本研究的主要目的在建立三視圖的假設性學習路徑，研究分三個階段進行。第一階段，藉由訪談，了解學生對於三視圖的自發性解題策略。第二階段，發展問卷，以調查五至八年級學生的積木方塊三視圖的表現。第三階段，形成三視圖的假設性學習路徑，並以教學實驗檢驗其有效性。
分述各階段的研究方法如下。第一階段的研究，是以半結構性訪談的方式進行，訪談對象是六位十年級學生，問題的變項包含：積木數量與可否懸空。第二階段的研究，是以問卷調查的方式進行，受測對象為551位五至八年級學生。共發展三項三視圖相關任務：「由視圖選立體圖」、「由底層和視圖計數」和「由視圖選出相容視圖」。其中，按給定的視圖資訊，「由視圖選出相容視圖」又分為三種題型：右俯、前俯，和前右。本研究探討了年級、操作物的有無、任務類型對學生表現的影響。第三階段的研究，是以單一組前後測的教學實驗方式進行，教學實驗的對象包含一個五年級、一個六年級，與一個十年級的班級。
研究結果發現，學生對於積木方塊三視圖的解題策略包含：視覺與局部推理，而使用的表徵包含：語言、圖像、模擬與心像。當積木數量由二立方增至三立方時，需積木模擬才能解題的學生增加，當積木由不懸空發展成可懸空時，局部推理的學生減少。可見二立方不懸空的情境下，學生可發展較多不同的解題方式。
三視圖問卷的發展，即鎖定二立方不懸空的情境，調查結果發現，年級和任務、積木和任務之間有交互作用。對於「由視圖選出相容視圖（前右）」任務，國小和國中學生表現有顯著差異。而提供操作物，對「由視圖選立體圖」表現沒有顯著影響。就主效果而言，有操作物時學生表現較無操作物時好。較高年級學生表現顯著優於較低年級學生。「由視圖選立體圖」和「由底層和視圖計數」表現顯著優於「由視圖選出相容視圖（右俯）」和「由視圖選出相容視圖（前俯）」，又顯著優於「由視圖選出相容視圖（前右）」（p<.01）。
三視圖的假設性學習路徑，包含：編碼、解碼，和空間推理三階段。編碼階段主要訓練學生以積木方塊的各種表徵（示意圖、三視圖、分層圖）進行溝通；解碼階段，是由視圖建造出相符的積木模型；空間推理階段，學生需在無積木的情境下論述積木方塊的組態。根據該假設性學習路徑，發展教學活動並進行單一組前後測，其結果為：五、六年級學生在「由底層和視圖計數」表現有顯著進步。十年級學生在「由視圖選出相容視圖」表現有顯著進步。
綜而言之，本研究可貢獻在三視圖學習與教學的三個面向：描述學生的解碼策略、區別學生的三視圖表現層次，並形成積木方塊三視圖的假設性學習路徑。
關鍵詞：方塊、假設性學習路徑、問卷、三視圖、空間推理

The study aimed to explore Taiwan fifth to eighth graders’ performance on orthogonal views of cubes in three phases：The spontaneous concepts, the accuracy with regard to different kinds of tasks, and the teaching effects.
The first phase, a survey was conducted on spontaneous concepts and decoding strategies among six 10th graders. The survey was based on a semi-structured interview with three tasks varied in numbers of the cube, and cube situation （with floating cubes or not）. The second phase, a questionnaire was developed which included three tasks：Choosing isometric, cube enumeration, and the compatible task. The sample included 551 5th -8th graders. The study aimed to investigate the effects of three variables：（concrete manipulatives, grades, and tasks） on students’ performance. The third phase, teaching experiments were conducted in a 5th, a 6th, and a 10th-grade class by pretest-posttest designs.
The results were as follows. Spontaneous concepts of orthogonal views included seeing one view as one layer. Decoding strategies（visual/analytic） and modes of representation（discursive/visualization/simulation/mental image） all influenced students’ decoding performance.
The effect of concrete manipulatives, grades, and tasks were as follows. There was an interaction between grades and tasks; and also between concrete material and tasks. As main effects, the elder outperformed the younger for each age group. The groups with manipulation outperformed the group without. And as for item difficulty, choosing isometric and cube enumeration were significantly easier than the compatible task（p<.01）.
The hypothetical learning trajectory （HLT） for orthogonal views of cube follows the teaching sequence：Coding, decoding, and reasoning. One teaching experiment conducted in a 5th and a 6th-grade class resulted in a significant improvement in cube enumeration. Another teaching experiment conducted in a 10th-grade class resulted in a significant improvement in the compatible task.
In conclusion, the study contributed the teaching and learning of orthogonal views of cubes in 3 faces：describing students’decoding strategies, distinguishing students’performance in different levels, and formulating HLT for orthogonal views of cubes.
Keywords： Cubes, hypothetical learning trajectory （HLT）, questionnaire, orthogonal views, spatial reasoning

第壹章、緒論 ………………………………………………………… 1
第一節、研究背景與動機 ………………………………………… 1
第二節、研究理念 ……………………………………………… 7
第三節、研究目的與待答問題……………………………………… 9
第四節、名詞解釋 ……………………………………………… 10

第貳章、理論背景 ……………………………………………………… 15
第一節、幾何物件的認知歷程 …………………………………… 15
第二節、三視圖與空間推理………………………………………… 18
第三節、積木方塊三視圖的任務與表現 ……………………… 22
第四節、假設性學習路徑 ……………………………………… 28

第叁章、積木方塊三視圖的解題策略 ……………………………… 35
第一節、理論背景 ……………………………………………… 35
第二節、研究方法和樣本 ……………………………………… 40
第三節、資料分析 …………………………………………………… 41
第四節、結果與討論………………………………………………… … 43

第肆章、積木方塊三視圖的評量 ……………………………………… 47
第一節、理論背景…………………………………………………… 47
第二節、前置研究 ……………………………………………… 49
第三節、研究工具 ……………………………………………… 50
第四節、研究方法和樣本 ……………………………………… 53
第五節、資料分析 …………………………………………………… 55
第六節、問卷的信效度……………………………………………… 55
第七節、三視圖的表現評估………………………………………… … 57

第伍章、三視圖的假設性學習路徑 ………………………………… 71
第一節、理論背景 ……………………………………………… 71
第二節、研究方法和樣本 ……………………………………… 74
第三節、資料分析 …………………………………………………… 74
第四節、結果與討論………………………………………………… 74



第陸章、結論與建議 ………………………………………………… 93
第一節、結論 ……………………………………………………… 93
第二節、建議 ……………………………………………………… 100

參考文獻 ……………………………………………………………… 105

附錄一、數學課程中空間內容的比較…………………………………… 113
附錄二. 前置研究的三視圖能力問卷（二立方）……………………… 117
附錄三. 前置研究的三視圖能力問卷（三立方）……………………… 129
附錄四、正式研究中的三視圖問卷……………………………………… 141
附錄五、三視圖的實徵研究……………………………………………… 157

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