대학생들의 미분 개념에 대한 표현 유창성 분석

External representations of mathematical concepts facilitate individuals’ thinking about those concepts. Multiple representations of a concept can convey different information and facilitate different cognitive processes. Students’ understanding of a mathematical concept can be revealed in their interactions with representations, and therefore the representational fluency was used as a multi focal lens to examine how and what students understand of it. In this study, 34 university students were assessed for representational fluency about the derivative concept, using items requiring translations from the input representation of function to the output representation of derivative that vary systematically algebraic, graphical, numerical and verbal representations. The students’ achievement levels of each item were analyzed, association analysis and hierarchical clustering were also conducted to infer the degree of connections between abilities to translate function-derivative representations as a connected knowledge network. Although there was high achievement of some items, the participants in this study got low achievement in several items, which involve verbal representation of derivative and applying derivative concept to functions represented non-algebraically, and there were few statistically significant associations between two translation abilities, which need to be interconnected. The finding showed students’ strength, weakness and characteristics of derivative understanding as a connected network of representational abilities. This analysis provided rich evidences about students’ conceptual understanding of the derivative and could contribute to explore the direction of improvement of calculus teaching and learning based on students’ understanding.