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題名：數學自由擬題之困境歷程與擁有感轉變
作者：黃佩岑
作者(外文)：Pei-Tsen Huang
校院名稱：國立中央大學
系所名稱：學習與教學研究所
指導教授：陳斐卿
學位類別：博士
出版日期：2020
主題關鍵詞：數學自由擬題；情意面；集體擁有感；擬題困境；同儕互評；free math problem-posing；affection；collective psychological ownership；difficulties；peer assessment
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培養學生喜愛數學是教育事業的核心關切。臺灣地區國民小學學生的數學國際排名名列前茅，然而學生在自信心與喜歡數學的程度卻持續殿後，似乎學生有著不錯的解題能力、卻不甚喜歡「解題」這種活動。因此，本研究採數學自由擬題的一題多磨活動，不僅了解學生的困境歷程為何，而且透過學生對題目的擁有感，進而探討擁有感的轉變為何。
研究場域為北台灣一間國小，研究對象為兩班五年級共56位學生。本研究設計採數學自由擬題的課程，施測時間各班總計長達十堂課，共計400分鐘。採用混合研究法，探討研究問題一：在數學自由擬題的困境歷程為何，分別有擬題、建議和修改等三類困境。資料蒐集有題目卷、建議卷、學習單、田野筆記以及訪談。資料分析以逐字稿為主，輔以相關文件；逐字稿的編碼系統採紮根精神的開放性編碼。探討研究問題二：在數學自由擬題歷程的擁有感轉變為何，以個人與集體擁有感問卷調查為主，統計分析採相依樣本 t 檢定、單因子重複測量變異數分析比較差異。
研究結果有五：一、知易行難的擬題困境：想法實現在題目、數學概念融入題目、出不來具挑戰性的題目，以及自擬的題目卻不會解題的困境。二、難掌握複雜型的建議困境：試答遇上數字大計算困難、自認能力不足、題目的字數多，以及題目問題多連帶計算步驟也多的困境。三、建議分歧不易採納的修改困境：自認能力不足、增加題目內容、難解題、數字安排的修改、琢磨題目的完整性，以及無幫助的建議內容。四、在數學自由擬題的四個階段內整體集體擁有感都顯著高於個人擁有感，且達顯著差異。五、個人擁有感的發展在第三階段開始有顯著上升；集體擁有感的發展則是在四階段無顯著差異。
本文的具體貢獻：本研究證實擬題活動打破學生不喜愛的刻板印象，雖然在擬題、建議與修改上會帶些困境和負面情感，但是從生產題目到答題來看，學生投入班級題庫與擂臺賽情形仍是熱絡參與，足以作為教學現場的參考與啟發。研究侷限在所難免，未來研究可以朝向擁有感量表的翻修與檢驗；探究給建議的鷹架設計。
Motivating students to enjoy learning math is one of the main objectives of compulsory education in Taiwan. A recent assessment conducted by the Trends in International Mathematics and Science Study (TIMSS) shows that primary school students in Taiwan are at a top-performed level, but their level of confidence in math is lower than most of other countries. This result implies that primary school students in Taiwan may not enjoy solving math problems even though they are able to apply their knowledge in a variety of relatively complex math problems. To further understand this challenge facing the students, this study adopts the free math problem-posing curricular – specifically, the one problem, many revised versions – activities in two classrooms and explores the transition of psychological ownership during the problem-posing.
The fieldwork, the free math problem-posing activities (10 classes, total 400 minutes to each class), was conducted in two 5th grade classrooms that comprise 56 students (the subjects) of a primary school in Northern Taiwan. Using mixed research methods, this study first explores the difficulties at the stage of posing, suggesting, and modifying during the problem-posing activities. Data was collected from the answered questionnaire, feedback notes, fieldwork memos, and it was analyzed mainly through the transcriptions from the interviews. Open coding system based on the grounded theory was adopted for the transcriptions. The second inquiry of this study is to examine the transition of psychological ownership during the free math problem-posing activities. Data analyzed was collected from the surveys of individual and collective psychological ownership during the problem-posing activities. As for the quantitative analysis, the dependent sample t-test and the one-way repeated measures ANOVA were conducted.
The main findings from this study: 1) Difficulties of problem-posing , students encountered difficulties in posing advanced math problems and embedding mathematic concepts into the questions they posed as well as in solving the problem they posed, even they have understood well those concepts; 2) Difficulties of making good suggestions, students were not able to give constructive suggestions especially when they encountered math problems that have large numbers for calculation, they lack of confidence to solve, have long descriptions as well as have a lot of sub-questions and need more steps to solve; 3) Difficulties of making the modifications, students faced difficulties to harmonize disagreements due to the lack of confidence, adding contents of the problems, difficult posed-questions, changing of numeric arrangements, consideration of the completeness of the problems, and useless feedback; 4) During the problem-posing, collective psychological ownership was significantly higher than the individual psychological ownership; 5) Individual psychological ownership significantly increased since phase three but collective psychological ownership remained unchanged throughout the posing activities.
In summary, as the main contribution, this study shows that although students experienced difficulties and challenges at the stage of posing, feedback and revision, they were still enthusiastically participating in the activities of generating and solving math problems. This finding is opposite from implication according to TIMSS that primary school students in Taiwan may not enjoy solving math problems, and can also serve as a reference for practitioners teaching the related subjects. It is also unavoidable of the research limitation in this study. For future study, it is suggested to improve and further examine the application of the questionnaire regarding psychological ownership as well as the research design for inquiring suggestion, and modification during the problem-posing.
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