國小學生在等值分數上的表徵轉換表現
摘要
本研究之研究目的旨在探討國小四、五、六年級學生在等值分數概念上，對各種表徵轉換的表現情形為何。研究樣本為國小四、五、六年級共729名學生，研究工具採紙筆測驗，並透過變異數分析方式求得學生在圖形與符號兩種表徵不同的轉換模式中的比較；於紙筆測驗後，採半結構的訪談研究方式，以求得學生由「符號表徵」轉換到「圖形表徵」、「操作物表徵」及「真實情境表徵」的各種表現。研究結果呈現在各種類型題目中，大致上都是六年級表現最好、五年級次之，四年級最差，其他結果表現則如下：
1. 對各年級學生而言，都是「圖形表徵轉換到符號表徵」表現優於「符號表徵轉換到圖形表徵」，若進一步將「圖形表徵內的轉換」加入比較，則又可發現，四年級學生在「圖形表徵內轉換」的表現和上述兩種轉換比較，表現最差，到了五、六年級，學生在「圖形表徵內的轉換」和此兩種表徵間的轉換則無顯著差異。
2. 四、五年級學生在三種分數意義的表現分別是「部分-全體」優於「集合子集」，表現最差的是「數線」題型，對六年級學生而言，數線雖仍是最困難的題型，但「部分-全體」、「集合子集」此兩者已無表現上的顯著差異。
3. 各年級在等值分數「部分-全體」的分數意義中，對圖形表徵中單位量分割份數的難度層次，分別是「單位量分割份數等於分母」表現優於「等於分母的因數」、「等於分母的倍數」、最難的都是「與分母無關」的題型。「集合-子集」意義也都是「等於分母」的題目表現優於「等於分母的倍數」題型；至於在「數線」的表現，四年級學生最能接受「等於分母」，其他類型無顯著差異，五年級學生最容易的題目也是「等於分母」，最困難的題目則是「與分母無關」，至於「是分母的因數」與「是分母的倍數」兩者則無顯著差異，六年級學生四種類型難度層次差異則較明顯，由易而難分別是「等於分母」、「等於分母的因數」、「等於分母的倍數」、「與分母無關」。
4. 四年級學生符號表徵中「求分子」的題目表現比「求分母」好，但對五、六年級學生而言，此兩種類型對學生而言無明顯的難度差異。
5. 透過訪談分析學生由符號表徵轉換到「圖形表徵」、「操作物表徵」及「真實情境表徵」的各種作答情形時，大致都可將作答類型分為三種，第一種是學生能利用這三種表徵表徵出等值分數的意義；第二種則是學生具備等值分數的符號運算技巧（符號運算公式），卻無法利用此三種表徵表徵出等值分數的意義；第三種類型則是學生不具備等值分數的運算技巧規則，也不能表徵出等值分數概念的意義。

Elementary School Student's Representation Translation Performance on Equivalence Fraction ABSTRACT
The purpose of this study was to investigate elementary school student's representation translation on equivalence fraction. Based on the questionnaire survey, the participants are 729 forth, fifth and sixth graders from 7 different elementary school. The study adopted questionnaire survey with the paper test firstly to collect students’ scoring performance based their translation between the two representations – “Picture” and “Symbol” representation on equivalence fraction, and compare the different performance of students via ANOVA; After paper test, the researcher also took the sub-construct interview method to 18 students from these participants, and expected to explore the performance of these students when they transfer from “Symbol representation” to “Picture representation”, “Manipulative representation” and “Real World Situations representation” . The study reveals that no matter what kind of the task types, sixth grade students always got the highest score, and forth grade students always got the worst. The other major results of this study are summarized as below:
1. For all students no matter the grade, the performance of “From Picture representation transfer to Symbol representation” is always better than of “From Symbol representation transfer to Picture representation”. If go a step further to compare with the “Translation between the Picture representation”, then the study revels that comparison with the grade 4 student’s performance on “translation within Picture representation” and above two translation, it’s the worst, but when they become fifth or sixth grade, there is no big difference among the students’ “translation within Picture representation” and the translation between the two representations.
2. To take one step ahead for analysis and compare the student’s scoring effect on the three fraction interpretations, and researcher found that forth, fifth grade students’ performance is “part-whole” better than “set-unit”; they got worst performance on “Number line” subject, however for the students of grade sixth, “Number Line” subject is still the most difficult question, but there is no significant difference on performance between “part-whole” and “set-unit” .
3. In the “part-whole” interpretation of equivalence fraction, to all students the degree of difficulty of the four numbers of what the Whole divided into Picture representation, is “numbers of what the Whole divided into equal to denominator” better than “numbers of what the Whole divided into equal to a factor of denominator”, “numbers of what the Whole divided into equal to a multiple of denominator”, the most difficult is “numbers of what the Whole divided into to have no relationship with denominator” Subject. For the “Set-Unit” interpretation performance of every grade is also , “numbers of what the Whole divided into equal to a factor of denominator” better than “numbers of what the Whole divided into equal to a multiple of denominator”; as to “Number Lines” performance, the fourth grade student can accept “numbers of what the Whole divided into equal to a factor of denominator” better, but no significant difference for other topics; the easiest subject for fifth grade is also “numbers of what the Whole divided into equal to a factor of denominator”, and the most difficulty question is “numbers of what the Whole divided into to have no relationship with denominator”. However there is no big difference between “equal to a factor of denominator” and “equal to a multiple of denominator”, the difference is more obvious for the degree of difficulty of sixth grade student, from simple to hard is “numbers of what the Whole divided into equal to denominator”, “numbers of what the Whole divided into equal to a factor of denominator”, “numbers of what the Whole divided into equal to a multiple of denominator”, “numbers of what the Whole divided into to have no relationship with denominator”.
4. The difference comparison between “to figure up the numerator” and “to figure up the denominator” under “translation within Symbol representation”, “to figure up the numerator” is better for “to figure up the denominator” for forth grade students. But there is no apparently difficulty difference between the two types for students of fifth and sixth grade students.
5. During Interview and analysis the students’ each answering circumstances from Symbol representation translation to “Picture representation”, “Manipulative representation” and “Real World Situations representations”, we can divided the student’s answering model to three types: The 1st type is students can take advantage of the three representations to represent the equivalence fraction’s meaning; the second type is , students have the operation skills of equivalence fraction(Symbol operation formula), but cannot use the three representations to represent the meanings of the equivalence fraction; the third type is that the students either don’t have any operation skills or rules of equivalence fraction, or cannot represent the meaning of the equivalence fraction concept.

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