圓形複合圖形面積解題學習軌道之教學實驗研究   全文下載

Teaching Students How to Calculate the Area of Complex Figures With Circles by Using Learning Trajectories

本研究利用學習軌道設計圓形複合圖形面積課程進行教學實驗，檢視學生學習情形及解題表現，瞭解學生在各教學環節產生的迷思概念及困難，教師透過教學省思檢討自身教學及思索如何進行有效教學，幫助學生建立完整及系統性的圓形概念，順利解題。研究樣本為臺灣中部地區某公立小學一班六年級學生及其導師。課程內容以現行九年一貫課程綱要─六年級數學科的圓形複合圖形面積概念為主，參考數學任務及教學指引自行設計。蒐集和分析的資料包含：一、師生課室互動；二、學生數學學習單之表現；三、課後訪談紀錄。以學生作答內容統計答題正確率驗證教學效果，並以質性方式分析學生錯誤及迷思概念。研究發現：一、依據學習軌道設計圓形複合圖形面積任務，包含「認識圓形的基本組成要素」、「理解圓面積公式的由來」、「運用分割和移補的技巧來計算面積」、「理解分配律，學習用符號標記面積」和「『填補』和『其他策略』」；二、學業成就表現高、低分組的學生於「圓形複合圖形面積任務」的表現有所差異，其反應促使教師省思與修正調整學習軌道中任務的設計安排；三、經由教學實驗回溯省思後，教師對於學生概念的理解、教學方法及課程設計等方面的信念產生改變。

A mathematics course teaching students how to calculate the area of complex figures with circles was designed using learning trajectories. We conducted a teaching experiment to examine the learning conditions and problem-solving performance of students and understand their misconceptions and difficulties in different areas of learning. Teacher journals review teaching techniques and effective teaching methods to assist students in forming complete and systematic concepts of circles and in solving problems. The subjects in this study comprised a sixth-grade class of students at a public elementary school in Central Taiwan and their homeroom teacher. The course content focused on the concepts of determining the area of complex figures with circles, as introduced in sixth-grade mathematics in the Grade 1-9 Curriculum Guidelines. We referred to math teaching materials and guidelines to design the course. The collected and analyzed data included: (1) teacher-student interactions during class, (2) the math worksheet performance of students, and (3) after-class interview records. We verified teaching effectiveness by using the worksheet scores and qualitatively analyzed the misconceptions of students. The findings of this study indicate the following: (1) the teaching materials designed using learning trajectories include the Basic Composing Elements of Circles, Understanding the Circle Area Formula, Cutting and Shifting Techniques to Calculate Area, Understanding the Distributive Law and Marking Areas with Symbols, and Filling in and Other Strategies, (2) students with different levels of cognition displayed varying performance on the area worksheets, and (3) after retrospective reflections on the teaching experiment, the teacher’s behavior changed with regard to understanding the conceptions of the students, teaching methods, and curricular design.