本研究的目的是希望探究透過一個月約16節課,每節50分鐘的教學,是否可以用波利亞的解題步驟來有效的教導國中比例單元。本研究以準實驗設計(quasi-experimental design)中的不等群前後測實驗設計(nonequivalent-group pretest-posttest design)方式進行。施測對象是臺北市某中下程度學校,兩班(每班22人)國二上學期的學生,一班被編作實驗組,透過波利亞的解題步驟來教導比例的種種概念,教學期間常明顯的強調波利亞「怎樣解題﹞@書中的策略;另一班則作為控制組,以傳統講述方式教學。在研究進行期間蒐集到的資料包括兩組學生在教學前後對比例的瞭解程度及對數學的態度、以及實驗組的學生經教學後對波利亞啟思法的態度等等。在資料分析方面,本研究從解題錯誤策略的考量,來對學生的答案作細部分析,並採用多變量變異數及相關的方法來分析形式推理能力、場地依賴/獨立以及教學方法等因素對學生表現的影響。在本研究中發現兩組學生在教學之前對於比例的先備知識、了解題意與畫圖能力、形式推理能力、場地依賴等層面均沒有差異。本研究的資料顯示波利亞的方法可被八年級的學生接受。關於波利亞解題步驟的第一步驟了解題意階段,經教學後,不管在了解題意或畫圖方面,實驗組的表現都優於控制組。具備形式推理能力的學生在「了解題意」及「畫圖」的表現上,有顯著優於沒有形式推理的學生。至於波利亞的第二及第三個步驟的教學效果就不如了解題意明顯。在教學前及教學後解題能力的提昇方面,兩組學生的差異並未顯著效果。從學生對數學的態度問卷中得知,多數的學生仍認為數學題目只有一個解法。至於回顧答案方面,實驗組有一名學生在後測中曾兩度使用兩個方法去解題,由於並沒有要求學生一題多解,故此可能實驗組經教學後比控制組有回顧問題解答的感覺(sence)。在一題多解的態度上,接受波利亞解題步驟教學的學生,對於能靠自己的能力想出新的方法所帶來的滿足感,其程度顯著高於接受傳統教學法的學生。一般而言,接受波利亞教學法的學生,在教學後對數學學習的態度上顯著優於接受傳統教學法的學生,雖然他們在教學前及後對數學學習態度的改變幅度並沒有顯著差異於控制組的學生,但卻普遍成正值。此外還發現場地依賴與形式推理對學生學習數學的態度有顯著的交互作用,在那些具備場地依賴的學生中,有形式思維者顯著高於沒有形式思維者。另一方面,具備形式推理的學生對「數學信念與作數學的態度」顯著優於控制組的學生。至於比例單元的結果,則發現兩組學生對濃度概念以及比與濃度之間關係的瞭解都比較弱。根據上述的研究發現與心得,本研究建議,教師宜針對易受誤導的學生訓練其了解題意的能力並且讓學生體會了解題意的重要性;而且宜讓解題貫穿整年的數學課,而且提供學生不同的解題經驗,並鼓勵教室討論的風氣。此外,教師宜教導學生建立一個有意義的比,並呈現數學概念時應與其他學科整合,更值得留意的是教師要培養學生表達數學的能力。至於未來研究方向,可以考慮延長一個月教導波利亞啟思法的時間,以及加入文獻中所提的後設認知去幫助學生學習,值得日後進一步研究者參考之。

The purpose of this study is to explore the effectiveness of one month (16 classes,each 50 minutes)of teaching junior secondary students to solve proportion problemsby means of Polya's problem solving heuristics. This study adopted a nonequivalent-group pretest posttest design on two classes of eighth graders(22 students each)that were taught by the same teacher in a medium-ability school in Taipei city.One of the classes,the experimental groupal,was taught proportion with frequent emphasis on various Polya's problem solving heuristics as were laid out in the book,"How to solve It."In contrast,the control group was taught by the traditionallecturing method. Various types of data were collected,including pri-and posttestof knowledge in proportion,questionnaire of students' attitude towards mathematicsbefore and after instruction,as well as the experimental group's attitude towardsPolya's heuristics. The questionnaire was found to consist of three factors,namely,attitude towards learning nathenatics,attitude towards doing mathematics,and attribution for success in examinations. Various data analyses were performed,including multivariate ANOVA of such independent variables as field independenceand formal thinking on various test/item scores,as well as detailed analyses ofstrategies used by students. It was found that the two groups were basically equivalentin terms of their prior knowledge of proportion,as well as starting abilities in understanding the problem,diagram drawing,field independence and formal thinking.The result from this study indicated that Polya's approach was readily acceptableby eighth graders.With respect to the first phase of Polya's approach,the experimentalgroup performed significantly better than the control group regarding their performancein "understanding the problem"and"drawing to facilitate problem solving.Moreover,students who handled more abstract thinking were found to perform better than thosewith lower abilities.As regards the second and third phase,the result was less obvious.The differences in problem solving ability before and after instruction were insignificant between the two groups.Though most students still view mathematicalproblems as cinstituting only one solution,one particular student in the experimentalgroup twice furnished more than one solution in the posttest. In addition,the extent of satisfaction derived from furnishing a new solution was significantlyhigher for the experimental group than in the control group. In general,students'attitude towards mathmatics learning was,after instruction,better for the experimentalthan the control group.Yet the enhancement of attitude from pretest time was notdifferent between the two groups.Also,it was found that field dependency and formalthinking had significant disordinal interaction effect on students' attitude towardslearning mathematics.In particular,among the field dependent individuals,those who were also formal thinkers held significantly better attitude over the lessformal group.On the other hand,formal thinkers had significantly better attitudetowards doing mathematics than nonformal thinkers,as did those instructed underPolya's approach rather the lecturing approach. As regards the learning of proportion,students in both classes were weak in the concept of concentration as well as its difference fromproportion.Based on the results of this study,it is suggested that teachers shouldhelp students, especially those who are easily mislead while problem solving,to appreciate the importance of understanding the problem.Students should be given chances to enculturate a rich problem solving experience.Problem solving shouldbe integrated into and diffused throughout the school curriculum.Besides,teachersshould enable students to construct meaningful proportion concept,preferably withrespect to other subject areas.Teachers should also encourage students to explaintheir procedures to make sure that they understand the mathematics concepts behind.Future study may consider extending the instruction period as well as including features from the metacognition literature.