高中生對拋物線、橢圓及雙曲線的心智模式類型研究

Abstract

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.