國小五年級學童在動態幾何軟體輔助下學習三角邊長關係臆測歷程之個案研究

本研究探討國小五年級學生在動態幾何軟體輔助下，如何進行三角邊長關係臆測任務。本研究以程序性反駁模式（F. L. Lin ，2006）作為中介理論架構，以此基礎發展數學臆測任務，佐以動態幾何軟體為「例子產生器」與表格資訊（鄭英豪、陳建誠與許慧玉，2017）來鷹架臆測歷程，目的是要了解教師如何融入動態幾何軟體於三角形邊長關係臆測任務以及促發學生臆測歷程，同時，深入了解國小五年級學生以此產生的數學臆測歷程與類型及學習困難。本研究採取質性研究中的個案研究，透過錄影、錄音逐字稿、螢幕錄影逐字稿以及學習單，觀察與分析六位新竹縣市國小五年級學生於幾何軟體Geogebra搭配表格資訊下，進行三角形邊長關係臆測任務時呈現的數學臆測歷程。本研究發現如下：（1）欲運用動態幾何幫助學生從圖形轉化成理論，需要教學設計前端鋪陳學生主動反思機會以及上升歷程與下降歷程轉換過程；（2）動態幾何軟體作為「例子產生器」，能幫助學生自主產生典型案例與非典型案例；（3）Geogebra量測值與圖形之間誤差將導致學生思考矛盾，教師須適時介入或於教學前說明；（4）結合動態幾何軟體與表格資訊下的臆測類型，在國小端需考慮「目標不對等」的臆測類型；（5）學生難以同時處理圖形改變、數值資料與表格資訊進行數學臆測，必要時將資訊簡化成數字規律呈現以利數學關係式的臆測；（6）學生於Geogebra連續量以及形狀變化的觀察，有機會發覺邊長「範圍」關係；（7）學生易針對「可見測量值」進行推測，而忽略未顯示的量測值。

關鍵字

動態幾何軟體；三角形；臆測任務；數學臆測歷程；程序性反駁模式；邊長關係

並列摘要

This study investigated how fifth-grade students conjecture triangle inequality in dynamic geometry software(DGS). Using the proceduralized refutation model as an intermediate theoretical framework and referring to DGS as an “example generator” combined with spreadsheets to support students to develop the conjecturing process. This study aims to getting know how to help students to induce conjecturing process by using DGS in triangle inequality conjecturing tasks, meanwhile, deeply analyzing the process and difficulties that fifth-grade students may have when conjecturing. A total of six fifth grade students participated in this case study research and observed the conjecturing process under GDS and spreadsheets by monitor video, recording, transcript and task. Based on the qualitative analysis, we demonstrated that (1) providing an opportunity for reflection and transferring between ascending and descending process in task design can help students to explore regularities from drawings to theory; (2) taking DGS as an “example generator” can help students spontaneously make typical and non-typical examples; (3)teacher could demonstrated the error of measurement in DGS in advance, otherwise, it may cause contradiction ; (4) another special conjecture approach, generating examples and conjecturing under goal asymmetry situation, being explored in elementary students; (5) it’s difficult for students to deal with drawing, measurement data and spreadsheet at the same time, if necessary, simplifying information to simple number list would help students to induce mathematical form; (6) students have tendency to explore length range relationship due to observing measurement data and drawing continuously changed in DGS; (7) it’s easy for students to conjecture with “visible measurement” and ignore measumrent that are not displayed.