在隨機應答的技巧之中，自華納(Warner,1965) 提出華納模式後，即有多位學者以華納的觀點為籃圖，展開一系列的縱橫發展，並插枝分葉出各式技巧來(Chandhuri & Makerjee,1988)。 本文在第一、二章內，將分別介紹其中最典型的兩種方式：華納模式(Warner Model)與不相關問題模式(Unrelated-Question Model)。在第一章中並圖示固定的參數設計p 之下，比較華納模式的隨機應答方式與一般的直接應答方式效率之優劣，隨受訪者的誠實度變化之情形。在第二章中，除介紹不相關問題模式的技巧與方法外，也圖示比較華納模式與不相關問題模式取樣結果的估計值有效性之優劣。在第三章中，則闡述保護受訪者隱私的立場與獲得有效估計的目的如何互相衝突、矛盾，而定下參數設計的準則，以求取二者之間的平衡。最後一章則因Greenberg(1977) 所定義的冒險函數未能由所給的數學式完全描述，故在此章提出一些較合理的修正。

本研究先藉由文獻分析法瞭解擴散性思考、數學問題發現與數學學業成就三者的內涵，繼而依據它們的內涵並佐以學習、問題解決的角度，建立起三者間關係的假設，並透過實證調查研究法來驗證這些假設。本研究之研究對象為台北縣市五所國中的318位國三學生，研究工具為「新編創造思考測驗」、「數學問題發現測驗」、「第一次數學科基本學力測驗」，並以次數統計、集群分析、相關分析、變異數分析、逐步迴歸分析進行資料分析。本研究主要的研究結果如下： 一、學生能夠發現各種思考產物類型與數學類型的問題。其中，關係性問題與發現性問題最多人提出，而單位性、類別性與驗證性問題則較少人提出。 二、學生的數學問題發現型態有個別差異。 三、擴散性思考與數學問題發現間為顯著中低度相關。 四、擴散性思考與數學學業成就多為顯著中低度相關。 五、數學問題發現與數學學業成就間為顯著中低度相關。 六、能問大量且層次高數學問題的學生其數學學業成就比較不會問數學問題的學生為佳。 七、擴散性思考之流暢力、數學學業成就、擴散性思考之變通力可以有效預測數學問題發現之問題數。 八、擴散性思考之流暢力、數學學業成就、擴散性思考之變通力可以有效預測數學問題發現之問題獨特性。 九、數學學業成就與擴散性思考之流暢力可以有效預測數學問題發現之問題品質。 十、數學問題發現之問題品質、數學問題發現之問題數可以有效預測數學學業成就。 本研究最後針對數學教育以及未來研究提出若干具體建議。 / First, this study probed into the contents of divergent thinking, mathematical problem finding, and mathematical achievement by literature review. Then the researcher made hypotheses of the relationships between divergent thinking, mathematical problem finding, and mathematical achievement based on the contents of them and the views of learning and problem solving, and designed survey research to examine these hypotheses. The subjects were 318 9th grade students from five junior high schools in Taipei county and Taipei city. The data- collection instruments included：(a) New Creativity Test; (b) Mathematical Problem Finding Test; (c) Basic Educational Indicator Tests of Mathematics. After utilizing frequency, cluster analysis, correlation analysis, ANOVA, and stepwise regression, the main results of this investigation are：(a) Students can find problems of all kinds of intellectual products and mathematics. Among them, problems of relations and problems to find were found most and problems of units and classes and problems to prove were found least ; (b) There are individual differences between mathematical problem finding styles; (c) The correlations between divergent thinking and mathematical problem finding are significantly positive; (d) Most of the correlations between divergent thinking and mathematical achievement are significantly positive; (e) The correlations between mathematical problem finding and mathematical achievement are significantly positive; (f) Students who can finds many high-level problems have higher mathematical achievement than those who can not; (g) Fluency of divergent thinking, mathematical achievement, and flexibility of divergent thinking can be used to predict the number of problems of mathematical problem finding effectively; (h) Fluency of divergent thinking, mathematical achievement, and flexibility of divergent thinking can be used to predict the rarity of problems of mathematical problem finding effectively; (i) Mathematical achievement and fluency of divergent thinking can be used to predict the quality of problems of mathematical problem finding effectively; (j) The quality of problems and the number of problems can be used to predict mathematical achievement effectively. Finally, the researcher brings up some suggestions on mathematical education and the future research.

在迪菲方塊中，我們將正方形的四個頂點皆填入數值，再利用相鄰兩頂點相減，再取絕對值的方式觀察其數列行為，發現四個頂點的數字最後皆會收斂至0。在本文中，我們將之推廣至五邊形，我們稱它為迪菲五邊形。我們套用同樣的運算模式後，發現亦有特殊的收斂行為。 / In Diffy box, we write down numbers on the four vertices of square, and then on the midpoint of each side write the difference between the two numbers at its endpoints. It is known that the numbers on the four vertices of a square will converge to zero finally. In this article, we use the same operations as Diffy box to discuss pentagons which we call" Diffy pentagon ". We find it will converge, too.

迎接21世紀的來臨與世界各國之教改脈動，自1997年起，台灣政府致力教育改革，以期提升國民素質及國家競爭力。因此，教育部依據行政院核定之「教育改革行動方案」，進行國民教育階段課程與教學改革。隨著「九年一貫課程綱要」的公佈，台灣正式開啟一連串的教育革新。而對岸的中國，也於2001年啟動第八次基礎教育課程改革。兩國教育改革的內容皆涉及到課程目標、課程內容、課堂教學、學生學習等各層面，因此教科書、教材教法皆產生了巨大的變化。現有關於兩岸教育相關研究相當多，但兩岸課程的比較研究多停留在國小階段；且針對教科書的研究多半是以教科書的制度、審定為主，幾乎未有針對兩岸國中階段教科書的內容比較。 本研究先簡略介紹兩國教育背景並進行九年義務教育課程目標的分析，再對台灣康軒版、中國人教版國中數學課本、習作、教師手冊做出比較，得出結果如下： 一、在課程目標方面，兩者都強調數學與生活、其他學科的連結，也強調學生創新能力的培養；不同的是，中國人教版較強調數學各主題中的深耕，台灣康軒版則偏向各主題間的橫向連結；而中國人教版也較注重傳統的數學運算能力，台灣康軒版則較重視數學在生活中的運用。 二、在數學課本課程內容中，台灣課程偏向主題集中學習，而中國課程則偏向主題分散學習，每一學年課程都有代數、幾何、資料統計與機率。除此之外，中國人教版不論是代數或幾何，習題數約是台灣康軒版的三倍，但台灣康軒版在代數部份，課程較為流暢；中國人教版在幾何部份題目題材較多元。 三、在課本編輯方面，兩者架構都相當類似，皆具備章前圖、多元正文欄目、重點整理、補充知識等，但台灣康軒版部分章節，列中字數過多，且沒有數學名詞的中英文對照 四、在習作排版方面，中國人教版像是一本濃縮參考書，但台灣康軒版全版彩色，計算空間大，版面整齊，但中國人教版排版密集，閱讀上較為辛苦。 基於以上結論，針對台灣教科書提出以下建議： 1.幾何部份可以增加動手操作的機會，如折紙。 2.可以增加課本中的習題數，讓學生有多一點練習的機會。 3.增加數學史課程，適當說明前人在解決日常問題中的數學智慧。 4.增加統整課程的題型，題目的題材可以結合理化、美術等，增加多 元類型，提供學生思考的空間。 / To welcome the 21st century and catch up with other countries’ educational efforts, the government in Taiwan has engaged itself to educational reform since 1997. The purpose is to enhance the quality of its people and the ability to compete with other countries. Hence the Ministry of Education has tried to reform the compulsory education curriculum and teaching according to the “Action Proposal for Educational Reform,” authorized by the Executive Yuan. After the publication of “Grade 1-9 Curriculum Guidelines,” Taiwan has formally initiated a series of educational reforms. Similarly, Mainland China has started its eighth reform on Basic Education Curriculum. The reforms in both countries include the curriculum target, the curriculum content, classroom teaching, and students’ learning. They unavoidably exert a huge impact on the textbooks and teaching materials. Related researches on the education in both countries abound. However, the comparison of curriculum in both countries basically focuses on elementary school curriculum. Also, most researches on textbooks focus on the system or authorization. Seldom do we find research that focuses on the textbooks in junior high school. This research will first briefly introduce the education background in both countries and then analyze the target of compulsory education curriculum. The Kang Xuan version of Taiwan and the People’s Education version of China are taken as examples. We get the following conclusion after comparing their mathematics textbooks, workbooks, and teacher’s manuals. First, in terms of curriculum target, both versions emphasize the connection between mathmatics and life or other subjects. Both aim to cultivate the students’ creativity, too. Differently, People’s Education version goes deeper within one single topic, while Kang Xuan version focuses on the horizontal connection between different topics. People’s Education version puts more emphasis on the traditional arithmetic, while Kang Xuan version puts more emphasis on the application of math in everyday life. Second, according to the content of these two versions, when a new topic is introduced, the curriculum in Taiwan prefers concentrative learning, while the other version favors learning at different phases. Take People’s Education for example, every school year students have to learn algebra, geometry, data statics, and probability. Besides, in terms of algebra or geometry, People’s Education version provides exercises three times than Kang Xuan version does. Kang Xuan version, however, has a smoother curriculum design in terms of algebra. As for geometry, People’s Education version offers more different types of exercises. Third, speaking of the editing, both versions have similar curriculum structure. We can find in both versions pictures before each chapter, multiple columns presenting the main idea, arrangements of important points, and appended knowledge. However, some chapters of Kang Xuan version have too many words in one line, and it lacks Chinese-English terminology of math. Fourth, in terms of the workbooks, People’s Education version looks like a condensed reference book, while Kang Xuan version is bigger and more colorful. Kang Xuan version provides space for calculation and has neater arrangement, while People’s Education version is too wordy and takes more time to read the questions. Based on the conclusion above, this study offers some suggestions for future compilation of the Taiwanese mathematics textbooks: 1. For geometry, a good mathematics textbook should provide students with more opportunities to have hands on experiences, for example, by way of paper folding. 2. There should be more exercises in the textbook, hence giving the students more chances to practice. 3. History of mathmatics can be added. Students may have a better idea how ancient people used their wisdom to solve everyday mathmatics problems. 4. Exercises related to integrated curriculum should be added, too. Different subjects like science, art, etc…, can be combined together to provide multiple elements and a wider space for students’ ideas.

壹、引言 近年來，訊號傳送的途徑，已擺脫了傳統上著重管線傳送的優勢；有愈來愈多的訊號 彌漫在廣闊的空間裡，而這種無線式的傳送所需面臨的問題是：不具有排它性，任何 有接收器材的非原始接收者都可以截聽到訊息，由於因應而生的保密技術格外受矚目 ，密碼學（CRYPTOGRAPHY）便是滿足此需要的學問。本論文所探討的排列多項式（PE RMUTATION POLYNOMIAL）是密碼學中重要的工具之一。 貳、論文主體 所謂排列多項式，即是佈於代數體上的多項式，把此多項式當成函數而作用於代數體 （FIELD ）上，如果此函數具有一對一的性質，則是排列多項式。即 ｆ（ｘ）＝ａ。+ ａ1 ｘ1 + ....ａnｘn ≡ Ｆq〔Ｘ〕且 ｆ（ａ）╪ｆ（ｂ），ａ，ｂ≡Ｆｑ，ａ╪ｂ． 在論文中，介紹先進學者對排列多項式的認識。如：LAGRANGE'S INTERPOLATION是利 用函數值來描繪多項式，著名的學者CARLITZ ，利用特殊多項式來合成出排列多項式 ，論文中有更進一步的合成法提出，而HERMITE 跟DICKSON 學者則提出Ft函數其冪次 的變化情形，來判別排列多項式之是否，是最通俗的判別理論。 此外，由吾人所蒐集的資料中發現，在祗有兩項的多項式中，被發現到其它更簡捷快 速的判別方法，故二項式的多項式的探討是本論文的第一主題，對於 k j Ｘ＋ｂｘ ≡Ｆｑ〔Ｘ〕， 給予固定類型的ｑ，ｋ，ｊ情形下，祗須檢定ｂ是否具特 殊性質就可決定是否為排列多項式，這是一種方法。另有學者並不固定ｑ，ｋ，ｊ， 反而從ｑ，ｋ，ｊ數字下手，找尋出某種關連性，其結果使得係數ｂ，只有當ｂ＝０ ，時才有機會是排列多項式，乘下單項式的判別過程，就很容易了。另外還有一種方 法也是找尋ｑ，ｋ，ｊ間的關係，不過其結果在找出：多項式為非排列多項式，是比 較特別的地方。上述三方法，本論文網羅大部份有關論文，綜合各家之長，並適當給 予一同於原作者的新觀點證明方法。 至於本論文第二主題是著名的CARLITZ'S CONJECTURE此預測敘述：對於任何具有最高 冪次是偶數的多項式，必定存在一個自然數ｋ，使得給定的代數體，其元素個數只要 超過ｋ，則此多項式必定不是排列多項式。此預測當degree n=10,12,14, and 2m 時 已被證實為真。本論文僅就n=2m，做系統地探討及重新證明。 參、結語 本論文所論的兩主題，對於佈於代數體上的多項式是否為排列多項式，在判別的過程 上應有相當的助益才是。

本研究主要目的在瞭解透過動態評量回饋系統輔助教學對提升國中數學學習成效及數學學習態度的影響。本研究針對新北市某國中八年級兩班學生共38人，每週進行五節，總計為六週之數學教學實驗。本研究採準實驗研究設計，自變項為教學模式，在實驗組以動態評量回饋系統融入學習之數學教學課程而在對照組使用一般傳統教學方法進行數學評量及回饋課程。依變項為「數學學習成效測驗」、「數學學習態度量表」，共變項為八上三次段考平均。以單因子共變數分析與二因子重複量數混合設計變異數分析進行統計分析，研究結果如下: 一、在數學科學習成效方面：以動態評量回饋系統輔助數學教學，能提升數學科學習成效。 二、在數學學習態度方面：在數學學習態度分向度之「外在動機」向度上，實驗組與對照組有顯著差異，證實以動態評量回饋系統輔助數學教學，能提升數學學習態度之外在動機。在數學學習態度整體面向並未有顯著差異，證實以動態評量回饋系統輔助教學並未能提升數學整體學習態度。 三、學生對動態評量回饋系統輔助教學的回饋： 1、學生普遍具有正向、積極的反應。 2、學生願意多花時間來加強數學的學習。 綜上所述，本研究針對結果提出具體建議，以提供教學設計、數學教學及未來相關研究之參考。 / The purpose of this study was to investigate junior high school students’ mathematics learning achievement by using a Dynamic Assessment and Feedback (DAF) system. The overall investigation process is designed as a quasi-experiment, where two classes of eighth-grade junior high school students in New Taipei County were randomly assigned to either the experimental(n=18) or the control groups(n=20). It took totally six weeks, and there were five classes per week to perform the instructional experiments. In the experiments, the independent variables were instructional strategies and the dependent variables were Mathematics Achievement Test and Mathematics Learning Attitude. The measurement instruments include Questionnaire for Mathematics Learning Attitude, Mathematics Achievement Test, and Questionnaire for Students’ Perception of Mathematics Instruction with DAF system. According to the students’ Mathematics Achievement Test, the statistical analysis was performed by one-way ANCOVA. By conducting t-test for the scores of the math learning attitude measurement, the analysis of math learning attitude was carried out by the two-way Mixed ANOVA. After analyzing the data, the findings of this study are summarized below: 1.Mathematics Achievement: The experimental group scored higher in the Mathematics achievement test than the control group did. 2.Mathematics Learning Attitude: No significant interaction between the teaching methods was found in attitude for learning Mathematics. A significant difference was found between the experimental group and the control group in their extrinsic motivation for Mathematics Learning Attitude. 3.The students’ feedback after the (DAF) system mode: I.Most students in the experimental group did enjoy the DAF system. II.Most students were willing to spend more time using the DAF system to practice Mathematics. Finally, based on these findings, this study proposes several suggestions for school administrators, teachers in junior high school math curriculum and future research are also discussed.

為瞭解國民中小學學生學習電腦相關行為及現況,本研究以台北市國民中學三年級學生1251人及國小六年級學生1201人為研究對象,經運用電腦經驗調查表.認知需求量表.數學焦慮量表.電腦態度量表及電腦素養測驗卷等研究工具,獲得所需的資料,再以chi-square考驗.皮爾遜積差相關. t-test.典型相關.多元(逐步)迴歸及因徑分析等方法進行統計分析,結果 有以下幾點發現: 一. 國民中小學學生電腦經驗.電腦資源.電腦態度與電腦素養現況方面 1. 國中與國小學生之電腦經驗有顯著的差異存在. 2. 國中與國小學生之電腦資源部份有顯著的差異. 3. 國中與國小男女學生之電腦經驗部份有顯著的差異存在. 4. 國中與國小男女學生之電腦資源部份有顯著的差異存在. 5. 國中學生與國小學生在電腦焦慮與電腦有用性兩電腦態度上有顯著差異. 6. 國民中小學男學生之電腦態度(焦慮與自信)顯著較女生好. 7. 國中學生之電腦素養(軟硬體.應用與影響.操作.倫理)顯著較國中學生佳. 8. 國民中小學男學生之電腦素養(軟硬體.應用與影響.操作)顯著較女生佳. 二. 影響國中小學生電腦態度方面 1. 電腦經驗與國中小學生之電腦態度有顯著之正相關. 2. 電腦資源與國中小學生之電腦態度有顯著之正相關. 3.認知需求與國中小學生之電腦態度有顯著之正相關. 4. 數學焦慮與國中小學生之電腦態度有顯著之相關. 5. 數學成就與國中小學生之電腦態度有顯著之正相關. 6. 認知需求.每星期使用電腦的時間.每星期閱讀電腦書籍及雜誌的.家中電腦週邊設備的情形.自己或與他人共同使用電腦以及是否有電腦學前經驗等六個變項對國中小學生之電腦態度有顯著的預測效果,預測解釋力為32%. 三. 影響國中小學生電腦素養方面 1. 電腦經驗與國中小學生之電腦素養有顯著之正相關. 2. 電腦資源與國中小學生之電腦素養有顯著之正相關. 3. 認知需求與國中小學生之電腦素養有顯著之正相關. 4. 數學焦慮與國中小學生之電腦素養有顯著之相關. 5. 數學成就與國中小學生之電腦素養有顯著之正相關. 6. 每星期閱讀電腦書籍及雜誌的時間.家中電腦週邊設備的情形.自己或與他人共同使用電腦.電腦焦慮.電腦自信以及電腦喜歡等六個變項對國中小學生之電腦素養有顯著的預測效果,預測解釋力約為52%.四.整合模式分析方面1. 國中學生之電腦經驗與電腦資源對其電腦素養之影響,主要是來自直接效果的影響(經驗:57.32%;資源:73.31%), 而認知需求對電腦素養之影響主要是透過電腦態度間接效果的影響(89.06%).2. 電腦經驗與電腦資源對國小學生電腦素養之影響, 來自間接效果的影響約佔(經驗: 67.55%;資源:52.32%),認知需求對國小電腦素養的影響主要是透過電腦態度的間接效果,佔61.29%. With the expansion of computer technology, it is important to prepare individuals for success in our increasingly computerized society. Therefore,factors that impact decisions regarding attitude and literacy of computer should be expored. Several factors should be considered: (1) individual characteristics, (2) computer experience, (3) computer resource, (4) cognitiveneed, (5) mathematics anxiety, and (6) mathematics achievement. The purposes of this study were : (1) To find out the situation of information education in junior secondaryand primary school at present; (2) To find out the situation of the computer attitude and literacy in junior secondary and primary school students at present; (3) To examine the relationship and effects toward computer attitudes and literacy of many factors. 1251 junior secondary and 1201 primary school students were measured by (1)individual characteristics and computer experience questionnare, (2) Cognitive Need Scale , (3) Mathematics Anxiety Scale , (4) Computer Attitude Scale, and(5) Computer Literacy Test. The mainly statistical methods are chi-square, pearson product moment correlation, t-test, canonical correlation analysis, multiple(stepwise) regression analysis and path analysis. The results of the analysis of data indicated the following : (1) There were significant sex and school level differences of attitude and literacy toward computer. Boys were better than girls. Primary school students were better than junior secondary school; (2) There were significant correlations between computer experience, computer resource, cognitive need, mathematics anxiety, mathematics achievement, and computer attitude; (3) There were significant correlations between computer experience, computer resource, cognitive need, mathematics anxiety, mathematics achievement, and computer literacy; (4) Computer experience, computer resourse, cognitive need and computer attitude were significant predictors for computer literacy. Computer attitude was the best significant predictor. Recommendations were made for information education , school , family,educational organization and future additional research , to improve the attitude and literacy toward computer of junior secondary and primary school students.

University of Macau / Faculty of Education

University of Macau / Faculty of Education

University of Macau / Faculty of Science and Technology / Department of Mathematics