近年來，問題解決普遍受到各領域的重視，更是數學教育的重心，由於問題表徵是問題解決中的重要成分，外在表徵是解題的重要輔助，因此，有關表徵的議題，自有其廣受研究與討論的價值。表徵之所以可以用來幫助解題，與其本質涉及到同構性有關，而且同一數學概念的不同表徵，彼此之間的聯繫與轉換也是藉由其同構性而來，因此，本研究的目的是想從同構的觀點出發，深入探究同構轉化這種解題表徵方式的特性，及其對解題的影響，此外，本研究還希望探討什麼因素會影響解題者選擇不同的解題表徵，以及不同解題表徵之間有何相關性。
本研究之對象為24名主要是國一的數學資優生，大部份來自大台北地區。整個研究進行約一學期，共12節課，每節為時約三小時，其中大部份時間為分組教學，24位學生共分為五組，本組有5位學生。前10節課雖有統一教材，但依各研究者所從事研究之主題而有不同的重點，最後兩節課則採個別訪談的方式進行，目的是讓各研究者能針對各自的研究主題對學生進行深入的瞭解與探究。故本研究資料包括兩部份：一是前十次學生在上課時的解題作業資料，二是後兩次的口頭訪談資料。在資料分析方面，由於本研究的人數不多，因此，本研究在量的研究方面以採用描述性的統計方式來進行分析，並以質的方式進行深入的個案研究分析，從而探討學生解題時有關表徵的表現情形。
本研究中的資料顯示：同構轉化對於解題有正面的影響，既可以學習亦值得培養，唯學生自發使用的優先性並不高、且不易養成習慣、其遷移性也不強。而此同構轉化能力可藉由具有階層性的同構提示來加以訓練達成。關於影響解題表徵方式選擇的因素，主要為個人特質，解題經驗、以及題目性質與呈現形式等三個因素，以最後一個因素為例，本組學生會依題目的深層結構，再考量題目的表面呈現形式來做出選擇。然而，無論上述三個因素如何影響表徵的選擇，都與學生的同構轉化能力有關。
本研究還發現本組學生選擇解題表徵方式的順序，大致上為先腦中思考、再畫圖、最後才是實作。對於比較習慣腦中思考的學生，抽象思考能力較優者，其解題效果亦較佳；至於以畫圖方式來進行解題者，若能發揮好的解題效果，其往往符合題目特性並具有好的畫圖品質；在實作方面若解題者的操作能力不好、題目難度不夠或題目類型不適當的話，其解題效果便不能發揮。
根據上述研究結果，對教學與研究建議教師宜嘗試運用同構轉化來教學，並培養學生同構轉化能力，唯須注意克服其運用上的限制；關於畫圖與實作等外在表徵的使用上，宜注意解題者以及題目對其解題效果所產生的影響，其中應提供學生充足、方便、利於操作的實作環境；另外教師在設計生活情境式的題目時，宜注意學生對其背後數學概念的熟悉程度以及題目敘述方式，否則可能會造成反效果，除此之外，數據之大小亦不宜忽略。
至於日後的研究方向，本研究建議多進行表徵教學的探究，研發如何利用內在與外在表徵，增進學生掌握題目深層結構以及解題關鍵的能力；另外，對於「解題表徵方式」、「題目」以及「解題者」三者之間關係的釐清也可做進一步研究。

In resent years, problem solving has become a key concern in many disciplines, especially in mathematics education. Since problem representation is an important component of problem solving, and the mastery of the use of external representation can be of much help in problem solving, so problem representation is a worthy area of research and discussion. The reason why representation can help in problem solving is because representation involved some form of isomorphism. Moreover, the relationship between different representations also depends on isomorphism to a certain extent. The purpose of this study is to explore and understand about how isomorphic transformation would influence on problem solving. In addition, this study investigated into how a problem solver would choose among various representations, and also into the relationship among different problem representations.
The subjects of this study included 24 mathematically gifted students, most of them being seventh graders. The duration of this study was about one semester with 12 sessions, each lasting for 3 hours. Most of the time instruction were conducted in a small group discussion setting. The subjects were divided into 5 groups. The first ten class materials were uniformly used across all groups, with each group having different emphasis. The last two sessions were conducted in a one to one interview format, so as to find out more about how students would approach solving problem. Because of the small sample size, data analysis were performed mostly by means of descripitive statistics, and qualitative case study method.
According to the data of this study, isomorphic transformation had positive influences on problem solving, but its use by students was not spontaneous. Moreover, students did not easily make a habit of using it, nor transfer its use on similar setting. However, The ability to use isomorphic transformation could be trained and fostered by means of solving isomorphic problems in a sequential way. The data also revealed that the factors that affected students' choice of representation included personal characteristic, experience of problem solving, and the formats of representation. For the last factor mentioned above, it was found that students tended to consider the deep structure of problem first, and then the format of the representation. Yet regardless of how the three factors would influence the choice of representation, it all depended on the students' ability of isomorphic transformation.
It was found that the order of choice of the representational format was mental computation, drawing, and then physical manipulation. As regards mental computation, it was found that the better the ability of the student to think abstractly, the better his/her problem solving ability. As for drawing, if the drawn figure matched the feature of the problem and had good quality, then it would facilitate the solving of the problem. As regards physical manipulation, it would not not help if either the problem solver was not handy in terms of manipulation, or the problem was not difficult enough, or if it was not of the appropriate problem type.
Based on the results of the study, it is suggested that teachers should try to utilize isomorphic transformation in their teaching. Moreover, they should foster the ability of students to perform isomorphic transformation, but yet notice the limits of its use. As regards the use of drawing and physical manipulation as the means of external representation, teachers should pay attention to how effective problem solving depended on the problem solver and the problem itself. Teachers should also try to provide a convenient and easy environment for students to solve problems by physical manipulation. Furthermore, teachers who design real-life problems should notice the extent to which students are familiar with the mathematical concept behind the problems as well as the way in which the problems are worded. Otherwise, it may cause opposite effect.
Future study may consider developing of programs to teach internal and external representation, so as to enhance students' ability to identify both the deep structure and the key points of the problems. Besides, more work should be devoted to study the relationship among the types of problems, their representations, and the problem solvers while solving problem.

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