以幾何面積為類比特教授國一代數乘法公式之研究

A Study of Using Calculation of Areas as Analogs to Teach the Concepts of Multiplicative Identities for the First Grade of Junior High School Students

江佳惠(Jia-Hui Jiang) ； 徐順益(Shun-Yi Xu)

《科學教育》 12期 (2002/08) Pp. 21-51

https://doi.org/10.6767/JSE.200208.0021

摘要

本研究主要目的在於將類比理論融入「代數概念－乘法公式的學習單元」中，以幾何圖形面積為類比物，幫助國一學生學習乘法公式新知識。并且分析學生在學習過程中的類經思考歷程。采用量與質的資料，詮釋學生之學習活動、學習成就與思考歷程。本研究結果發現：
量的資料方面
一、「基礎知識」、「邏輯推理能力」、「乘法公式單一法則」、「乘法公式雙項法則」、「乘法公式多項法則」與「乘法公式總合」之Pearson相關考驗，皆達到顯著相關水準。顯示學生的「基礎知識」與「邏輯推理有力」是影響乘法公式學習成就的主要因素。
二、個案學生的乘法公式學習成就測驗分數高於平均數者，在基礎知識測驗分數亦高於平均數。這也說明了「基礎知識」是影「乘法公式學習成就」的主要因素。
質的資料方面
一、以具體幾何面積拼圖作為類比物，透過小組討論的類比學習，有利於提昇學生學習數學的興趣。
二、在四個學習主題中，全部學生都能夠「确認」與「檢索」源領域知識，學生最感困難部分是類比思考機制的「映射」與「評估」步驟。而且「差的平方公式」與「平方差公式」的機體面積操作較其他兩個主題來得困難。
三、以圖形面積拼圖為類比物的類比學習，可有效地提昇學生對於抽象代數乘法公式的理解，并能破除乘法公式迷思概念、降低學生在思考上的困難。
四、邏輯揄能力與基礎知識是影響學生在小組討論中發言品質的因素。
五、以圖形面積拼圖為類經特的代數乘法公式教學，有助於發跡學生已有的基礎知識乘法公式之迷思概念。

關鍵字

類比 ； 數學 ； 代數 ； 乘法公式

並列摘要

The main purpose of this study is to put the theory of analogy into the learning unit of multiplicative identities in order to use calculation of areas as analogs to help the first grade junior high school students learn the knowledge of multiplicative identities and analyze their thinking process of analogy. This study adopts quantitative and qualitative data to explain the learning activities, learning achievement and thinking process of students. The conclusions of this study are as follows:
In the aspect of quantitative data
1. The correlation coefficient testing of the basic knowledge, the logically reasoning ability, the single rule of multiplicative identities, the bilateral rule of multiplicative identities, the multiple rule of multiplicative identities and the summation of multiplicative identities have reached the prominent level which means students' basic knowledge and logically reasoning ability are the main factors affecting the learning achievement of multiplicative identities.
2. If the case students' test score of the learning achievement of multiplicative identities is higher than the average, their test score of the basic knowledge is also higher than the average. This explains that the basic knowledge is the main factor affecting the learning achievement of multiplicative identities.
In the aspect of qualitative data
1. Using concrete puzzles of calculation of areas as analogs to learn through group discussion is helpful to arouse students’ math learning interest
2. In the four learning topics, all students are able to identify and retrieve the source domain knowledge. The most difficult parts for students are the steps to map and evaluate in the thinking organism of analogy; moreover, the operation in calculation of areas of (a-b)^2=a^2-2ab+b^2 and (a + b)(a — b)=a^2-b^2 are harder than the other two topics.
3. Using the calculation of areas as analogs in the analogical learning is not only an effective way to promote the understanding of multiplicative identities of abstract algebra of students but also a good way to break the misconceptions of multiplicative identities and lower students' difficulties of thinking.
The logically reasoning ability and the basic knowledge are the factors affecting students' utterance quality in the group discussion. Using the calculation of areas as analogs in the teaching of multiplicative identities is advantageous to change the misconceptions of the basic knowledge and the multiplicative identities students had before.

並列關鍵字

analogy ； analogs ； mathematics ； algebra ； multiplicative identities