本研究旨在探討國小五年級學童以反例促使幾何圖形概念改變之結果及其認知歷程，特別關注幾何知識與推理能力各自所扮演的角色，並針對長方形包含正方形的關係進行研究。前導研究以131名三至五年級學童檢驗幾何知識與推理能力之自編測驗的品質，並以11名五年級學童確認實驗流程。正式研究包含概念改變實驗與定義改變實驗，前者目的是改變學童概念低度外延之情形和調整內涵性，後者則是在概念外延性改變之後，更精緻化概念內涵性；前者材料是正例和反例的圖卡，後者則是增加了非例的圖卡。以圈圖題自台北市與新北市五年級學童336名中選出明顯低度外延者為研究對象，再以自編的幾何知識測驗和推理能力測驗篩選出三組學童，分別是「高知識高邏輯組」、「高知識低邏輯組」和「低知識高邏輯組」，有效人數依序為16、17和16名。概念改變實驗的結果，首先，接受反例和概念外延性改變方面，「低知識高邏輯組」人數少於「高知識高邏輯組」，顯示幾何知識較差者，較難接受反例和調整概念外延性。其次，整體而言，有近半數的學童概念內涵性進步，並且近七成的學童可發現必要屬性──四角直角。最後，在共同屬性的歸納上，「高知識高邏輯組」和「高知識低邏輯組」的長方形描述數量多於「低知識高邏輯組」，且「高知識高邏輯組」在推理正方形可歸類為長方形的歷程中，想出較多的幾何性質且多為推理有效的方式，顯示幾何知識和推理能力影響了歸納共同屬性的表現。定義改變實驗的結果是接受實驗處理的學童有四分之一的概念內涵性轉為正確，不過「高知識高邏輯組」概念為精緻化者屬於少數。研究者推測工作記憶的限制是學童無法將概念精緻化的因素。研究者分析學童的概念改變歷程表現，認為幾何知識於幾何圖形概念改變中所扮演的角色是提供學童進行推理時的可用的背景知識，至於學童是否可善加運用背景知識，最後獲得正確概念則需仰賴推理能力的程度。

This study is to understand the results and the cognitive processes of geometric conceptual change from the 5th grade elementary pupils who face counter-examples of quadrilaterals, especially to focus on the roles that geometric knowledge and reasoning ability play in the conceptual change. The inclusion relationship, square included in rectangle, was discussed further. 131 grade 3rd to 5th elementary pupils were selected for the pilot study to examine two self - designed tests of the geometric knowledge and reasoning ability. In addition, grade 5th elementary pupils were selected simultaneously to confirm the accuracy of the experimental procedure. The formal study contained two experiments: the conceptual change experiment and the definition-changed experiment. The former aimed to change the children’s performance of under-extension and to adjust the performance of conceptual intension; the latter aimed to refine the performance of conceptual intension when children were able to change the performance of performance of conceptual extension. As far as materials were concerned, positive instances and counter-examples were involved in the former; except for the materials mentioned above, the negative instances were added in the latter. The subjects were divided via the 2(geometric knowledge)*2(reasoning ability) way; then, three groups were formed, exclusive of the group with those had the worst geometric knowledge and reasoning ability. The valid samples for each group were 16, 17 and 16. All the subjects were with under-extension. The results of the conceptual change experiments were addressed as follows. First, the “worse geometric knowledge and better reasoning ability” group performed significantly better than the “better geometric knowledge and reasoning ability” group in accepting the counterexample and changing the performance of conceptual extension. It reveals that the poor geometric knowledge subjects have, the more difficult is for them to accept the counter-example and adjustment conceptual extension. Secondly, nearly half of the children are progressive in understanding the conceptual intension, and nearly 70% of the subjects found that the necessary attribute is “four angles are right angles.” Finally, the performance of inducting common attributes for positive instances and counter-examples of all subjects are shown as follows. the “better geometric knowledge and reasoning ability” group and the “better geometric knowledge and worse reasoning ability” group described more rectangular properties than the “worse geometric knowledge and better reasoning ability” group. Moreover, the “better geometric knowledge and reasoning ability” group could come up with more geometric properties, which can be regarded as effective reasoning concerning why square can be included in rectangular. It demonstrates that the performance of inducting common attributes is affected through the geometric knowledge and the reasoning ability. The result of the definition-changed experiment was that quarter of subjects turned their wrong conceptual intension correct after the experimental treatment. However, very few of the “better geometric knowledge and reasoning ability” group were able to refine the performance of conceptual intension, so the researcher speculated working memory limits subjects to refine the performance of conceptual intension. The researcher found that the role of the geometric knowledge in geometry conceptual change is to provide background knowledge for reasoning. Eventually, the school children rely on the reasoning ability to manipulate background knowledge and obtain the correct concepts.

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