正負數量表徵的心理數線發展

The Development of the Mental Number Line in the Representation of Positive and Nagative Numbers

林寶玲(Pao-Lin Lin)

指導教授 ： 柯華葳

國立中央大學/文學院/學習與教學研究所/碩士(2012年)

摘要

本研究旨在找出國內不同階段學童在正負數量表徵的發展狀況。研究方法分二個實驗，分別為實驗一：(1) 小二、小四、小六、國二及國二學習障礙學童進行數量範圍0~100、0~1000、0~10000、0~100000的數線評估作業（number-line estimation task）；(2) 上述小六及國二學童增加進行負數數量範圍－10~10及－100~100的作業；(3)以上各組受試皆進行問卷針對其學習狀況的調查並收集數學成就評量成績以做為背景分析。實驗二：將實驗一中小六隨機選取14名及國二隨機選取24名進行正負數量大小判斷作業。
研究結果發現，在正數範圍的數線評估作業誤差率的部分，小六和國二的數線評估誤差率無顯著差異，但其他各年級間皆有顯著差異。國二學障生只與小四無顯著差異。在負數範圍的數線評估作業誤差率部分，小六、國二及國二學障生無顯著差異。在數線評估的數量表徵型態的迴歸分析結果發現，小二在0~100的數量範圍逹線性表徵，小四、小六、國二及國二學障生在0~100000的數量範圍逹線性表徵；在負數數量範圍的部分，小六、國二及國二學障生在－100~100的數量範圍逹線性表徵。小二在數量100以內逹到線性表徵的結果與國外研究相近，但小四生在數量100000以內逹到線性表徵則比國外研究指出要到成人才能逹到線性表徵的情況要早。此結果推估因國內數線教學於小三開始，可能是小四學生已較熟悉數線和大數量有關。在負數數量表徵的部分，小六、國二及國二學障學生在數量－100以內逹到線性表徵，此結果與國外研究相同，唯國外研究對象為成人，顯示國內從小六學生即逹負數線性表徵，推估小六學生可藉由對正數的概念自行推估0左端的負數概念。而國二學障生在正負數量100以內的小數量可逹線性表徵，而在100000範圍的大數量則可能為其弱勢。在正負數量大小判斷作業的結果顯示小六和國二學生的錯誤率並無顯著差異，經評估學生在兩數皆為負數的作業反應時間顯著較長，小六生偏向使用以性質符號和數字分開存取的成份表徵型態，而國二生偏向使用合併存取的整體表徵型態。最後，在兩個實驗間的相關分析結果發現，小六的正負數量大小判斷的錯誤率和數學成就評量間有顯著相關，但在正負數量大小判斷錯誤率和數線評估誤差率間則無相關，國二則三者均無相關，此結果的意義在文中有進一步討論。

關鍵字

數線評估作業 ； 數量大小判斷作業 ； 概算能力 ； 負數 ； 數量表徵 ； 心理數線

並列摘要

This study examined the representation of the positive and negative numbers in students during different stages of development in Taiwan. The sample included second graders, fourth graders, sixth graders, eighth graders and the learning disability students.
In the first experiment, all children were evaluated with number-line estimation task, which incorporated four kinds of ranges, expanding from 0-through-100000. The last three groups in the sample were also evaluated by the same task for the -10 through 10 and -100 through 100 ranges. In addition, students’ learning backgrounds were also collected by using questionnaires and their average mathematics achievement in the semaster. In the second experiment, sixth and eighth graders were chosen from the first experiment to access with the additional numerical comparison task.
The results indicate that in the number-line task, the main improvement occur in the fourth grade from the number 0 through 100000. The performance score is similar to how the adult college students performed in other countries. Second, sixth grade also show significant improvement in the negative range number-line task, which indicates that was some strategies maybe used by the students when solving the estimation task with negative numbers despite the fact that students have not learned the negative numbers yet. Third, eighth grade learning disabilities student also showed significant improvement in the small range numbers in between -100 and 100, however, had some difficulties in range above 100000. This could be a potential explanation of what causes the students failed the mathematic achievement. Forth, the results of numerical comparison task indicate that sixth graders may had presented with components representation in mind and that the eighth graders can presented both the component and/or the holistic representations when needed for the task.
We also test the correlation between the two tasks and the math grades; the result shows that the correlation was significant under the sixth grade but not in the eighth grade and learning disabilities students. Overall, the results suggest that children should learn more to solve the more difficult mathematical problems when get into the junior high school, even they have improved in the numerical representation.

並列關鍵字

numerical comparison task ； estimation ； negative numbers ； numerical representation ； number-lime estimation task ； mental number line

參考文獻

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