國小六年級學生運用一般化基模進行圖形規律問題解題之研究   全文下載

Application of Generalization Schemas to Solve Figural Pattern Problems on Sixth Graders

本研究旨在探索學生對圖形規律問題一般化歷程產出的基模，以掌握學生一般化認知結構，理解其一般化運作情形；並探討學生一般化歷程基模的轉換，建構一般化解題模式，以提升代數思考教學的成效。研究樣本為小學3 名六年級學生，參與研究者設計之三十二項圖形規律問題，並從學生對作業的操作與訪談蒐集資料，資料分析與說明則採質性方法呈現。綜合研究發現，獲得以下三階段結果：學生在發想階段利用「整體圖形關係」和「部分結構要素」概念基模計畫解題，在連結階段則運用「圖形特徵與圖次比對」與「物件計數與圖次比對」概念基模進行圖形與圖次關係的連結，在歸納階段採用「單位組合」與「圖形結構」概念基模，協助其進行解題。在一般化歷程上，也利用「加法」、「乘法」與「實用」等運算基模協助整合規則或算式。學生因圖形結構的性質與一般化解題經驗與知識，讓其在一般化歷程上產生基模與解題策略的改變與轉換，使其朝向更精細代數思考心智模式的運用與圖形整體結構關係之整合。以一般化基模運作與發展作為基礎，可建構出「利用圖形結構」與「利用數字序列」兩種一般化解題模式。研究者根據發現結果提出學生解題模式與建議，提供未來代數思考教學與研究參考。

The purpose of this study is to enhance our understanding of the generalization process by examining the generalization schemas of figural pattern problems and capturing the cognitive structure of generalization, and by examining the ways in which to improve the generalization schema transformation to construct models for solving generalization problems and promote the effects of algebraic thinking. Three Grade six students in a teaching activity setting completed 32 tasks related to figural pattern problems, and the worksheets and interviews data were collected. The data were analyzed qualitatively, and three stages were considered: (1) Students used both the "relationship of whole figure" and "elements of part structure" concept schemas for problem-solving planning in the abductive stage; (2) both schemas were applied during the connection stage on the "figural characterizes contrast with figural terms" and "objects count contrast with figural terms" to combine the relationship between figures and terms; (3) students used both the "unit combined" and "figural structure" concept schemas to solve the figural pattern problems during the generalized stage. Students used "addition," "multiplication," and "practical" operation schemas to integrate the rules and expressions for resolving the figural pattern problems. The change and transformation of the schemas during generalization were influenced by student knowledge, experiences, and characteristics of the figural structure. Researchers constructed both the "utilize the figural structure" and "utilize the number alley" models for problem-solving generalization learning based on students' generalization schema operation and development. The findings such as the models of problem-solving generalization support teachers' instruction, engaging students in algebraic thinking and implementing algebraic teaching with figural pattern problem solving.