數學擬題活動的合作效果─五年級學童之經驗

The collaborative effect of the mathematics problem posing activity ─ the fifth graders' experiences

江家瑋(Chia-wei Chiang)

指導教授 ： 陳斐卿

國立中央大學/文學院/學習與教學研究所/碩士(2014年)

摘要

近年來，擬題活動逐漸受到國際重視。出一個數學題目的歷程，即為數學擬題；其中又以自由擬題的出題形式最無限制。在台灣學童數學學習成就高、興致卻低落的情況之下，數學自由擬題可能是一改善的管道，也可能觸發更主動、深入的思考。

另一方面，長久以來，學術研究者認為合作學習好處多，然而教師在現場卻因為學生的異質性、課程進度壓力或經營的障礙，所以對合作的效果有所保留。因此，在對於擬題與合作效果的兩造關切之下，本研究將探討，在數學自由擬題的合作歷程當中：1)學童所擬題目的進展如何？2)合作效果又是如何？

本擬題活動特色有四─自由、合作、學生主導、一題多磨。研究對象為小五學童。每位學童僅擬「一」個題目，每個題目歷經五個版本的出題，歷經三階段的合作設計，並鼓勵學童互相建議與協助。第四版本出題後，教師會介入引導，故以第一版本至第四版本為「同儕協助階段」，以第四版本至第五版本為「教師引導階段」。

研究方法採質量混合方式。量化部份，自擬「擬題品質評量規準」與「建議品質評量規準」為工具，藉此對題目與建議之品質進行評量；質化部份，採內容分析法以分析學童所擬題目、建議、學習單與訪談內容，進而探索學生擬題之進展與合作效果。搭配兩個研究問題進行資料分析與詮釋。

研究結果發現分成三部分：自由擬題成品特徵、題目進展情形以及合作效果進展。首先，自由擬題的題目成品特徵有三：字數多、數字大又繁以及外部連結性強。第二，學生主導的擬題活動，題目品質於各向度皆有明顯進展。同儕協助階段的進展在生活性、精緻性與年級層次；教師引導階段則在可解性與可讀性。第三，合作時機與建議對象不同會帶來不同的合作效果。自時機來看，「第一次建議」的建議品質最高，「試答後建議與批改歷程」的效果多元，「轉燈建議」最能感受群眾數量壓力。

擬題成品與研究結果為學術與教學現場激起許多討論與應用上之建議。擬題成品的呈現，展現了自由擬題與生活情境題的價值：使學童更投入、主動、深入地學習，並自然而然地發展數學外部連結。合作時機與建議對象帶來的不同效果與互動歷程，也提供教學現場實施之圖像。而兩規準之應用，除了加強數學內部連結之外，建議規準的應用也可使數學溝通能力更有進展。

關鍵字

數學 ； 擬題 ； 自由擬題 ； 合作 ； 生活情境題

並列摘要

Problem-posing and collaboration have been recently considered as two important issues by mathematics education researchers around the world. However, academic researchers address the benefits of collaborative learning while school teachers point out that the students’ widely varied abilities and teachers’ time constraints will have an influence on the quality of the collaborative effects. In addition, “high achievement but low interest” is still an unsolved problem in mathematics education in Taiwan. Therefore, the two research questions are: 1) To what extend do student’s problem-posing improve? 2) What collaborative effects do suggestions from peers and teachers have on the quality of students’ problems?

The major characteristic of this problem-posing activity is “one problem, many revised versions”. There are two stages: the “peer-help” stage is from the first to the fourth version, and the “teacher-guide” stage is between the fourth and the fifth version. Each fifth grader is encouraged to give suggestions to his or her classmates and to help each other pose a mathematic problem. After the fourth version is completed, the teacher will provide guidance to problems that need further improvement.

This study used a mixed method. Two sets of rubrics were generated to evaluate the quality of the problems by posers and suggestions by peers. The results were analyzed in a quantitative fashion. Problems between versions, interviews, and worksheets data were used to analyze qualitatively the progress they made.

The result shows that: 1) The free problem-posing activity produced problems presented with wordiness, inconvenient numbers, and high relations with students’ life experience or other subjects. 2) The student-posed problems made significant progress throughout the five-version process. Each received different kinds of help from the “peer-help” stage and “teacher-guide” stage. In particular, the teacher had a significant influence on solvability and readability of the problems, while peers helped on the connection to their daily life, the refinement, and the concept level of the problems.

Debates and implications for school teaching are provided. The free problem-posing has its virtues in engaging students in more active and deeper thoughts, but has its drawbacks in developing lengthy and context-rich problems which goes against the current practice among professional mathematicians. The different collaborative design and its effects on peer collaboration provide a promising picture for teacher practitioners. Future study can focus on the refinement of the two sets of rubrics. The potential of problem posing on student’s mathematics communication ability is also discussed.

並列關鍵字

mathematics ； problem-posing ； free problem-posing ； collaboration ； context-embedded problem

參考文獻

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