探討雙重鷹架在多重表徵的動態幾何環境中對解題過程之影響－以二元一次方程式為例

Abstract

培養學生具備問題解決的能力，是學習數學的主要目的。透過多重表徵的使用，除了能幫助學生理解概念，更能進一步協助問題解決的過程。故本研究先利用學習單作為學生理解多重表徵的鷹架，再使用多重表徵以作為問題解決的鷹架，進而分析不同數位教材與學習單的使用下，不同先備知識學生的學習表現與解題過程之差異。其中表徵方式分為常用表徵（方程式、靜態圖形、數對）與附加表徵（常用表徵、表格、動態圖形）；而學習單分為一般型（概念理解、情境問題解決）與轉譯型（概念理解、表徵轉譯、情境問題解決）。 本研究以「二元一次方程式」單元進行設計，並利用GeoGebra動態幾何軟體進行教學，以探討113位八年級學生於教學過程中，透過不同表徵多樣性與學習單的使用，不同先備知識學生在概念理解、表徵轉譯、問題解決的學習表現。並挑選24位學生進行教學後晤談，以了解不同表徵多樣性與學習單的使用對於學生解題過程之影響。資料收集與分析主要為表現測驗（前、後測）與半結構式晤談。 研究結果顯示，在學習單與先備知識交互作用下，使用轉譯型學習單的學生在表徵轉譯與問題解決的學習成效上，達到鷹架的目的，其中以高先備知識學生的學習成效最為明顯；對於低先備知識學生，其學習成效則無明顯差異。在表徵多樣性與先備知識交互作用下，雖然對於學生在問題解決的表現有所不同，但未有明顯差異，因此附加表徵似乎未能達到鷹架的目的。對照量化與晤談結果發現，不同學習單對於表徵轉譯與問題解決有不一樣成效，其中轉譯型學習單可能對於理解問題、分析目標、發展計畫等三個階段比較有效果，而表徵多樣性對於問題解決沒有明顯成效。此外，對於學生整體的解題表現，轉譯型學習單則是搭配常用表徵進行教學下較有效果。

The purpose of this study is to investigate how the use of worksheets and multiple representations scaffold students’ learning about linear equations in two variables and to understand whether students’ prior knowledge interacts with the use of designed worksheets and multiple representations. Two types of worksheets were used: General type (including conceptual understanding and situational problem-solving items) and Translation type (including the items in the general type and the translation of representations). Also, two sets of multiple representations created by GeoGebra software were developed: Common representations (including equations, static graphics, and ordered pairs) and Additional representaions (including the common representations, tables, and dynamic graphics). A 2x2 factorial research design was employed with the participation of 113 eighth-grade students divided into 4 groups. To learn the topic, each group received instruction by using one type of the worksheets with either common representations or additional representations. Their performances in problem solving, conceptual learning, and representation translation were examined by the pretest and posttest. After the instruction, 24 students with different levels of prior knowledge were interviewed in order to understand their problem-solving process. According to the three-way ANCOVA analysis, the translation worksheets can scaffold high prior knowledge students’ learning performances in problem solving and representation translation, but the effect of different sets of multiple representations were not found. Additionally, the analysis of the interviews showed that students who used the translation worksheets tended to perform better on problem comprehension, goal analysis, and plan development. The findings suggested that the translation worksheets could better support students’ learning performances when they were used with common representations.