本研究的目的是對於高中學生在拋物線、橢圓及雙曲線的心智模式類型進行研究，進而了解學生學習過圓錐曲線單元後，對於拋物線、橢圓及雙曲線所形成的概念，以及在這些概念之下所形成的心智模式。

本研究為質性研究，使用半結構性晤談。研究之初，研究者先對幾位科教所不同背景的學生進行訪談，以瞭解本研究是否可行。接下來，研究者先以一般性的問題對學生進行紙筆測驗，並與任課老師討論後，挑選出適合晤談的學生。訪談的過程中，發現學生的表現與一般性的問題表現並無太大的正相關，因此研究者與指導教授討論後，再挑選另一所PR值無太大差異的國立高中，由教師挑選不同能力及不同組別的學生進行晤談，在正式施測時，兩所學校的研究對象總共為自然組及社會組學生各6位。

透過質性資料的分析，本研究發現學生對於拋物線的心智模式類型有火箭筒型、類拋物線型；對橢圓的心智模式類型有操場型、類橢圓形；對雙曲線的心智模式類型有雙曲線型、類雙曲線型。另有一心智模式，受圖形外觀影響較大的，本研究稱為受圖像型。在這些類型中，研究者歸納出學生判斷拋物線、橢圓、及雙曲線可略分為六個概念，分別為「開口」、「無限延伸」、「弧度」、「對稱」、「漸近線」及「幾何定義」。這六個概念中，「開口」為教師們在圓錐曲線單元的教學的常用語；「無限延伸」為描述圖形的發展；「弧度」與高中教材所定義的弧度(radian)不同，為學生為描述一曲線彎曲的程度所使用的日常用語，且不同的學生在使用上所表達的概念不盡相同；「對稱」為圓錐曲線中一重要的概念；「漸近線」為學生觀察雙曲線的重要概念；「幾何定義」對學生而言，為不太容易了解的概念。

本研究發現學生學習過圓錐曲線單元後，仍舊有許多與正確模式相異的心智模式存在。建議教師在教學的過程中，強調拋物線、橢圓及雙曲線的幾何定義與學生所熟知的各概念，如「開口」、「對稱」等作連結。本研究另發現不同學生口中的「弧度」不盡相同，因此建議教師教授本單元時，可利用離心率切入，使學生在理解拋物線、橢圓及雙曲線具有融貫性。

The main purpose of this study is to find out what conceptions senior high school students in Taiwan may have in relation to parabola, ellipse and hyperbola after formal instruction. A further purpose is to identify the kinds of mental models that students may perceive about these three mathematical objects.

This study adopted a qualitative analysis approach and was executed in three stages, each with its specific purpose. This study began by interviewing with graduate students in science education with different background to help formulate its research question. This formed the explorative stage of the present study. During the second adjustment stage, in order to help focus the research direction, a general test on conic sections was compiled. After administering the test to a group of senior high school students, a number of them were recommended by their math teacher to be interviewed by the present researcher. However, it was later found out that their performance in the clinical interview did not quite related to those presented in the written test. After discussing with an expert, it is decided to pick up an extra senior high school the average abilities of its students was no difference from the previous one. 12 students with different genders and different academic abilities from the two senior high schools were selected by the present researchers as participants in the third stage, the formal stage. The students were given diagrams with portions of curves from different conic sections and were then probed for various mathematical judgment.

After data coding and analysis, it was found that some of the participants did reveal certain mental models when they thought of different conic sections. For the parabola, the mental models found include the rocket model and a parabola-like model. The mental models for ellipse include the playground model and an ellipse-like model. As for the hyperbola, the mental models include the two-parabola model and a hyperbola-like model. Besides these mental models, there is an extra one known as the graphic model with which the participants were affected by the shape of the curves. It was also found that there were six concepts that the participants used to distinguish between various curves. They were the concept of opening, infinity, “radian,” symmetry, asymptote, and mathematical definitions of the mathematical objects. In particular, the way the concept “radian” was used was different from the formal definition. Here, it was used to describe the degree of bending of a curve and was used as a daily term. In general, it was found that many participants could not master a deeper realization of the definitions of conic sections.

The results revealed that many participants still held different mental models regarding the mathematical objects about which they were being instructed. This study suggested that mathematics teachers should focus on the definition of parabola, ellipse, and hyperbola and try to enhance students’ understanding regarding their differences. Moreover, they may consider introducing the concept of eccentricity to help clarify the differences between parabola, ellipse, and hyperbola.

中文部份
石宛臻 (2004)：反例對國小五年級學童四邊形幾何概念調整的影響。國立臺北師範學院數理教育研究所碩士論文。
沈佩儀 (2007)：高二學生解拋物線題目之研究。國立政治大學應用數學研究所碩士論文，未出版，台北市。
陳吟汝 (2005)：台南地區高二學生圓錐曲線單元錯誤類型之分析研究。國立高雄師範大學數學系碩士論文，未出版，高雄市。
黃美玲 (2005)：橢圓標準式解題歷程與錯誤類型之分析。國立彰化師範大學科學教育研究所碩士論文，未出版，彰化市。
溫安榮 (2007)：GSP融入數學教學對高二學生數學學習成效影響之研究 : 以「圓錐曲線」單元為例。國立高雄師範大學數學系碩士論文，未出版，高雄市。
鄭英豪 (2000)：圓錐截痕與二次曲線：一個數學老師的無聊之舉。數學傳播，23(3)，21-33。
蔡志仁 (2000)：動態連結多重表徵視窗環境下橢圓學習之研究。國立台灣師範 大學數學研究所碩士論文，未出版，台北市。
左台益和蔡志仁 (2001)：高中生建構橢圓多重表徵之認知特性。科學教育學刊，9(3)，281-297。
蘇惠玉 (2007)：從正焦弦看圓錐曲線。高中數學電子報，18。
許志農 (2008)：普通高級中學數學第四冊。台北縣：龍騰文化。
教育部 (2003)：國民中小學九年一貫課程綱要。台北：教育部。
教育部 (2008)：普通高級中學課程綱要。台北：教育部。
教育部國語推行委員會編纂 (1994)：重編國語辭典修訂本。台北：教育部。
進入兒童心中的世界 (謝如山譯) (2004)：台北市：五南圖書(原著出版年：1997年)。
數學學習心理學 (陳澤民譯) (1995)：台北市：九章出版數(原著出版年：1987年)。
陳彥廷，柳賢（2005）：合作學習情境中學生數學概念學習之研究-以Posner 概念改變模式與Toulmin 論證模式分析為例。科學教育研究與發展季刊, 第三十九期, p.80-105

英文部分
Janvier, C. (1987a). Representation and Understanding: The Notation of Function as an example. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 67-71). Hillsdale, NJ: Lawrence Erlbaum.
Janvier, C. (1987b). Translation Processes in Mathematics Education. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 27-31). Hillsdale, NJ: Lawrence Erlbaum.
Kaput, J. J. (1987). Representation Systems and Mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp 19-26). Hillsdale, NJ: Lawrence Erlbaum.
Mason, H. J. (1987). What do Symbols Represent? In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 73-81). Hillsdale, NJ: Lawrence Erlbaum.
Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. Mathematical Teaching, 77, 20-26.
Vinner, S. (1983). Concept Definition, Concept Image and Notion of Function. International Journal of Mathematical Education in Science, 14(3), 293-305.
Vinner, S. (1991). The Role of Definitions in the Teaching and Learning of Mathematics. In D. Tall (Ed.), Advanced Mathematical Thinking.(pp.65~81). Dordrecht: Kluwer Academic Publishers.
Vosniadou, S. (1994). Capturing and Modeling the Process of Conceptual Change. Learning and Instruction, Vol. 4, 45-69.
Vosniadou, S. (2006). The Conceptual Change Approach in the Learning and Teaching of Mathematics: An introduction. In Novotna, J., Moraova, H., Kratka, M., & Stehlikova, N. (Eds.). Proceedings 30th Conference of the InternationalGroup for the Psychology of Mathematics Education, (Vol.1, pp. 155-184). Prague: PME.
Biza, I., Souyoul, A.& Zachariades, T. (2005). Conceptual Change in Advanced Mathematical Thinking. Proceeding 4th Congress of the European Society for Research in Mathematics Education.
Lesh, R., Post, T., & Behr, M. (1987). Representation and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.
Merenluoto, K and Lehtinen, E. (2002). Conceptual Change in Mathematics:
Understanding real numbers. In M. Limon & L. Mason (Eds.), Reconsidering conceptual change. Issues in theory and practice. Kluwer.
Merenluoto, K. amd Lehtinen, E. (2004). Number Concept and Conceptual Change: Towards a Systemic Model of the Processes of Change. Learning and Instruction, 14 (5) 519-534.
Merenluoto, K. and Lehtinen, E. (2006). Conceptual Change in the Number Concept: Dealing with Continuity and Limit. In J. Novotna, H. Moraova, M. Kratka, and N. Stehlikova (Eds.) Proceedings of the 30th Conference of the International group for the Psychology of Mathematics Education. Prague: Charles University, (pp. 163-165).
Tall, D.O. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics, with Special Reference to Limits and Continuity. Educational Studies in Mathematics, 12, 151-169.
Ustun, I. & Ubuz, B. (2004). Student’s Development of Geometrical Concepts through a Dynamic Learning Environment. Paper presented at the meeting of the 10th International Congress on Mathematical Education, Copenhagen, Denmark. [Topic Study Group TSG-16] Retrieved February 20, 2006. from http://www.icme-organiser.dk.
Van Dooren, W., Verschaffel, L., & Onghena, P.: (2002). Impact of Preservice Teachers' Content Knowledge on their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems. Journal for Research in Mathematics Education, 33(5), 319-351.
Vinner, S. ＆ Dreyfus, T. (1989). Images and Definitions for the Concept of Function. Journal for Research in Mathematics Education, 20(4), 356-366.