國一學生一元一次不等式錯誤類型分析之研究

Abstract

本研究的主要目的是探討國一學生在一元一次不等式的錯誤類型，並分析錯誤原因。\n 本研究的設計採用調查研究法，共分成兩個階段進行，第一階段為準備階段，主要工作為文獻探討、分析國中數學教材、自編「一元一次不等式錯誤類型分析研究」試卷，進行試卷的施測，預試樣本共56名國三學生，抽樣方式非隨機取樣，採方便取樣進行，再由預試結果經修改編製正式施測之試卷。第二階段為正式施測與分析階段，正式施測樣本共30名國一學生，男生12名、女生18名，根據施測結果，依成績分成高分組、中分組、低分組三組，再隨機抽取男女生各2名進行半結構的晤談，以瞭解學生答題的想法，分析學生錯誤的原因。\n 本研究的主要結果如下：\n一、不等式答題表現在文字問題的錯誤比率、空白比率最高。\n二、在一元一次不等式的錯誤類型為：\n （一）同義詞的轉換：1.不等號的同義詞概念；2.不等號的符號認知。\n （二）範圍解與圖示：1.數的運算；2.無法判斷範圍解；3.數線的認知；4.不等號的圖示認知；5.不等號的方向。\n （三）解不等式：1.不等號改變方向；2.去括號；3.數的運算；4.遺漏或增加符號；5.移項的錯誤；6.等量公理的誤用；7.未化簡；8.不等號概念的錯誤9.多項錯誤；10.抄錯題目或答案；11.胡亂猜測答案。\n （四）文字問題：1.無法理解題意；2.列式錯誤；3.三角形面積公式錯誤；4.忽略題目的已知條件；5.答案遺漏或錯誤；6.不等號的概念；7.數的運算；8.多項錯誤9.不等號的同義詞概念。\n三、一元一次不等式的錯誤原因：1.先備知識的不足；2.資料使用錯誤；3.新舊學習經驗的互相干擾；4.錯誤的使用運算規則；5.由題目所給數字直接產生答案；6.不清楚題目設計或文字敘述而產生錯誤；7.忽略題目所給條件或答案不夠完備而產生錯誤；8.沒有從離散量概念延伸至連續量概念。

The main purpose of this study is to investigate “seventh grade students` misconceptions and error types of linear inequalities in one unknown”and analyze the causes of the errors.\nThis study adopts survey research and includes two phases. The first stage is the preparation phase, including literature review, analysis of mathematics textbooks, self-compiled test papers on “misconceptions and error types of linear inequalities in one unknown,” and the pretest. The pretest adopts convenience sampling- totally 56 students from ninth grade. The results were later revised to compile the formal test papers. The second stage is the official survey and analysis phase. The 30 samples are seventh grade students, 12 boys and 18 girls. According to the results of the test, these sample students are divided into three groups-high, medium and low performances. Out of each group, two boys and two girls are randomly sampled and conduct semi-structured interviews to analyze the causes of the errors.\nThe findings of this study are as follows:\n1.Most of the errors are due to text problems.\n2.Error types of linear inequalities in one unknown are:\n(1)The conversion of synonyms: a. the concept of synonyms; b. symbolic cognitive.\n(2)The range of solution: a. calculation; b. to determine the range of solution; c. number line; d. notation of inequality sign; e. direction of inequality sign.\n(3)Problem-solving in inequality: a. to reverse symbol; b. to remove bracket; c. calculation; d. to omit or add symbols; e. transposition errors; f. isometric axiom errors; g. lack of simplification; h. the misconception of inequality sign; i. multinomial errors; j. to copy wrong questions or answers; k. to speculate answers.\n(4)Text problem: a. not to understand the questions; b. mistakes in formulating expressions; c. triangle area formula errors; d. to ignore provided conditions; e. omission or wrong answers; f. notation of inequality sign; g. calculation; h. multinomial errors; I. the concept of synonyms.\n3.Cause of errors: \n(1) lack of prior knowledge; (2) to use wrong data; (3) the mutual interference of old and new learning experiences; (4) to use wrong calculating rules; (5) to generate answers from given numbers of the questions; (6) not to understand description of the questions; (7) to ignore the provided conditions or the answers are not complete; (8) not extend to continuous volume.