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題名:數學建模活動下國小五年級學生代數思考及其發展歷程之研究
作者:陳冠州
作者(外文):Kuan-Jou Chen
校院名稱:國立彰化師範大學
系所名稱:科學教育研究所
指導教授:段 曉 林
秦 爾 聰
學位類別:博士
出版日期:2009
主題關鍵詞:代數思考算術思考數學建模活動符號化algebraic thinkingarithmetic thinkingmathematical modeling activitysymbolization
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本研究主要的目的在數學建模活動下,探索學生代數的解題表現,並透過Tall的E-S-F認知模式(Embodiment, Symbolism, and Formalism cognitive model)了解學生代數思考的發展歷程。研究的設計是以32位五年級學生為研究對象,由研究者研擬建模活動實施的方法,藉由閱讀文章、預備活動、陳述問題以及策略分享等過程,透過建模教學模式讓學生在引模、探模與修模的不同階段去針對「視覺型樣式活動」與「文字情境的代數問題」的數學建模活動進行討論、解題。整個建模活動一共分成8個單元,每個單元進行時間為2~3節不等,歷經一個學期的教學實施,除了針對所有的課室教學進行錄影外,其他所收集的研究資料還包括學習單、學生任務報告單、教室觀察表和學習日誌等。研究者將上述多元的質性資料加以編碼分析,並結合課室錄影之影像分析結果進行三角校正,據此作深入的分析詮釋,以獲致研究的結論。從學生代數解題表現中發現,在建模活動中學生不僅能以文字或符號表徵代數問題,使用符號表示未知數以及求出代數方程式的解,也可以透過察覺問題情境中關鍵性要素,與發現基本樣式中的規律與關係,最後順利提出模式並解決問題。研究結果顯示,學生的代數解題表現可分成:從理解基本樣式的要素到運用一般化算術,再由運用一般化算術到使用符號的轉變,最後由使用符號到察覺代數結構的等價性等三個階段。以E-S-F認知模式來看,透過知覺、行動以及心智壓縮等過程,五年級學生的代數思考歷程似乎能夠藉由文章閱讀、語言溝通和活動操作逐漸進入具象化階段;另外,學生可以經由圖形-文字-算術-符號的階層性的心智壓縮,而從具象化階段進到符號化階段;隨後,在前面兩個階段的基礎下,學生透過視覺-空間思維、算術或代數形式進入到形式化的階段。研究者根據上述結果,在數學建模活動下,除了發現五年級學生具有算術一般化和代數一般化的表現外,從E-S-F認知模式的觀點,本研究對於其代數思考的發展歷程亦提出了一個可能的演化路徑。最後研究者則提出關於教學、課程、學習和研究方法等方面的蘊涵與建議,以供未來研究參考。
This doctoral dissertation aims at investigating the fifth graders’ algebraic thinking and its developmental process through Tall’s E-S-F (Embodiment, Symbolism, and Formalism) cognitive model under the context of mathematical modeling activity. The research subjects are thirty-two fifth graders. The implementation of modeling activity includes article reading, preparation activities, problem description, and strategy sharing, which allow children to discuss and solve visualization pattern and textual algebraic problems in the modeling-eliciting, modeling-exploring, and modeling- adapting stages. The whole modeling activity consists of eight units which have been implemented for a semester. Including the video recording of all the classroom teaching, the other collected data are worksheets, task-reports, students’ journals, and classroom observation records. These qualitative data are triangulated with the video analysis of the classroom teaching records. It can be observed from children’s performances in solving algebraic problems, they can not only use words or symbols to represent algebraic problems and solve the algebraic equations by representing the unknowns with symbols, but also successfully propose a model and solve the problem by detecting the critical elements in the problem context and discovering the rules and relationships within the basic patterns in the modeling activity. The results of this study show that children’s performances in solving algebraic problems can be divided into three stages: comprehending the components of basic patterns to generalized arithmetic, transition from generalized arithmetic to generalized algebra, and from generalized algebra to appreciation of the equivalence of algebraic structures. From the viewpoint of the E-S-F cognitive model, the fifth graders’ algebraic thinking seems to be active in the embodiment phrase through article reading, oral communication and manipulation activity; then, it might be shifted from embodiment phrase to symbolism phrase by way of mental compression of pictures, words, arithmetic and symbols. In addition, under the basis of the former two phrases, it might reach the formal phrase through visuo-spatial, arithmetic, algebraic modes. According to these results, the researcher proposes a hypothetical development trajectory of children’s algebraic thinking from Tall’s E-S-F viewpoint. Finally, some relevant implications and suggestions of instruction, curriculum, leaning and research methodology are addressed.
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