國中生在二次函數概念上的主要錯誤類型及其補救教學之研究

Abstract

本研究目的在探討國中九年級學生在學習「二次函數」單元後，有哪些錯誤類型。本研究採用二階段評量來診斷並透過訪談整理歸納成為錯誤類型，再進行錯誤類型的成因分析，然後根據成因設計補救教學教材，並進行補救教學活動。 根據本研究，國中九年級學生在二次函數概念的主要錯誤類型可分成以下四大類，共11種：(一)對二次函數代數式解釋的錯誤：(1)不瞭解二次函數中「二次」的意義；(2)將二次函數y=ax^2+bx+c與一元二次方程式ax^2+bx+c=0混淆；(3)在做一般式y=ax^2+bx+c轉換成標準式y=a(x-h)^2+k、假設標準式y=a(x-h)^2+k將a當成1或從標準式找對稱軸發生之錯誤。(二)對二次函數圖形解釋的錯誤：(4)只關心圖形看得到的部分，忽略圖形隱含的解析性質；(5)認為拋物線的部分圖形是線性；(6)對稱軸概念的錯誤。(三)二次函數代數式表徵與圖形表徵之間轉換的錯誤：(7)不瞭解y=ax^2之a與圖形之關係；(8)不瞭解圖形的左右平移與代數式y=a(x-h)^2+k中h、k之關係。(四)二次函數的特殊點（與x、y軸的交點、頂點）的錯誤：(9)認為二次函數的頂點都在y軸上；(10)不瞭解二次函數y=ax^2+bx+c中b^2-4ac與x軸交點個數的關係；(11)不瞭解二次函數y=a(x-h)^2+k的頂點坐標(h,k)與y=ax^2+bx+c的關係。 就補救教學的成效而言，在經過補救教學活動之後，後測各題的答題正確率皆高於前測。在所有的試題中，其答題正確率全部均提高35%以上，其中有6題後測答題正確率超過85%。參與補救教學的學生，其後測的答題正確率皆高於前測。就錯誤類型的變化情形來看，學生所犯錯誤類型數量皆低於前測。可見二次函數概念的補救教學活動對於改善學生在二次函數概念常犯的錯誤類型有顯著的成效。從後測和延後測的結果來看，學生在後測與延後測的答題情形差異不大，顯示學生對於二次函數概念補救教學的學習具有保留效果。The objective of this study is to investigate the error patterns in learning quadratic function among the ninth graders, and to develop the associated remedies for students to better understanding the course materials. This study adopts the two-tier assessment to identify and organize the error patterns of quadratic functions among the ninth graders. The causes of these error patterns are identified by using the technique. Based on these results, the materials and activities of remedy are then developed, and we use these materials to have a teaching experiment. Four major error patterns of quadratic functions are found: 1. the wrong interpretation of the algebra form of quadratic function, 2. the wrong interpretation of the graph of quadratic functions, 3. the mistakes in the transition between the algebra and graph representations of the quadratic functions, and 4. the errors in some special points of quadratic functions, i.e. the vertex, and the intersection points with X and Y axes respectively. From the results of the teaching experiment, it shows that the rates of accuracy in all of the questions was significantly increased from the pre-test to post-test with respect to the remedy activities. The accuracy rates were increased over 35% in all questions. Among them, six of the thirteen questions have over 85% accuracy rates in the post-test. All of the participants got better accuracy rates in the post-test than in the pre-test. The students also showed their great improvements in all of the identified error patterns. Therefore, we conclude that the remedy activity can significantly mitigate the occurrence of the major error patterns among the students. Comparing the results of post-test and extended-post-test, the retention of teaching experiment is effective.