探討臺灣高中生使用邏輯連接詞的錯誤：以解一元二次方程式與不等式為例

Taiwanese High-School Students' Mistakes in the Use of Logical Connectives when Solving Quadratic Equations and Inequalities

本研究探討高中一年級學生在一元二次方程式與不等式的解題過程中，使用邏輯連結詞的錯誤情形。本研究參與施測學生有146人，施測題目有6題，包含4題二次不等式與2題二次方程式。研究者透過錯誤分析（errors analysis）以及持續比較法，先歸納出四個答題類別：正確解答、正確的代數運算但是過程或解答不完整、代數運算上的錯誤導致錯誤的解答以及未答。接著，依據正確的代數運算但是過程或解答不完整這類別，分別歸納出使用邏輯連接詞（且、或）的錯誤類別。研究結果顯示：（1）「且」有3種誤用類別：忽略邏輯連結詞「且」並且以逗號（，）取代、以「或」取代「且」以及不正確地詮釋邏輯連結詞；（2）「或」有5種誤用類別：忽略邏輯連結詞「或」、忽略邏輯連結詞「或」並且以逗號（，）取代、忽略邏輯連結詞「或」並且以頓號（、）取代、以「且」取代「或」以及不正確地詮釋邏輯連結詞；（3）在忽略邏輯連結詞並且以逗號（，）取代這個類別上，「或」的誤用比例比「且」誤用比例高；（4）邏輯連結詞「或」會被頓號（、）取代這個類別的發生，說明學生的中文語言經驗對邏輯連接詞使用的影響。

The aim of this study is to examine the mistakes made by students in the use of logical connectives while simplifying and solving algebraic equations and inequalities. The study was conducted among 146 first-grade high-school students. The data was collected from a questionnaire for students comprising items containing inequalities or equations. The data was analyzed according to interpretative theory. The findings identify mistakes in the way students use the mathematical logical connectives when manipulating algebraic expressions. Firstly, the mistakes of using AND include ignoring the AND and replacing it with a comma (,), replacing the AND with OR, and incorrect interpretation of the AND. Secondly, the mistakes of using OR include ignoring the OR, ignoring the OR and replacing it with a comma (,), ignoring the OR and replacing it with a slight-pause mark (、), replacing the OR with AND, and incorrect interpretation of the OR. Thirdly, in the category of ignoring the logical connective and replacing it with a comma (,), the frequency of the mistake of using OR is higher than the frequency of the mistake of using AND. Fourthly, this category of ignoring the OR and replacing it with a slight-pause mark (、) reveals the effect of students' language background on the use of logical connectives.