探討五年級學生使用圖像表徵解題在擴分、約分和通分上的表現
摘要
本研究以個案研究方式探討五年級學生使用圖像表徵解題在擴分、約分和通分上的表現，研究對象選取方式為使用獨立樣本T檢定選出同質性最高的兩個班級，共48人，透過自編「國小五年級擴分、約分和通分測驗卷」按照學生五上數學成績表現排序，交錯給予A、B試卷作測驗，再將兩卷測驗成績以統計分析方式整理與隨機選取測驗成績低、中、高分者和及平時數學成績表現低成就但在本測驗表現高分者等共十六位深入訪談，收集學生在解題計算過程時有無附圖像表徵之解題情形與解題想法，並了解學生解題時與表徵的思考連結過程及分數概念的理解。
研究結果如下:
一、附圖像表徵試題能使低成就表現學生成績明顯提升
圖像表徵如同學生解題的引信，能引起學生對解題的意願、興趣與自信，即使是低成就的學生對於不容易理解的文字題，也會想仔細觀察附圖思考與題意的關聯性，透過主動性的思考與觀察，自行建構解題的能力，進而從附圖表徵重新理解題意，並正確解題，使成績表現提升至高分組。
二、訪談過程中適當的引導學生觀察圖像表徵，學生可以透過圖像表徵自我修正
透過引導再予以圖像表徵做參考，學生由圖像表徵的觀察與思考，便能自行發現並澄清自己在算式上的錯誤。這些解題失敗的學生，大部分都能經由引導在圖像表徵上找到解題的關鍵，重新做出正確的解題。
三、學生於有附圖題型測驗之成績表現未必完全優於無附圖題型測驗之成績表現
第四章的資料顯示，無論是分數擴分，約分和通分，從低、中、高分組成績表現觀察，雖然A組學生整體附圖題型測驗表現較明顯高於B組無附圖題型，但對於高分組的兩組學生之間測驗成績表現並無較明顯差異。


關鍵詞:擴分、約分、通分、圖像表徵、低中高分組

Abstract
This study conducted case studies to explore the performances of the 5th graders on using diagrams to solve problems of expansion, reduction, and reduction to a common denominator. In this study, 48 students from two classes with the highest homogeneity were selected based on an independent sample T-test. Self-designed“Test Paper of Expansion, Reduction, and Reduction to Common Denominator for 5th Graders” were used on student subjects as experimental material to conduct the A/B test in alternating order according to their math performance ranking obtained from the first semester of their fifth year of study. Then, the results of these two tests were organized through statistical analysis to randomly select a total of 16 subjects who scored high, medium, and low marks, as well as those who generally did not perform well at math but scored high marks in this experiment to carry out in-depth interviews. The interviews collected subjects’ opinions on solving problems with and without diagrams and the related problem-solving processes as well as explored how students connect the diagrams with questions during these processes and their understandings for fractions.
The results of this study are as follows:
1.Test papers with diagrams can significantly enhance the performances of low-performing students.
Diagrams were like problem-solving cues for students and could trigger their willingness, interests, and confidence regarding this process. When it comes to plain-text questions, even the low-performing students would observe the attached diagrams and think about how they are related to the questions. Through proactive thinking and observation, student subjects managed to construct problem-solving abilities independently, rethink the meanings of the questions based on the diagrams, and resolve the problems correctly, making their scores on par with the high-performing groups.
2.Students can make corrections independently through diagrams when appropriately guided to observe the diagrams during the interviews.
With the guidance and referencing diagrams, the students were able to discover and correct their calculation mistakes based on their observation and thinking of the diagrams. Most of the students who failed on the problems could find the key to problem-solving and resolved them correctly through the diagrams when guidance was provided.
3.Students' performances at tests with diagrams will not necessarily be higher than those they get at tests without diagrams.
The data provided in Chapter Four suggested that regarding the observation yielded from low, medium, and high performing groups on questions related to either expansion, reduction, or reduction to common denominator, students from Group A had overall performed significantly better than those from Group B at the tests with diagrams. However, there are not any significant differences among high-performing students from both groups.
Keywords: expansion, reduction, reduction to common denominator, diagrams,low, medium, and high performing group

目錄
第一章 緒論 1
第一節 研究背景與動機 1
第二節 研究目的 2
第三節 名詞解釋 3
第四節 研究範圍與限制 4
第二章 文獻探討 6
第一節 分數的學習 6
第二節 數學表徵 15
第三節 表徵與分數學習之相關研究 20
第四節 表徵與數學解題之相關研究 25
第三章 研究方法與過程 30
第一節 研究架構與流程 30
第二節 研究方法與工具 32
第三節 研究對象 33
第四節 研究工具 34
第五節 資料整理與分析 41
第四章 研究結果分析 43
第一節 學生於附圖像表徵測驗之擴分、約分和通分解題表現 43
第二節 學生於無附圖像表徵測驗之擴分、約分和通分解題表現 74
第三節 比較學生於有無附圖像表徵測驗之擴分、約分和通分成績表現 99
第五章 結論與建議 103
第一節 結論 103
第二節 建議 104
參考文獻 106
中文文獻 106
英文文獻 108
附錄 112
附錄一 有附圖像表徵測驗 112
附錄二 無附圖像表徵測驗卷 114

 
表目錄
表2-1-1 分數與國小各年級相關的學習內容及對照之學習表現 8
表2-1-2 108年數學綱領中五年級學習重點 9
表2-2-1 國外研究 16
表2-2-2 國內研究 19
表2-3-1 表徵發展層次之特徵，各層次及其表現特徵 22
表3-3-1 X班與Y班五上數學總成績之分析結果(獨立樣本t檢定) 34
表3-2-1 教科書分數學習內容分析 34
表3-2-2 K版本分數學習擴分、約分和通分單元教材內容對照 36
表3-2-3 A卷(有畫圖)，半結構式訪談大綱 40
表3-2-4 B卷(不要求畫圖)，半結構式訪談大綱 40
表3-4-1 圖像表徵類型與題目舉例 41
表3-4-2 半結構是訪談代碼 42
表4-1-1 不同表徵:面積表徵、離散量表徵、線段表徵解題情形統計表 43
表4-1-2 擴分題型於面積表徵之做答分析 44
表4-1-3 約分題型於面積表徵之做答分析 47
表4-1-4 通分題型於面積表徵之做答分析 51
表4-1-5 擴分題型於離散量表徵之做答分析 56
表4-1-6 約分題型於離散量表徵之做答分析 59
表4-1-7 通分題型於面積表徵之做答分析 61
表4-1-8 擴分題型於線段表徵之做答分析 65
表4-1-9 約分題型於線段表徵之做答分析 67
表4-1-10 通分題型於線段表徵之做答分析 70
表4-2-1 無附表徵:解題情形統計表 74
表4-2-2 擴分-無附面積表徵圖題型之作答情形 75
表4-2-3 擴分-無附離散量表徵圖題型之作答情形 76
表4-2-4 擴分-無附線段表徵圖題型之作答情形 77
表4-2-5 約分-無附面積表徵圖題型之作答情形 82
表4-2-6 約分-無附離散量表徵圖題型之作答情形 83
表4-2-7 約分-無附線段表徵圖題型之作答情形 84
表4-2-8 通分-無附面積表徵圖題型之作答情形 90
表4-2-9 通分-無附離散量表徵圖題型之作答情形 91
表4-2-10 通分-無附線段表徵圖題型之作答情形 92
表4-2-11 B卷訪談六位解題錯誤學生，提供圖像表徵後的解答情形 98
表4-3-1 有無附圖像表徵測驗成績描述性統計量 99
表4-3-2 有無附圖像表徵測驗成績之分析結果(獨立樣本t檢定) 99
表4-3-3 擴分題型A卷(附圖)和B卷(無附圖)答對率 100
表4-3-4 約分題型A卷(附圖)和B卷(無附圖)答對率 100
表4-3-5 通分題型A卷(附圖)和B卷(無附圖)答對率 101
表4-3-6 有附圖像表徵測驗成績低、中、高分組 101
表4-3-7 無附圖像表徵測驗成績低、中、高分組 102

 
圖目錄
圖2-2-1 Lesh、Post & Behr 表徵關係圖 18
圖3-1-1 研究架構 30
圖3-1-2 研究流程 31
圖4-1-1 A卷第1題，As01面積表徵-擴分之計算過程 45
圖4-1-2 A卷第2題，As02面積表徵-約分之計算過程 48
圖4-1-3 A卷第4題，As03面積表徵-通分之計算過 52
圖4-1-4 A卷第3、4題，As04面積表徵-通分之計算過程 54
圖4-1-5 A卷第5題，As05離散量表徵-擴分之計算過程 57
圖4-1-6 A卷第6題，As06離散量表徵-約分之計算過程 60
圖4-1-7 A卷第7、8題，As07離散量表徵通分之計算過程 63
圖4-1-8 A卷第9題，As08線段表徵-擴分之計算過程 66
圖4-1-9 A卷第10題，As09線段表徵-約分之計算過程 69
圖4-1-10 A卷第11、12題，As10線段表徵-通分之計算過程 72
圖4-2-1 B卷第1題 Bs01無附圖-擴分之計算過程 78
圖4-2-2 A卷第1題面積表徵-擴分 79
圖4-2-3 Bs01 B卷第1題修正後之計算 79
圖4-2-4 Bs01 B卷第5題無附圖-擴分之計算過程 80
圖4-2-5 Bs01 B卷第9題無附圖-擴分計之算過程 81
圖4-2-6 A卷第5題附圖離散量表徵-擴分 81
圖4-2-7 A卷第9題附圖線段表徵-擴分 81
圖4-2-8 Bs01 B卷第5題修正後之計算 81
圖4-2-9 Bs01 B卷第9題修正後之計算 81
圖4-2-10 B卷第6題無附圖-約分 85
圖4-2-11 A卷第6題附圖離散量表徵-約分 85
圖4-2-12 Bs03 B卷第10題無附圖-約分之計算過程 86
圖4-2-13 Bs04B卷第2題約分之計算過程 87
圖4-2-14 A卷第10題線段表徵-約分 88
圖4-2-15 A卷第2題面積表徵-約分 89
圖4-2-16 B卷第03題 無附圖-通分之計算過程 93
圖4-2-17 B卷第03題面積表徵-約分 94
圖4-2-18 Bs06B卷第10題 無附圖-通分之計算過程 96
圖4-2-19 Bs06繪製圖像表徵 96
圖4-2-20 Bs06文字說明 96

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