異。

The main direction of this research is to explore the differences in the performance of ninth grade students in solving the three types of geometric problems, as well as the students' mathematics self-concept relating to geometric properties, and finally the cross-analysis between conceptual understanding and problem-solving performance.
Through the questionnaire survey method, the "Convenient Sample-Simple Random Assignment" method was used to conduct the "Geometric Properties-Cognitive Understanding" and "Geometric Algebra, Geometric Calculation, Geometric Proof" questionnaires.
The subjects of this test were ninth grade students from seven national high schools in Taipei City, New Taipei City, Taoyuan City, and Hsinchu County, with a total of 378 students. The test results have been analyzed and researched by statistical analysis methods such as descriptive statistics, category variable-relational analysis, and chi-square test.
The results of the research show that among the three types of questions, students performed best with the “geometric calculation” type, and significantly less well with “geometric algebra” and “geometric proof” questions.
In the problem-solving performance of "understanding of self-concept" and "conceptual understanding", students seem to find that "symmetrical angle properties" and "corresponding angle properties" are geometric properties that are easier to understand, while "circle tangent properties,” "triangle exterior angle theorem" and "the property of staggered angles within parallel lines" are geometric properties that are more difficult to understand. When describing the content of geometric properties, students have good problem-solving performance using "Symmetrical Angle Properties" and "Corresponding Angle Properties.” However, in the "Triangle Outer Angle Theorem," "Circle Tangent Line Segment Properties" and "Parallel Line Internal Staggered Angle Properties," the performance is less satisfactory. The cross-analysis of "mathematics self-concept" and "understanding of concept" shows that students who have different performances in mathematics self-concept have significant differences in their description of nature and content of mathematics questions'.
In the correlation analysis between "concept understanding" and "problem solving performance," students have different levels of conceptual understanding of key properties, and there are significant differences in the problem solving performance of "geometric algebra" and "geometric proof".

參考文獻
中文部分：
1. 李政貴(2002)。 從數學難，數學美談數學教育。科學教育月刊，(253)，19-27。
2. 鄭英豪（2010）。國小學生視覺化幾何性質圖形的困難。數學暨資訊教育研討會。
3. 左台益、梁勇能。(2001)。國二學生空間能力與 van Hiele 幾何思考層次相關性研究。師大學報： 科學教育類。
4. 黃佩岑、陳斐卿(2020)。國小學生數學自由擬題困難之初探。弘光學報，(85)，59-80。
5. 李怡君(2014)。探討國中資優數學幾何圖形之輔助線。中原大學應用數學研究所學位論文。
6. 薛育青(2013)。國小獨立研究常用的研究方法。檢索日期:2020.7.10。取自 World Wide Web:http://163.23.67.198/ttjhswp/research/files/2013/08/問卷調查法.pdf
7. 黃文仙(2002)。顧客互動強度、顧客知識管理能力及顧客特性對新產品績效影響之研究－以台灣資訊軟體產業為例。中原大學企業管理研究所碩士論文。
8. 國家教育研究院(2018)。十二年國民基本教育課程綱要總綱。臺北市。
英文部分：
1. Anderson, J. R. (1983). Acquisition of proof skills in geometry. In Machine learning (pp. 191-219). Springer, Berlin, Heidelberg.
2. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. Handbook of research on mathematics teaching and learning, 420-464.
3. Duval, R. (1998). Geometry from a cognitive point of view. Perspectives on the Teaching of Geometry for the 21^< st> century, 37-52. Dordrecht: Kluwer.
4. Fey, J. T. (1984). Computing and Mathematics. The Impact on Secondary School Curricula. Report of a Conference Sponsored by the National Science Foundation (College Park, Maryland, October, 1982). National Council of Teachers of Mathematics, 1906 Association Dr., Reston, VA 22091.
5. Hsu, H. Y., & Silver, E. A. (2014). Cognitive complexity of mathematics Instructional tasks in a Taiwanese classroom: An examination of task sources. Journal for Research in Mathematics Education, 45(4), 460-496.
6. Piaget, J., & Inhelder, B. (1967). The child's conception of space (F.J. Langdon & J.L. Lunzre, Trans.). New Tork: W.W. Norton.
7. Piaget, J., Inhelder, B. & Szeminska, A. (1960). The child's conception of geometry. London: Routledge and Kegan Paul.