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題名:國小六年級學童未知單元概念及其進展之個案研究
作者:陳維民
作者(外文):Wei-Ming 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:甯平獻
王國華
學位類別:博士
出版日期:2010
主題關鍵詞:未知數概念代數思考線性組合根本建構主義教學晤談法conceptions of the unknownalgebraic thinkinglinear combinationradical constructivismteaching interview
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本研究利用教學晤談法來探討一位六年級兒童,黃小凱(假名)的未知單元解題類型及其進展與未知單元概念及其進展。研究資料則來自於十七次晤談的錄影帶所轉譯與編輯而成的晤談原案。
根據原案分析及結果,本研究主張小凱在初始階段解未知單元問題的解題類型可以分為以下幾項:藉由逆算、藉由等價關係解一個未知數的問題,利用合併常數項以結算常數項、利用加減逆算以結算常數項解未知數的合成問題,利用乘法對加法的前分配性、利用乘法對加法的後分配性解未知數的分配問題,分割未知數作為新的單位量、嘗試約估符合條件的整數解未知數的分割問題。進入最終階段,小凱在解題類型的進展如下:透過逆算常數項進行關係轉換、透過逆算常數項及合併未知數單位進行關係轉換解未知數的合成問題,透過移除常數項以回溯未知數單位進行關係轉換、透過追加常數項以回溯未知數單位進行關係轉換解未知數的比較問題,利用當量除解未知數的分割問題,藉由比較未知數單位的差量、藉由方程式的比較解未知數的兩側問題。
根據原案分析的結果,本研究主張小凱在初始階段的未知單元概念具有:未知數概念的活動內嵌性、未知數概念的可被計數的性質、未知數概念的可被分割性質、未知數線性組合的內嵌性、未知數線性組合的分配性等五項性質。進入最終階段,小凱的未知數概念有下列的進展:未知數線性組合的脫嵌性質、未知數概念的有向數性質、以等量加減法公理調整關係、分割有向數未知數後喪失有向數性質。
最後本研究提出「未知單元概念的可能模型」,模型具備五種特性:未知數的前置概念(I)、未知數的前置概念(II)、起始未知數、有向未知數、關係未知數。
The main purpose of the study was to explore a sixth-grader’s problem solving patterns and the concepts of the topic of the unknown and their progresses. Seventeen teaching interviews had been conducted with a sixth-grader, Shaw-Kai (pseudo name), to collect data of his performance via video-taping.
According to the analysis and results, the study concludes that Shaw-Kai’s problem solving patterns of the topic of the unknown in the initial stage can be divided into the following items: 1) solve a problem where the unknown appears just once on one side of the equation via inverse operations or via equivalent relations; 2) solve the synthesis problems of the unknown via combining constants to account for constants or via inverse addition and subtraction to account for constants; 3) solve the distribution problems of the unknown via the multiplication on the addition’s pre-distribution and post-distribution properties; 4) solve the segmentation problems of the unknown via the segmentation of an unknown as a new unit or via an eligible integer based on try and estimation. In the final stage, Shaw-Kai’s problem solving patterns have the following progresses: 1) solve the synthesis problems of the unknown via the relation’s transformation resulting from inverse operation for the constant terms or from inverse operation for the constant terms together with the combinations of the unknown; 2) solve the comparison problems of the unknown via the relation’s transformation resulting from removing or adding the constant terms so as to go back to the units of the unknown; 3) solve the segmentation problems of the unknown via the equivalent division; 4) solve the problems where the unknown appears on two sides of the equation via comparing the difference of the units of the unknown or by comparison of equations.
The study also concludes that Shaw-Kai’s concepts of the topic of the unknown in the initial stage included the following five properties: 1) the unknown is embedded by activities; 2) the unknown is countable; 3) the unknown can be divided; 4) the embedded property of linear combinations of the unknown; 5) the distribution of linear combinations of the unknown. In the final stage, Shaw-Kai’s concepts of the topic of the unknown have the following progresses: 1) the disembedded property of the linear combination of the unknown; 2) the directed property for the concepts of the unknown; 3) to adjust the relationship by the equivalent axiom of addition and subtraction; 4) After the splitting of the directed unknowns, the directed property will disappear.
Finally, this research presented a possible model for the concepts of the topic of the unknown. The model has five properties: the pre-concept I of the unknown, the pre-concept II of the unknown, the initial unknown, the directed unknown, and the relational unknown.
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