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。

在統計學上，我們常使用皮爾森相關係數(Pearson’s Correlation Coefficient)來表達兩變數間線性關係的強度，同時也表達出關係之方向。傳統之相關係數所處理的資料都是明確的實數值，但是當資料是模糊數時，並不適合使用傳統的方法來計算模糊相關係數。而本研究探討區間模糊樣本資料值求得模糊相關係數，首先將區間型模糊資料分為離散型和連續型，提出區間模糊相關係數定義，並提出廣義誤差公式，將相關係數作合理的調整，使所求的出相關係數更加精確。在第三章我們以影響數學成就評量的因素，作實證研究分析，得出合理的分析。而此相關係數定義和廣義誤差公式也能應用在兩資料值為實數或其中一筆資料值為實數的情況，可以解釋更多在實務上所發生的相關現象。 / In the statistic research, we usually express the magnitude of linear relation between two variables by means of Pearson’s Correlation Coefficient, which is also used to convey the direction of such relation. Traditionally, correlation coefficient deals with data which consist of specific real numbers. But when the data are composed of fuzzy numbers, it is not feasible to use this traditional approach to figure out the fuzzy correlation coefficient. The present study investigates the fuzzy samples of interval data to find out the fuzzy correlation coefficient. First, we categorize the fuzzy interval data into two types: discrete and continuous. Second, we define fuzzy correlation with interval data and propose broad formulas of error in order to adjust the coefficient more reasonably and deal with it more accurately. In Chapter Three, we conduct empirical research by the factor which affects the evaluation of mathematical achievement to acquire reasonable analysis. By doing so, broad definition of coefficient and formulas of error can also be applied to the conditions of either both values of the data are real number or one value of the data is real number, and can explain more related practical phenomenon.

在試題難易度分析上，傳統方式常以答對率或得分率的高低來認定試題的難易度。但答對率或得分率高低並不能真正表達應試者在答題時的難易感受程度。過去研究指出影響數學科試題難易度的主要因素有三：評量的數學內容、解題時的思考策略及解題所需的步驟數。 本論文針對影響數學科試題難易度的三個主要因素進行分析統計。以模糊統計的角度，提出非固定權重因子之二維模糊數。應用模糊數絕對距離的概念以摒除族群中異端值，最後針對不同族群進行難易指標分析與檢定。 / On the basis of difficulty analysis, the hit rate or the scoring rate are considered as the index of the difficulty of the questions traditionally. However, these rates can not represent how test takers feel when taking tests. According to the previous papers on this subject, they note three factors to affect the difficulty analysis on math questions the content of the evaluation, the strategies of question solving, and the required steps to solve a question. This research is focused on the three major factors which influence the difficulty of math examinations. With the angle of fuzzy statistics, a two-dimension fuzzy number will be presented with non-fixed weighted factors. By applying the absolute distance concept of fuzzy number, the extreme values are excluded. Consequently, the indices of difficulty from different groups can be tested and analyzed.

University of Macau / Faculty of Education

本研究主要是針對高中數學課程中「信賴區間」的這個單元，依據99 課綱中的課程規劃，設計出一套專題式的研究教材，並以筆者所任教高中的高二及高三學生作為研究對象，進行專題性的課程授課，且對其學習成果進行評量。主要研究結果如下： 一、高三學生雖已進行過「信賴區間」及其先備知識之授課，但前測的成績並不理想。 二、高二與高三學生經由筆者授課教學後，其後測成績均較前測成績有非常明顯之進步，不過高二與高三學生的後測成績並無顯著差異。 三、高二自然組與高二社會組學生經由筆者授課教學後，其學習成效亦無顯著差異，但社會組學生學習上普遍較為認真，後測成績稍高於自然組。 四、高三自然組與高三社會組學生經由筆者授課教學後，其後測成績具顯著差異，而進步成績的學習成效亦具有顯著差異，自然組優於社會組。 五、依高中數學學習成就分成高分群、中分群與低分群三群，雖然在前測與後測成績表現上顯著不同，但進步的成績則三群並無顯著差異。 此外，筆者於本次研究中也對學生問卷調查一些筆者有興趣的相關議題，並進行問卷分析，得到以下結果： 一、對於本研究所編撰「信賴區間」之課程教材，學生普遍能夠接受且瞭解，並知曉「信賴區間」在生活上的用處，且能解讀其資訊。唯實務面上，他們對「信賴區間」之學習則持可有可無的態度。 二、本次研究的授課方式對於自然組與社會組學生的接受程度是具有差異的，其中自然組學生較能接受本次非傳統型的授課方式。 三、學生普遍認為高中數學中，「非統計類數學課程」是比較有趣的，「統計類數學課程」則在學習上具相對困難性。而在統計的課程中，「信賴區間」倒是比較感興趣的這單元。 整體而言，本次研究對學生進行信賴區間的教學結果，是具有學習成效的。 / Based on the 99 Curriculum Guidelines for the Senior High School Math, a special set of study material for Confidence Interval was composed. Eleventh and twelfth grade students from a girl’s senior high school were recruited voluntarily and lectured, and their learning performance were evaluated before and after the completion of the lecture. The primary findings are as the following: 1. Though twelfth grade students have already studied Confidence Interval before the lecture, their pre-test scores were still low. 2. On the average, both eleventh and twelfth grade students performed better after the lecture, and no significant differences were observed between them. 3. For the eleventh grade students, no significant differences were observed between social science and natural science groups. However, students in social science group appeared to work harder, and their post-test results were slightly better than those in natural science group. 4. For the twelfth grade students, significant differences were observed between social science and natural science groups. Natural science group students appeared to outperform their counterparts in social science group. 5. Among the top third, the middle third, and the bottom third of all the participating students, although their pre-test and post-test scores differed significantly, the differences between the two tests were not significant. In addition, some secondary issues were also explored, and the related findings are summarized as follows: 1. Students showed appreciation for the study material, understood the concept of Confidence Interval better after the lecture and even realize how to apply the concept to their daily life. Surprisingly, however, they didn’t think learning Confidence Interval would make any difference in their life. 2. Students in the natural science group appeared to have greater acceptance toward the unconventional teaching method than those in the social science group. 3. For the topics covered in senior high school math, students generally considered those unrelated to statistics more interesting, and thought that statistics-related topics were more difficult to learn. However, among the statistics-related topics, Confidence Interval was the most intriguing one. In conclusion, this study reveals that the experimental teaching approach concerning Confidence Interval are apparently positive and effective.

本研究主要目的是探討合作學習融入數學教學對於偏遠地區學校國一學生在學習國中數學第一冊的學習成效。以屏東縣一所偏遠地區學校國一學生兩班共55人為研究樣本，其中一班為實驗組，實施「合作學習融入數學教學」；另一班為對照組，實施傳統講述法，來探討學生接受不同的教學方法之後，在數學學習成就、態度及保留三方面的差異性。實驗結果，可得以下之結論： 一、在衡量數學學習成就指標上，實驗組與對照組之間有顯著差異且實驗組優 於對照組。 二、在衡量數學學習態度指標上，實驗組與對照組之間有顯著差異且實驗組優 於對照組。 三、在衡量數學學習保留指標上，實驗組與對照組之間有顯著差異且實驗組優 於對照組。 最後本文再對上述統計檢定推論提出建議，以供教師實務上教學及後續相關研究之參考。 / The main purpose of this study is to apply “cooperative learning in mathematics” in remote area to explore its effects of learning performance by seventh graders. Two classes, which have a total of 55 students, were sampled from a junior high school in a remote area of Pingtung County. One class students were assigned as the experimental group and the other students as the control group. The first class was taught using “cooperative learning in mathematics”, while the second one was taught using traditional method. In order to find if there are differences on learning achievements, learning attitudes, and learning retention of mathematics between two teaching methods, statistical tests were conducted. The following conclusions are reached︰ 1.Based on mathematics learning achievement indicator, there is a significant difference between two teaching methods. The effect by the experimental group is significantly better than that by the control group. 2.Based on mathematics learning attitude indicator, there is a significant difference between two teaching methods. The effect by the experimental group is significantly better than that by the control group. 3.Based on mathematics learning retention indicator, there is a significant difference between two teaching methods. The effect by the experimental group is significantly better than that by the control group. Finally, suggestions for practical teaching are provided and future possible researches are also discussed.

數學能力向來被視為解決問題能力的重要基礎。隨著知識經濟時代的來臨，數學愈加受到重視，數學教師的教學效能亦面臨新的挑戰。為了協助師培生發展更具創意與多元的教學方式，本研究在中學數學科教材教法之課堂中，提供學生以知識翻新（knowledge-building）(Scardamalia, 2002) 教育理念為核心的課程設計，並使用知識論壇(Knowledge Forum™)數位學習平台作為本課程的線上輔助學習環境。研究主要目的在利用教室中與論壇上的知識翻新活動以提昇師培生對數學相關信念之反思與瞭解。研究對象為9位師培生。本研究透過混合研究設計以蒐集資料，資料來源包括：（1）整學期師培生在「知識論壇」平台上之討論與貼文；（2）期初與期末的數學信念開放式問卷；與（3）師培生的試教。資料分析方式如下：(1)平台上的貼文主要使用知識論壇分析工具(Analytic Toolkit for Forum)探討師培生在平台上的互動與知識翻新歷程；（2）數學信念開放式問卷主要以兩種觀點進行內容分析並交叉檢證；與（3）試教過程(錄影檔)以開放式問卷所產生之編碼進行影片內容的分析。 研究結果發現：（1）以知識翻新理論為原則之課程設計有助於師培生發展更建構取向之數學信念；（2）使用知識論壇平台有助於師培生分享知識與自我反思；（3）數學信念與教學實踐呈現相互影響的關係；以及（4）透過知識翻新的課程設計，師培生逐漸理解多元與彈性的教學對學生學習的重要性。本研究根據研究結果提出以下幾點建議以供未來師培教育之參考：（1）數學師資培育必須將專業學術知識的學習與成熟信念的養成作更密切的銜接；（2）師培機構進行課程設計時，應提供更多元開放的教學方式來幫助師培生學習；（3）師培教育應鼓勵師培生發展更能適應未來學生學習之教學方法。 / The purpose of this study was to investigate the impact of knowledge building pedagogy and technology on preservice teachers’ views on the nature of, and teaching practices in, mathematics. Participants were nine preservice teachers who took a university course titled “High-School Mathematics Teaching.” A software program called Knowledge Forum was employed to enable an online knowledge building environment for preservice teachers to explore, reflect, and discuss about the nature of, and teaching practices in, mathematics. Data mainly came from three sources: (1) preservice teachers’ online posting and discussion recorded in a KF database, (2) a survey about mathematical beliefs with eight open-ended questions, and (3) preservice teachers’ teaching practice videos. To analyze, (1) online discussion records automatically recorded in Knowledge Forum were analyzed using descriptive statistics; (2) open-ended survey questions were content-analyzed following an open-coding procedure; and (3) two cases of preservice teachers’ teaching practice were further explored by employing video analysis techniques. The main findings were as follows: (1) engaging in knowledge building was found to help preservice teachers develop more constructivist-oriented mathematics beliefs; (2) use of Knowledge Forum was helpful for preservice teachers in sharing their knowledge and reflecting on their teaching; (3) mathematics beliefs and teaching practice could mutually influence each other; (4) after engaging in knowledge building for a whole semester, preservice teachers were able to realize the importance of capitalizing on more diversified ways of teaching to enhance student learning. Building on the results, this study made the following three suggestions: (1) teacher education program should pay more attention to integrating the learning of pedagogical content knowledge into the development of more mature mathematical beliefs among preservice teachers; (2) teacher education program should encourage more diversified and flexible ways of teaching practice when designing its courses; and (3) teacher education program should encourage preservice teachers to develop more adaptive teaching practices to help students learn in the future.

為達成「精熟學習」的教與學過程中強調的個別化教學、學生學習的回饋、校正學生學習錯誤等高效能的教學理想。本研究自行發展一套數位化診斷系統，在不改變學校政策、班級作息、教室管理實施情況下，對準備參加科大入學測驗高工三年級的學生，在考前六週進行數學科總複習的「精熟學習配合數位化診斷」實驗教學。經由學生的作答反應，本診斷系統完成以下功能： (1)能快速的篩選出優良試題，並提供試題的難度、鑑別度、答對率與評鑑試題的品質。 (2)能診斷出個別學生與試題的差異性，並完成試題卷的信度、效度評鑑與S-P分析表。 (3)能提供完整的學習成就遷移記錄，並可歸類出學生學習潛能與整體學習表現。 (4)能彙整出學生學習成效與具體教學建議。 教師藉診斷系統所提供的資訊，對學生進行個別化校正或充實教學，期以達成精熟學習的教學理想。並在實驗教學後分析學生學習回饋單，大多數學生對實驗教學法持正面態度，且自認學習態度更積極，並建議若能提早實施精熟學習配合數位化診斷教學，應會有更好成績表現。 本研究並探討「實驗教學」的實驗A組學生，分別對「不同科別」的實驗B組學生與對「傳統教學」的控制C組學生的數學學習態度、學習成就、學習成就絕對均差的差異性，經單因子共變數分析，排除前測影響後，具體結論如下： (1)實驗A、B組與控制C組學生在數學學習態度後測得分並無明顯差異。 (2)實驗A組與控制C組在數學學習成就後測成績有明顯差異，且實驗A組分數明顯高於控制C組。 (3)不同科別實驗組的A組電機類與B組機械類學生，在數學學習成就後測成績有明顯差異，且電機類分數明顯高於機械類，但機械類排除未完成實驗的學生後，則不同科別的實驗A組、實驗B組，在數學學習成就後測成績無明顯差異。 (4)實驗A組與控制C組學生在數學學習成就後測成績的絕對均差有明顯差異，且實驗A組絕對均差小於控制C組。 最後為使實驗教學能更順利進行，建議教學前先營造愉快與舒適教學環境與良好的師生互動情誼再實施，這將會是實驗教學成功的關鍵 / Mastery Learning Strategy is a highly effective teaching method, emphasizing the importance of individualized tutoring, student feedback and learning error correction. In this study, a self-devised computerized diagnostic system was developed for the third-year vocational high school students who were preparing for the technological university entrance examination. An experimental teaching method using Mastery Learning Strategy with computerized diagnostic system was given to the aforementioned students as a math review lesson six weeks before the entrance examination, following the original school policy, class schedule or class management. From test responses, the diagnostic system achieved the following: (1)Quick selecting of good quality test items. It also evaluates the test items and provides the difficulty, discrimination, and correction rates of them. (2)Individualized diagnosis of each student vs. test items. It also completes the reliability, validity and S-P charts of the test. (3)Offering a comprehensive record of the learning process. It also analyzes the learning process of the students and sorts out their learning potential. (4)Analyzing students’ learning effects and coming up with teaching suggestions. In this experiment, the data gained from the diagnosis were used to provide each student with individualized instruction or error correction with the hope of attaining the goals of Mastery Learning. The analysis of the students’ feedback after the experimental teaching method showed that most students thought positively of the experiment, admitting that they developed a more aggressive learning attitude, and suggested that this computerized diagnostic system, had it been implemented earlier, would have helped them achieve even better grades. The students in this study consisted of three groups. Group A and B were both experimental groups except that group A were students from electrical engineering department while group B were students from mechanical engineering department. Group C was the control group receiving only conventional teaching method. Group A was compared with group B and group C respectively in terms of differences in math-learning attitude, math-learning achievement, and in the absolute mean deviation of learning achievement. By means of one-way analysis of covariance, with pre-test effects eliminated, the conclusions were as follows: (1)None of the groups exhibited significant grade differences on the math-learning attitude post-test. (2)Group A scored significantly higher on the math-learning achievement post-test than did Group C, the control group. (3)Group A of electrical engineering students scored significantly higher on the math-learning achievement post-test than did Group B of mechanical engineering students. However, if the grades of the Group B students who didn’t complete the experiment were unaccounted, no grade difference was noticeable on the math-learning achievement post-test between Group A and Group B. (4)Group A, the experimental group, had significantly lower absolute mean deviation on the math-learning achievement post-test than did Group C, the control group. Finally, it was suggested that creating a comfortable environment for teaching and learning as well as an advance establishment of a good teacher-student rapport are the key to a smooth-going and successful experimental teaching method.

University of Macau / Faculty of Education

University of Macau / Faculty of Education