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English Abstract

Geometric learning is to foster students’ logical reasoning and spatial imagination ability; congruent triangles is an important topic in the plane geometry, which is the foundation to the advanced geometric learning. This study will focus on the learning of congruent triangles among junior high school students, investigating the differences from gender and grades, studying the causes of difficulties and errors in learning the geometric concept. Hopefully, it will also provide some insights for teaching. A researcher-developed assessment was mainly based on the van Hiele model of geometric thinking, the subjects were junior secondary students. There were 454 students in total, and a few was selected to do the individual interview which helped the research knew more about the learning processes and the errors, and the reason of failure to access the advanced levels of geometric thinking. Through the data collection, data analysis, and literature review, researcher came up the following conclusions: 1. Most of the students achieved the congruent triangles concepts at the level of intuitive cognition, the domination and application of determination rules are undesirable, proof writing is particularly difficult. The number of students achieving van Hiele model of geometric thinking level III was the most, followed by level IV and level II. 2. While doing the analysis of variance regarding to gender, the girl’s average score is higher than the boy’s, and the individual difference is less than the boys’. In the van Hiele model IV of geometric thinking, the percentage of the girls achieving level I is significantly higher than the boys’. 3. While doing the analysis of variance regarding to grades, the knowledge related to congruent triangles are more well-grounded among the grade 9 students. In the van Hiele model of geometric thinking. In ratio, more the grade 9 students achieved at level III and level IV, the ratios were significantly higher than the grade 8 students. 4. The reasons related to the students unable to access to the next level of geometric thinking may be: they do not understand that level of mathematical language and expressions, answer question by memory alone, do not form abstract thinking, and the basic ability of proof is relatively weak. 5. The errors made by students on congruent triangles werre related to graphical visualization, geometric symbolization, knowledge confusion, mastering the method of determining congruent triangles, written proof and logical reasoning questions. The reasons were related to cognitive intuition; unable to deal with the diversified determination methods; discrepancy between graph and given conditions. Keywords: congruent triangles; the van Hiele levels of geometric thinking; understanding; error analysis

Chinese Abstract

幾何的學習是培養學生邏輯推理、空間想像的能力,而全等三角形則是平面幾何 的一個重要部分,為往後幾何圖形的學習奠下了基礎。本研究將探討初中學生學習全 等三角形的情況,從性別上,年級上瞭解其差異,並探討學習全等三角形的困難或錯 誤及其原因,從而為以後的教學提供可參考的數據。 根據 van Hiele 幾何思維層次的前四個層次以及相關資料來編制測試卷,測試對象 為初中學生,共 454 個測試者,再從中選取被測試者進行訪談,進一步瞭解學生在學 習過程中容易出現的錯誤,及學生不能進入下一個幾何思維層次的原因。通過收集數 據,整理資料,文獻分析,得到以下結論: 1. 大部分學生對全等三角形的認知都停留在直觀和性質的層面上,對於判定定理 的掌握和應用較為不理想,書寫證明過程尤其困難。學生達到 van Hiele 幾何思維層次 三的人數最多,其次是層次四和層次二。 2. 在性別上的差異檢驗中,女生在測試卷上各題的平均分均比男生略高,而且女 生的個體差異没有男生大。在 van Hiele 幾何思維層次中,只有在層次一,女生的達到 率顯著性地高於男生。 3. 在年級上的差異檢驗中,初三學生掌握全等三角形的知識較為鞏固,在 van Hiele 幾何思維層次上,初三學生在層次三和層次四中都比初二學生顯著性地高。 4. 學生不能進入下一個幾何思維層次的主要原因是學生不理解該層次的數學語言 II 和表達方式;單憑記憶來做題;沒有形成抽象思維;證明的基礎知識比較薄弱。 5. 學生學習全等三角形的錯誤在於圖形的視覺化;幾何圖形的符號表示;知識點 的混淆;全等三角形的判定方法的掌握;證明題的書寫以及邏輯推理。其錯誤的原因 是學生感觀的認知容易出現誤差;判定方法多樣化,使學生造成困擾;圖形跟已知條 件脫離。

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Faculty of Education




Geometry, Plane -- Study and teaching (Middle school)

平面幾何學 -- 學習及教學 (高級小學至初級中學)

Cognitive learning theory




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