分數表示式對等比例概念建構影響之研究

The Influence of Fractional Expression on Conceptual Construction of Proportion

等比例之分數型式「a/b=c/d」，容易造成學生對等比例與等值分數的混淆。雖然，關於分數與比例的研究很多，卻顯少探討到此混淆所造成的影響。故本研究透過個案的研究來探討等比例之分數表示式對等比例概念建構的影響。本研究個案在解題表現中除了比例問題外，在計算題與對等問題中皆正確解答。經研究後所得結論為：（一）受試者對分式等式「a/b=c/d」與比例式「x:y = a:b」的符號角色產生混合，造成比例推理時同時並存絕對性關係的思考與相對性關係的思考。（二）由於「a/b=c/d」之型式，使得受試者以分數概念的「商」和「部分－全體」來思考「比值」概念，而將「比的等價」與「分數的等價」產生混合，因而忽略比是意指兩量比較關係之意義。最後研究者根據結論提出相關建議。

A fractional expression of the proportion makes students confuse equal fractions with equal ratios. Although lots of proportional and fractional researches were found, few papers concerned the influence from their confusion. The purpose of this study is to inquire the influence of fractional expression on conceptual construction of the proportion. The subject, whose performance in calculating problems and equivalence problems were good, failed in solving ratio problems in the task. Two conclusions were found in this study: ( I ) because of the mixed role of the equivalent expression “x/y=a/b ” with the proportional one “x:y = a:b” , both relative thinking and absolute thinking were presented simultaneously when the subject reasoning proportionally; ( I I ) the expression “a/b=c/d” makes the subject interpreted the concept of rate with “quotient” and “part-whole” fraction , so that they got confused equal fractions with equal ratios and neglected what the ratio means is to compare two quantities. Finally, some suggestions resulted from this study are proposed for mathematical teaching and future studies.