臺灣博碩士論文加值系統

兒童的分數學習困難，因此研究兒童在分數概念與運算的錯誤類型與錯誤原因，有助於教師有效的教學，提昇學生的學習成效。本研究的目的有二：1、瞭解國小五年級學童在分數概念與運算的錯誤類型；2、探討國小五年級學童在分數概念與運算發生錯誤的原因。
研究方法採筆紙測驗與晤談二種方式相互配合進行，藉由筆紙測驗調查學生在分數概念與運算的表現及犯錯情形。經由筆紙測驗的結果整理、歸納學生的錯誤情形，從中抽取具有代表性錯誤的學生接受晤談，深入瞭解學生解題的歷程、想法以及運算規則。綜合測驗與晤談的資料歸納成各種錯誤類型並探討學生可能犯錯的原因。本研究的研究樣本為高雄縣市五所國小五年級十二個班級，合計422位學童。研究工具有兩項自編工具，一項是「國小五年級教師對『國小五年級學生分數概念與運算錯誤類型』調查問卷」，另一項是「分數概念與運算測驗」。
本研究結果為：
1、分數概念的錯誤類型有：缺乏部份與全部的概念、缺乏等分概念、單位量指認錯誤、認為分數不是一個數、把數線「分割點」當成是「分隔數」、數線的標分數出現兩個或三個箭頭、把分數是「數線上的線段長」當成是「數線上的一點」、把數線的左端整數或單位段當分母、以讀取小數的方式讀取數線上的分數、數與量概念無法辨別、缺乏「分數是兩數相除的結果」的概念、缺乏「分數是一個比值」的概念、缺乏分數稠密性概念、缺乏等值分數概念、缺乏分數的約估能力。
2、整數加分數的錯誤類型有：「將被加數加上加數的分子成為答案的分子」、「運算符號的錯誤」、「將加數的分子減被加數成為答案的分子」、「計算錯誤」、「假分數化成帶分數的錯誤」、「將被加數分別加上加數的分子、分母成為答案的分子、分母」。
3、分數減法的錯誤類型有：「將減數的分子減被減數成為答案的分子」、「整數化成分數錯誤」、「運算符號的錯誤」、「借位的錯誤（向整數借1卻減2、向整數借1後沒有減1）」、「計算錯誤」、「假分數化成帶分數的錯誤」、「答案忘了寫上分母」、「將被減數減減數的分子當成答案的分子」、「減數的整數部份用減法計算，真分數部份用加法計算」、「被減數與減數倒置」、「將被減數分別減去減數的整數、分子、分母成為答案的整數、分子、分母」、「被減數的分子與減數的分子倒置計算」、「答案忘了寫上整數部份」、「減數的整數部份用減法計算，真分數部份不減」。
4、整數乘分數的錯誤類型有：「乘數的整數部份用乘法，真分數部份用加法」、「將被乘數分別成乘上乘數的整數、分子、分母成為答案的整數、分子、分母」、「被乘數、乘數都化成假分數後，分子乘分子，分母則不相乘」、「只做真分數部份的乘法，整數部份則不相乘」、「乘法用加法算則計算」、「帶分數化成假分數的錯誤」、「假分數化成帶分數的錯誤」、「計算錯誤」。
5、整數除整數的錯誤類型有：「沒有用分數作答」、「把商數當成分母，除數當成分子」、「把被除數當成分母」、「把商數當成答案，餘數則省略」、「計算錯誤」、「商數與除數倒置」、「假分數化成帶分數錯誤」、「把商數當成分子，餘數當成分母」、「答案沒寫上商數部分」。
6、分析分數概念與運算錯誤形成的原因有：不瞭解題意、過於依賴連續量的部份－全部模式、「分數是數線上的一點」與「分數是數線上的線段長」的概念混淆、將先前知識做錯誤的類推（十進位系統的類推、正整數的類推、帶分數的類推、小數的類推）、語言的影響（受分數從分母讀起影響、受直式中文順序影響）、缺乏先備知識（小數可以除以大數的概念、分數是一個數的概念、整數四則運算的技能）、概念間缺乏連結（整數的倍數概念與分數的倍數概念的連結、真分數倍的意義與乘法算式的連結、除法與分數的連結）、缺乏合理性的檢驗策略、學生的自我想法（文字或圖形的解讀、依據猜測作答、錯誤的簡易原則）。
根據研究結果加以討論，並提出下列建議提供教師在教學及教材編寫設計上之參考。

A Research of the Error Patterns of Concept and Computation of Fraction to the Fifth Grade Students of Elementary School
ABSTRACT
Some problems exist when children are learning the fractions. Thereafter to study the error patterns and the causes of concept and computation of fraction , is helpful to secure the teaching efficiency of the teachers and the learning achievement of the students. There are two purposes of this study: 1. To realize the error patterns of concept and computation of fraction happened to the fifth grade students of primary school. 2. To investigate the reasons why the fifth grade students made the mistakes in the concept and computation of fraction. This study was conducted in parallel through both paper-pencil test and interview as well to investigate the error patterns of concept and computation of fraction happened to the students. An interview as further test was administered to the students who had the representative error through the results of the paper-pencil test collected. To sum up the information gathered throughout the results of both paper-pencil test as well as the interview, we may have the chance to conclude the error patterns, and to investigate the reasons why the students committed the error. A total of 422 students in 12 classes were samples from five elementary schools of Kaohsiung Hsien and Kaohsiung City respectively. Two self-made instructions had been adopted in the study. One of them was “the questionnaire toward the teachers in the fifth grade classes of elementary school concerning ‘the error patterns of concept and computation of fraction to the fifth grade students of elementary school’ ”, and the other one was the “concept and computation of fraction test”.
The results revealed as follows:
1. The error patterns of conceptual fraction are : The shortage of concept in both complete and partial fractions, the shortage of concept of partition, the mis-identification of the volume of the units, the fraction is not a number, misunderstood the “separating point” as “separated number”, misunderstood the fraction as two or three numbers, misunderstood the fraction as two or three arrow heads appeared on the number line, taking “fraction is length of the section line” as “fraction is one of the point on the section line”, taking the integer in the left end of the number line or the unit line section as the denominator, taking the way of reading the decimal number as the way in reading the fraction on the number line, incapability to tell the number from the volume, insufficient concept in the fractional integration, insufficient concept in the equal fraction, and the insufficient sense in number, insufficient capability in estimating the value of the fraction, shortage of the sense of figures, etc.
2. The error patterns in the integer pluses fraction are : taking the sum of numerators of addend and addencand as the numerator of the answer, using the wrong computational signals, to subtract the addencand from the numerator of the addend as the numerator of the answer, the computation error , mis-conversion from the improper fraction to the mixed fraction, adding the numerator and denominator of the addend to the addencand as the numerator and denominator of the answer respectively, etc.
3. The error patterns of the fractional subtraction are : subtracting the minuend from the numerator of the subtrahend as the numerator of the answer, error in the conversion from integer to fraction, use the wrong computational signs, error in backward (backward of 1, but with the subtraction of 2; backward of 1 from the integer without substraction of 1), computation errors in substraction, mis-conversion from the improper fraction to the mixed fraction, forgot to put the denominator on, subtracting the numerator of the addend from the numerator of the addendhend as the numerator of the answer, to perform the subtraction with the integer part of the addend, while to perform the addition with the proper fraction, mis-placement between addend and addendhend, to subtract the integer, numerator and denominator of the addend from the addendhend as the integer, numerator and denominator of the resulted answer respectively, invert computation between the numerators of addend and addendhend, forgot to put on the integer, using subtraction with the computation in the integer part of the addend, whilst no subtraction with the proper fraction, etc.
4. The error patterns in the multiplication between integer and fraction are : Using the multiplication with the parts of the multipliers, whilst, using the addition with the parts of proper fractions, To multiple the integer, numerator, and denominator of the multiplier with the multiplicand as the integer, numerator, and denominator of the answer respectively, The multiplication between numerators, while no multiplication was made between the denominators after the conversion as improper fractions of both multiplier and multiplicand, Use multiplication only with proper fractions, while without integers, To compute the addition with multiplication, The errors in converting the mixed fraction to improper fraction, The errors in converting the improper fraction to mixed fraction, Faulty multiplication, etc.
5. The error patterns in dividing the integers are : Not using the fraction as the answer, To take the quotient as the denominator, and the divisor as the numerator, To take the dividend as the denominator, To take the quotient as the answer, and the balance to omit , Computation errors in division, Faulty placement between quotient and divisor, The error to convert the improper fraction to mixed fraction. To take the quotient as the numerator, and the balance as the denominator. The answer without quotient, etc.
6. The reasons to the formation of the analysis of concept and computation of fraction are : Without fully understanding the meaning of the problem, Too much reply on the portion of continuous part—whole model, Conceptual confusion happened between “fraction is one of the points on the number line.” and “fraction means the length of one section line on the number line”, Faulty comparison with prior knowledge (comparison of decimal system, comparison of positive integer, comparison of mixed fraction, comparison of decimal), Lingual affects (reading the fraction from the denominator, vertical sequence of Chinese characters), Insufficiency of prior knowledge (the concept that the smaller number can be divided by the larger number, fraction is one of the conceptual number, the computational skill with 4 rules of computation), The shortage of conceptual linkage between the concepts (the linkage between the multiple concept of the integer and the multiple concept of the fraction, the linkage between the true meaning of proper fractional multiple and multiplication, the linkage between division and fraction). The insufficiency of reasonable testing strategies. Self-thinking by the students themselves (the understanding of the characters or diagrams, to answer by guess, incorrectly concise principles). etc.
To discuss throughout the researching results as above, it is suggested to the teachers in their teaching manners and the composition of instructions.

國小五年級學童分數概念與運算錯誤類型之研究
目 錄
第一章 緒論
第一節 問題背景與研究動機 ……………………………………………1
第二節 研究目的 …………………………………………………………4
第三節 研究範圍與限制 …………………………………………………4
第四節 名詞界定 …………………………………………………………5
第二章 文獻探討
第一節 數學概念與運算學習的理論 ……………………………………6
第二節 兒童的分數概念發展與分數的意義……………………………11
第三節 分數概念與運算錯誤情形之研究………………………………15
第四節 分數概念與運算錯誤原因之探討………………………………23
第三章 研究架構與方法
第一節 研究設計…………………………………………………………27
第二節 研究樣本…………………………………………………………28
第三節 研究工具…………………………………………………………28
第四節 實施步驟…………………………………………………………33
第五節 資料處理與統計…………………………………………………37
第四章 結果與討論
第一節 學生在「分數概念與運算測驗」之錯誤情形…………………38
第二節 分數概念與運算的錯誤類型……………………………………46
第三節 分數概念與運算的錯誤原因……………………………………82
第四節 綜合討論 ………………………………………………………103
第五章 結論與建議
第一節 結論 ……………………………………………………………121
第二節 建議 ……………………………………………………………125
參考書目
一、中文部分 ……………………………………………………………128
二、英文部分 ……………………………………………………………130
圖 次
圖2-1 Markle ＆ Tiemann（1970）概念學習的三種錯誤類型 ………7
圖2-2 圓錐形的概念模型 ………………………………………………7
圖2-3 Skemp（1980）概念形成的流程圖 ………………………………8
圖3-1 八十二年國小新課程的分數學習架構圖 ………………………30
圖3-2 實施過程的流程圖 ………………………………………………36
表 次
表3-1 研究樣本人數之基本資料 ………………………………………28
表3-2 八十二年國小新課程分數教材內容分析表 ……………………29
表3-3 「分數概念與運算測驗」內容分析之雙向細目表 ……………32
表4-1 「分數概念與運算測驗」犯錯的情形統計表 …………………40
表4-2 教師問卷結果與學生答題錯誤率對照表………………………44
附 錄 次
附錄一 國小五年級教師對『國小五年級學生數學科分數概念及運算
錯誤類型』調查問卷……………………………………………139
附錄二 分數概念與運算測驗之難度及鑑別度…………………………142
附錄三 「分數概念與運算測驗」各題錯誤答案或錯誤類型…………143
附錄四 面談原案…………………………………………………………160
測 驗 工 具
測驗工具一 ……………………………………………………………208
測驗工具二 ……………………………………………………………215

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