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。 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。 本研究的主要結果如下： 一、不等式答題表現在文字問題的錯誤比率、空白比率最高。 二、在一元一次不等式的錯誤類型為： （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。 （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。 （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。 （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。 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. This 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. The findings of this study are as follows: 1.Most of the errors are due to text problems. 2.Error types of linear inequalities in one unknown are: (1)The conversion of synonyms: a. the concept of synonyms; b. symbolic cognitive. (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. (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. (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. 3.Cause of errors: (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.

本研究旨在探討高一學生於「複數與複數平面」單元教學完畢後，解題時發生錯誤的主要類型，並探討發生錯誤的原因，再根據這些錯誤的原因，設計補救教學課程，以進行補救教學活動，並分析補救教學活動的成效。 根據本研究，學生於「複數與複數平面」單元解題主要錯誤類型有下列十一種：對複數的定義了解不清、不知二根式乘除的運算規則、不熟悉i的次n方或不具「級數」觀念無法順利利用 乘冪的特殊性質、作複數的四則運算時易發生錯誤、不知複數絕對值的意義或觀念仍停留在國中「負的變正的，正的還是正的」的錯誤觀念、不清楚共軛複數的定義或共軛複數與複數絕對值產生混淆、不會利用複數絕對值的運算性質、無法將複數正確地對應到複數平面上、平面幾何能力不足、尚未建立複數概念、習慣以整數思考。發生錯誤的主要因素有：定義不清，似是而非、受直觀影響，產生錯誤類推、無法將複數與複數平面連結、先備知識不足。 補救教學成效方面：空白率不論由整份試卷或個別試題做檢定，均呈現明顯下降、答對率以整體學生來看，大部分試題答對率均上升，以高低分組來看，無論是高分組或低分組學生，成績均明顯改善。保留效果以整體學生來看，答對率並沒有顯著差異，全體學生後測與延後測成績呈高度正相關。特別針對低分組學生做檢定，發現低分組學生的學習保留得很好，後測與延後測答題差異並不顯著。此外，學生普遍認為以PowerPoint 與 GSP 為工具的補救教學，學習內容與教材更能吸引學生的注意，增進學生的學習成效，因此學生對補救教學大多抱持著正面的看法。 / The research aims to explore the main error types of first grader’s in a senior high school when solving the unit of complex number and complex plane after the instruction, including the possible reasons of their errors. The course designs of the remedial instructions are followed according to those reasons. At the end of this study, some analyses of the effects of remedial instruction activities are provided. In this study, there are eleven main error types from the first graders when proceeding to do the unit of complex number and complex plane : vague understanding about the definition of complex number; lacking necessary knowledge for the operation rules with the multiplication and division of two radicals; unfamiliar with handling the power of i and sum of the them; constantly making mistakes when doing the fundamental operations of arithmetic with complex number; confusing the meaning or conception concerning the absolute value of complex number under the false ideas existed in junior high about “A negative becomes a positive , and a positive is still a positive ”; not knowing the definition of conjugate complex number or confusing conjugate complex number with the absolute value of complex number; incapable of using the idea of operating the absolute value of complex number; failing to relate complex number to complex plane; inability in plane geometry; unable to construct the concept of complex number; habitual thinking with integer. The primary factors of making the above errors, according to this research, are as follows: ambiguous definition; paradoxical; influenced by instinct; making false analogy; failing to effectively connect complex number with complex plane; lacking prerequisite knowledge. As for the results of this remedial teaching, the rate of blank in students’ answer sheets, regardless of the whole test sheets or individual tests, has obviously decreased; the learners’ rate of accuracy as a whole is mostly higher; the improvement of grades, both in the brilliant group and in the poor group, is significantly improved; the effect of retention, in view of the whole participants’ rate of accuracy, varies insignificantly; more important, the results of students’ post-tests and postponed post-tests are in highly positive correlation. In testing the poor group, we find their retention of learning excellent. The variations of answers in the post-tests and postponed post-tests are not significant. Besides, most learners pay much attention to the contents and materials of employing Power Point and GSP and consider them as useful tools in the remedial teaching. In conclusion, the participants make great progress under this experiment. Therefore, most learners take positive perspectives from the remedial instruction.

本研究旨在探討小學生在數學基本運算的歷程中，所產生的錯誤類型，由文獻中尋找出特定的錯誤類型加以驗證，並利用與受試晤談的方式來了解兒童的思考歷程，試圖找出這些錯誤類型發生的原因，同時探討不同性別在這些錯誤類型上是否有差異存在。 　　本研究以國小二、三及四年級的學生為取樣對象，有效樣本為1782人（男926，女856人）。研究工具為自編之「加法運算能力測驗」、「減法運算能力測驗」、「乘法運算能力測驗」及「除法運算能力測驗」：資料分析則以信度考驗、次數分配、皮爾遜積差相關及單因子多變量變異數分析來進行。 　　本研究的主要結果如下： 　　（一）在加法的基本運算中，學生最常犯的錯誤為「相加時未加上進位數」，但此類型錯誤多為隨機產生的。在與學生晤談過程中，即可發現是因學生對進位概念不夠深刻而導致不小心犯錯。比較具有系統性規則可循的錯誤類型為「個位數未進位至十位」及「不對稱相加」二種，這二種錯誤的產生與學生誤用對齊相加的概念有關。這四種錯誤類型彼此之間具有顯著的正相關存在。 　　（二）在減法的基本運算中，最明顯也具有系統性的錯誤類型為「大數減小數」及「０減任何數為０」，這二種類型錯誤可能原因與兒童自行建構錯誤算則及對０的概念不清楚有關，由學生的晤談中可加以佐證；其他的類型（由左而右運算及未退位相減）的系統性偏低，是屬於隨機性發生的，這四種錯誤類型之間具有顯著的正相關存在。 　　（三）在乘法的基本運算中，「進位數直接與十位相乘」、「個位數直接相乘，十位往左乘」及「未乘十位數」等三個錯誤類型具有系統性規則可循，而且這三種錯誤的產生與進位數位置干擾有關；其餘二個類型錯誤（進位數重複相加及未加進位數）則屬於隨機發生的錯誤，此五類型之間亦具有顯著正相關存在。 　　（四）在除法運算中「商數未補０」及「商數多加０」二種錯誤類型是系統性較強的錯誤，這二種錯誤與學生的位值概念不清楚有關；而「單獨相除」及「倒置商數」二個類型錯誤則多為隨機性產生的，此四種類型間具有顯著正相關存在。 　　根據研究結果加以討論，並提出若干建議，以供教學輔導之參考。