Abstract The main purpose of this study is to investigate, through the ¡§Series¡¨ unit in the first volume of high school mathematics, the impact of the updated-version textbooks on the learning effectiveness of students so as to provide references to be used by future revisers of teaching materials. The research subjects were drawn from Kaohsiung City high schools. Three schools were selected (two public and one private). These three high schools, from the 1999 academic year, began to use mathematics textbooks of three different versions, one to each school. A first-year class and a second-year class were drawn from each school, and there were six classes in total. In total there were 237 students receiving the experimentation. The students from the same school, no matter they were first-year or second-year, had been taught by the same teacher in respect of the unit that was included in the study. Among the three schools, School ¥Ò®Õ admitted for the 1998 and 1999 school years new students with scores higher than 580 on the joint entrance examination, School ¤A®Õadmitted for the 1998 and 1999 school years new students with scores from 530 to 580 on the joint entrance examination, and School ¤þ®Õadmitted for the 1998 and 1999 school years new students with scores lower than 530 on the joint entrance examination. The three new versions used in the study were published by Nan-I Book Store, Ta Tung Information Enterprise, Co., Ltd., and Han Lin Publisher respectively. They are high school mathematics textbooks examined and approved by the Ministry of Education (excluding Teacher¡¦s Manual, supplementary materials, journals, reference books, etc). The outdated version was published by National Bureau of Editing and Translation. Mathematics Test written by the researcher was used as the testing tool, and the t-test method was adopted. Moreover, because the variances of the file data collected from a school fluctuated tremendously and the sample space was unequal, a single-factor ANOVA analysis was conducted to indicate whether there existed a significant difference between classes in terms of their learning achievement. The ¡§Questionnaire on Mathematics Opinions¡¨ was administered and the results were represented by ¡¥5¡¦, ¡¥4¡¦, ¡¥3¡¦, ¡¥2¡¦, ¡¥1¡¦, which indicated their scores. An ANOVA analysis was conducted to investigate the differences in students¡¦ learning attitude. After synthesizing the analytical results of this study, the author has come up with the following conclusions: ¤@¡BIt is known from t-test and ANOVA analysis: in School ¥Ò®Õ, there existed a significant difference between the first-year and the second-year students in respect of their learning effectiveness; in School ¤A®Õ, there existed no significant difference between the first-year and the second-year students in respect of their learning effectiveness; in School ¤þ®Õ, there existed no significant difference between the first-year and the second-year students in respect of their learning effectiveness. ¤G¡BAccording to the correct percentage of each question made by students and the error pattern, the test questions were classified into three types: concept, calculation, and comprehensive application (including questions that integrated comprehension, analysis, and application abilities). It was found from the classification of questions that: (1) Textbooks have no impact on both the well-established (School ¥Ò®Õ) and the weak (School ¤þ®Õ) students in respect of the effectiveness of learning concept questions. There is no difference between the outdated and the updated-version textbooks. However, as far as the students with ordinary achievement (School ¤A®Õ) are concerned, the updated-version textbooks have a significant impact on learning effectiveness. (2) Textbooks have an impact on both the well-established (School ¥Ò®Õ) and the weak (School¤þ®Õ) students in respect of the effectiveness of learning calculation questions, and the first-year students have benefited more from the updated-version textbooks than have their second-year counterparts in terms of the effectiveness in learning calculation questions. However, as far as the students with ordinary achievement (School ¤A®Õ) are concerned, the updated-version textbooks have a significant impact on the effectiveness of their learning to reduce cycle fractions, but have no significant impact on their learning other calculation questions. (3) Textbooks have an impact on the effectiveness of learning comprehensive application questions, as far as both the well-established (School¥Ò®Õ ) and the students with ordinary achievement (School¤A®Õ) are concerned. Moreover, the first-year students have benefited less from the updated-version textbooks than have their second-year counterparts in terms of the effectiveness in learning comprehensive application questions. However, as far as the weak students (School¤þ®Õ) are concerned, the updated-version textbooks have a significant impact on the effectiveness of their learning comprehensive application questions. In particular, the updated-version textbooks have an especially significant impact on the effectiveness in learning the comprehensive application questions of regression principle. ¤T¡BFrom investigating the difference between attitude toward and effectiveness in learning mathematics as far as different groups are concerned, it has been found: There existed a significant difference between learning attitude and learning effectiveness as far as the first-year and the second-year students of School ¥Ò®Õ are concerned. There existed no significant difference between learning attitude and learning effectiveness as far as the first-year students of School ¤A®Õare concerned. There existed a significant difference between learning attitude and learning effectiveness as far as the second-year students of School ¤þ®Õ are concerned. There existed no significant difference between learning attitude and learning effectiveness as far as the first-year and the second-year students of School ¤þ®Õ are concerned. ¥|¡BFrom the correlation between the learning attitude (consisting of five sub- categories: learning habit, learning process, learning desire, learning method, and test preparation) and the learning effectiveness of the first-year and second-year students of the three schools, it has been found: ¡]1¡^ As far as the correlations between their learning effectiveness and their learning desire and between their learning effectiveness and their learning method are concerned, there existed a significant difference between the first-year and the second-year students. Because both learning desire and learning method are related to textbooks, the updated-version textbooks had a significant impact on the learning effectiveness of the first-year students of School ¥Ò®Õ. ¡]2¡^ On the average, there existed a significant correlation between the learning effectiveness of the second-year students of School¥Ò®Õ , School ¤A®Õ, and School ¤þ®Õ and each of the five sub-categories of learning attitude (learning habit, learning process, learning desire, learning method, test preparation). Their performance on each sub-category of learning attitude is more or less related to textbooks. Therefore, there existed a significant correlation between the updated-version textbooks and each of the five sub-categories of learning attitude. ¡]1¡^ The correlation between learning effectiveness and learning method presented by the first-year students of School ¥Ò®Õ and School ¤þ®Õ (especially School ¥Ò®Õ) is more significant than that presented by their second-year counterparts. ¡]4¡^ The correlation between learning effectiveness and test preparation presented by the first-year students of School ¥Ò®Õ, School ¤A®Õand School ¤þ®Õ is more significant than that presented by their second-year counterparts.

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。