中文摘要
幾何是數學中重要的分支，而幾何圖形的心像在幾何學習上扮演重要的角色。心像操作的本質始終存在著爭議，且目前心理學界尚未清楚心像運作的細節內容。本研究嘗試設計各種不同形式剛性變換下的兩個相似三角形，以探究國中生如何建構心像來知覺其相似性。本研究首先提出心像相關的四個層面：心像的保量情形、心像對作圖歷程的影響、相似圖形作圖的錯誤類型、以及圖形的「位置」對作圖歷程的影響，並以尚未學習過相似概念的五十位八年級學生為樣本，進行施測。
針對國中生心像的保量情形，研究結果清楚指明國中生在直角三角形直角保量的能力較好，對鈍角三角形鈍角保量的能力較差，同時對鈍角三角形次長邊長度保量的能力較強，並對鈍角三角形交疊時鈍角保量的能力更弱，以及國中生對鈍角三角形交疊時子圖個數保量的能力較弱。再者，本研究也分析國中生心像對作圖歷程的影響，結果顯示其歷程大致可區分為四種。另外，關於國中生相似圖形的作圖結果，研究也發現在各種伸縮旋轉變換下作圖的錯誤類型可區分為二十種類型，在各種伸縮鏡射變換下作圖的錯誤類型可區分為十四種類型。
再就圖形的「位置」對作圖歷程的影響而言，研究結果顯示（1）當兩圖形的位置「相離」時，兩相似鈍角三角形的作圖答對率為最低；（2）當兩圖形的位置「共點」時，一般來講，國中生在伸縮鏡射的答對率都比伸縮旋轉的表現來得好。（3）當兩圖形的位置「共邊」時，從訪談中得知學生也會受到某種程度的干擾，例如兩個圖形整個一起看，導致無法在心像上旋轉或翻轉。（4）當圖形的位置「交疊」時，子圖個數會影響對作圖的答對率，一般而言交疊時的子圖個數愈多，則答對率一般愈低。最後，研究結果將提供對數學教學以及對後續研究的相關建議。

Abstract
Geometry is an important branch of mathematics, and the mental imagery of geometric figure is very important during the learning of geometry. The nature of the imagery operation is always controversial, and it is not yet clear about the content details of the imagery operation in psychology. We tried to design two similar triangles of different forms under rigid transformation, to explore how middle school students perceive similarity by forming mental imagery. To this end, we first proposed four main aspects of mental imagery, including conformal invariance of mental imagery, mental imagery's influence on sequential apprehension, error types of similar graphics drawing and the influence of the position of the figure on sequential apprehension. We constructed a survey and administrated it to fifty 8th graders, who have not learned the concepts of similarity before.
There are five finding about conformal invariance of mental imagery in this study. First, middle school students’ ability to maintain conformal of right angle is better. Second, middle school students’ ability to maintain conformal of obtuse angle is worse. Third, middle school students’ ability to maintain congruent of second long side is better. Fourth, middle school students’ ability to maintain conformal of obtuse angle is worst, when two obtuse triangles are overlapped with each other. Fifth, middle school students’ ability to draw the same number of subfigure with original figure is worse, when two obtuse triangles are overlapped with each other. Furthermore, we find four processes about mental imagery's influence on sequential apprehension in this study. Moreover, regarding the error types of similar graphics drawing, we find twenty error types under stretch rotation, and we also find fourteen error types under stretch reflection.
Concerning the influence of the position of the figure on sequential apprehension, there are four finding in this study. First, the correct rate of drawing two similar obtuse triangles is lowest when two triangles are disjoint. Second, general speaking, the correct rate under stretch reflection is better than under stretch rotation when two triangles have only one common point. Third, when two triangles have only one common edge, we find some degree of interference, which is that subjects may look two figures together as entire one, cause subjects to be not able to rotate or flip the image. Fourth, when two triangles are overlapped with each other, the number of subfigure will affect the correct rate of drawing. Generally, the more the number of subfigure is, the lower the correct rate is. Accordingly, we provide some suggestions regarding mathematics teaching.

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