van Hiele幾何發展水準之量化模式—以國小中高年級學生在四邊形概念之表現為例

The Quantitative Model of the Van Hiele Levels of Geometric Thought and Its Application

何森豪(Sen-Hau Ho)

《測驗統計年刊》 9期 (2001/12) Pp. 81-129

https://doi.org/10.6773/JRMS.200112.0081

摘要

本研究旨在試圖結合無母數試題反應理論及模糊理論，針對van Hiele幾何思考模式提供另一種方法論，目的在將van Hiele幾何思考模式量尺化。並以國小中高年級學生為研究對象，以幾何概念中之四邊形概念為實例分析之素材，提供此方法論之應用。
根據理論發展與實例研究之結果，本研究的主要結論可歸納如下：
一、在理論發展方面：本研究所發展之「van Hiele幾何發展模式之量化模式」以無母數試題反應理論來代替傳統「全有全無」的測驗模式，可獲得較精確的能力估計值；以模糊理論之模糊聚類分析法來代替以答對率為通過標準的真實水準，不但能更精準指派學生的van Hiele思考水準，也能以量尺來連續描述之。因此，這三種理論的結合具合理性與可行性。
二、在實例應用方面：結合模糊聚類分析分群之結果與冉母數試題反應理論所得的學生能力值，對應van Hiele思考水率，得到本研究之量化模式。
（一）學生能力值在-0.7473~-2.4225、0.4925~-0.7271、2.1542、0.4997 間，則指派水準分別為0、1、2。
（二）根據所得的學生能力值範圍，結合van Hiele水準，可繪製成一把連續的量尺，為van Hiele幾何發展量尺。教師可利用這把量尺，根據學生的能力值來指派學生的van Hiele水準。
（三）總計有25.43%的兒童處於水準0階段，有40.39%的兒童處於水準1階段，有34.18%的兒童處於水率2階段。

關鍵字

van Hiele幾何發展論 ； van Hiele水準 ； van Hiele幾何發展量尺 ； 模糊聚類分析 ； 模糊隸屬度函數 ； 無母數試題反應

並列摘要

The purpose of this study is to develop a new quantitative model to represent the students' van Hiele levels. Being different from the previous researches, this new model provides an alternative paradigm to evaluate the acquisition of the van Hide levels, which is based on the non-parametric item response theory and fuzzy theory. The subjects for the application of this new model consisted of 315 fourth-grade students and 314 sixth-grade ones. The quadrilateral geometric conception test is used to measure the van Hiele levels of geometric thought of the subjects. The main findings of the study are as follow: 1. From the theoretical points of view, we use the alternative paradigm with non-parametric IRT to replace dichotomous scoring method. So we can obtain more accurate ability score. And we use fuzzy membership function to replace the passing rate in a given subtest. Then we can obtain more accurate van Hiele levels to keep showing them continuously. 2. From the practical application points of view, the combination of the result of fuzzy membership function and the ability score comparing with van Hiele levels to get the quantitative model. We can find: (1) The ability score of subjects from to are posted to L0. The ability score of subjects from to are posted to Li. The ability score of subjects from to are posted to L2. (2) According to the range of ability score and the van Hide levels, we can get a continuous scale, which is called van Hide scale. Therefore, according to the ability score teachers can use the van Hiele scale to post the students' van Hiele levels. (3) There are 25.43% of subjects at LO, 40.39% at Li, and 34.18% at L2.

並列關鍵字

van Hiele geometric thought ； van Hiele levels ； van Hiele scale ； fuzzy clustering analysis ； membership function ； non-parametric IRT