HKIER

摘要

本研究旨在探索學生在圖形樣式作業之一般化的表現，理解他們：如何利用姿勢與言辭對一般化各階段物件辨識出共通性；如何對樣式物件進行結構關係的連結；如何對一般化不同算式的等價進行認知；對代數的演變持何種觀點。本研究採用個案研究法進行探究，樣本來自台灣南部某公立小學兩名六年級學生；利用攝影、訪談與寫作方式蒐集資料，並採質性方法予以分析。研究發現，學生於：（1）發想階段運用視覺化圖形要素、比對分析物件變化的數量與配合項次數字形成規則，配合圖像的、直證的與比喻的姿勢知覺物件產出的共通性特質。（2）連結階段大多運用直證的和比喻的姿勢與語意做連結，配合數字、符號算式等對問題結構進行關係的連結。（3）以比喻的姿勢配合運算結果的驗證、算式結構的比對、物件關係的比對等策略，以作認知不同算式等價的基礎。（4）對一般化路徑中代數概念的演變，以指示樣式物件的數量、圖形項次的數字序號、與圖形結構關係的未知數等觀點持續發展。作者並針對學生一般化歷程代數概念發展與符號指示間的關係、多元表徵應用等議題提出建議，以作精進學生代數思考的教學參考。

關鍵詞：一般化；代數思考；姿勢；圖形樣式

Abstract

This study investigated how students generalize graphic patterns, aiming to understand how they utilize words and gestures at each phase of generalization, how they identify links between problems and recognize the equivalence of generalized equations, and how their understanding of algebra evolves. We conducted a case study of two sixth-grade students from a public elementary school in southern Taiwan, using video recordings, interviews and writing to collect data, which was then analyzed qualitatively. The study found that: (a) In the concept phase, students form rules by using visual graphics to compare changes in objects and item numbers, in accordance with the common elements of iconic, deictic and metaphoric gestures and perceptions. (b) In the linking phase, most students use deictic and metaphoric gestures and semantics, along with numbers and equations, to identify relationships in the problem structure. (c) Students employ strategies such as verifying metaphoric and computational outcomes, and comparing equation structure and object relationships, to comprehend the equivalence of different equations. (d) Students continue to develop their understanding of algebraic concepts based on item numbers, numbering of symbols, and the unknowns of relationship diagrams. Recommendations on diverse permeation and the relationship between development of algebraic concepts and symbol indication were provided. They can serve as references in advancing algebraic thinking.

Keywords: generalization; algebraic thinking; gesture; graphic pattern